From cube-lovers-errors@mc.lcs.mit.edu Tue Aug 26 21:22:54 1997
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Mail-from: From walsmith@erols.com Tue Aug 26 21:03:14 1997
Message-Id: <34037BAB.7923@erols.com>
Date: Tue, 26 Aug 1997 20:58:19 -0400
From: Walter Smith
Reply-To: walsmith@erols.com
To: cube-lovers@ai.mit.edu
Subject: Got a new shape...?
On 8/15/97 David Goyra asked for ideas for simulated puzzles.
Obviously there are infinite possibilities. If you want a source of
inspiration for simulated or real puzzles, I recommend the following
book:
Shapes, Space and Symmetry
by Alan Holden
Dover Publications, Inc.
I got mine at Boarders Bookstores. It is a book about three dimensional
shapes. It discusses symmetry and other properties with a minimum of
mathematical terms. It gives instructions (and pictures) on
constructing many shapes from cardboard or wire.
Any solid shape could be cut (or cuts) parallel to the sides, between
opposite corners, between opposite edges, along edges or any combination
of the foregoing. You will see the shapes of the common puzzles and
ideas for hundreds more.
Walt Smith
WALSMITH@EROLS.COM
From cube-lovers-errors@mc.lcs.mit.edu Wed Aug 27 14:39:14 1997
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Mail-from: From reid@math.brown.edu Wed Aug 27 14:33:56 1997
Message-Id: <199708271830.OAA29527@life.ai.mit.edu>
Date: Wed, 27 Aug 1997 14:36:47 -0400
From: michael reid
To: cube-lovers@ai.mit.edu
Subject: minimal maneuvers for "continuous" isoglyphs
i finished computing minimal maneuvers for the "continuous" isoglyphs.
some may be in a different orientation than herbert gave them.
i also give a maneuver that is simultaneously minimal in both the
quarter turn metric and the face turn metric when there is such a
maneuver.
*.*
*** (type 01)
***
1. (girdle 3-cycle)
F R' L U' R' U R L' B' R F' B (12q, 12f)
2. (distorted girdle 3-cycle)
U R U D' F2 U' D R U' (10q, 9f)
*.*
.** (type 02)
***
3. edge hexagon of order 2
U B2 U' F' U' D L' D2 L U D' F D' L2 B2 D' (20q, 16f)
4. edge hexagon of order 3
U' D L' B D B' U2 D' B' R' B R U' L D' (16q)
F L B U F2 B2 R F2 B2 L' U' B' L' F' (14f)
5. (off-girdle 3-cycles)
B' U F2 L' F2 U' F' B L B2 U B2 L' F (18q, 14f)
6. (distorted off-girdle 3-cycles)
F L B R D' F B2 L' F' B L' F' D R F R' F' (18q)
U R2 D F' L U2 D2 R' U2 D2 F D' R2 U' (14f)
*.*
**. (type 03)
*.*
7. (plummer's C's)
F U' F B' D2 B' U' D R B2 R L' B R' F U' D R' (20q)
L2 U2 R' B' U' D B2 D' R' D L D2 F U2 D L2 (16f)
*.*
.*. (type 04)
*.*
8. pons asinorum
U2 D2 F2 B2 R2 L2 (12q, 6f)
9. checkerboards of order 3
F B2 R' D2 B R U D' R L' D' F' R2 D F2 B' (20q, 16f)
10. checkerboards of order 6
R' D' F' D L F U2 B' L U D' R' D' L F L2 U F' (20q)
R2 L2 U B L2 D' F B2 R L' F' B R D F2 L' U' (17f)
***
*** (type 10)
**.
11. meson
U F' D F U' F' L' U' L D' L' U L F (14q)
D F2 D' R B2 R' D F2 D' R B2 R' (12f)
*.*
*** (type 11)
**.
12. (meson & girdle 3-cycle)
F' L' B' D2 B' D' B D' R F' R F R2 B L F (18q, 16f)
***
**. (type 12)
*..
13. two twisted peaks
F B' U F U F U L B L2 B' U F' L U L' B (18q)
F D2 B R B' L' F D' L2 F2 R F' R' F2 L' F' (16f)
14. exchanged peaks
F U2 L F L' B L U B' R' L' U R' D' F' B R2 (19q)
F2 R2 D R2 U D F2 D' R' D' F L2 F' D R U' (16f)
*.*
.** (type 12)
**.
15. (meson & girdle 3-cycles)
F B' R F' U L U' F B' D' B D L' B D' R' D F' (18q, 18f)
*.*
**. (type 13)
*..
16. (plummer's Y's)
R U' R B' R F R' U D' R L' B' L F L' F' R F' (18q)
L F B' U' R' B' R' L' U2 L D' R F R2 B2 L' F (17f)
*.*
.*. (type 14)
*..
17. (plummer's cluster & girdle 3-cycles)
R U' F U F' D' R F D' R L' F B' D' R F' L F' (18q)
F B2 U R L2 B' L F D' L' B L B' U L' U' D2 (17f)
18. (christman's cluster & girdle)
DL DB DR DF UL UB UR UF LB LF RB RF DLB URB UBL ULF DRF DFL UFR DBR
F U R' U' R U2 R' B' R' F R' D R' L U F D' F B' R' (21q)
F2 U R2 L' U2 D' F2 U' B R L' B' U2 B U' R B' L (18f)
.*.
*** (type 30)
.**
19. (plummer's rabbits)
F L' F R' U R U' F' L U R' U' R F' (14q, 14f)
.*.
.** (type 31)
.**
20. twisted cube edges, orthogonal bars
F L' U L U' R' U F' L F L' U' R F' (14q, 14f)
...
.** (type 32)
.**
21. cube in a cube
F L F U' R U F2 L2 U' L' B D' B' L2 U (18q, 15f)
.*.
**. (type 32)
..*
22. twisted duck feet
U2 F' B D B' U D2 L U2 F L F U' R' B' R F' (20q, 17f)
23. exchanged duck feet
U F R2 F' D' R U B2 U2 F' R2 F D B2 R B' (21q, 16f)
...
**. (type 33)
..*
24. (plummer's bend)
F R B' R U R F D' L' F2 R U' F R' B' R F' (18q)
F' R U2 L' F B U B2 R' U R2 D' R2 U' L' U (16f)
...
.*. (type 34)
..*
25. twisted chicken feet
D2 R U L' F2 R F' U F' U' B' U F D' L F' (18q, 16f)
26. exchanged chicken feet, cherries
F L' D' B' L F U F' D' F L2 B' R' U L2 D' F (19q, 17f)
.*.
*** (type 40)
.*.
27. christman's cross
U R L' F2 U2 F2 R' L U2 F2 U (16q, 11f)
28. plummer's cross
U' D2 R B2 D' R' U D' R L' D R F2 D' R2 L (20q, 16f)
...
*** (type 41)
.*.
29. four way street
L U2 F' U F L2 U L F' D' F2 L' D' L D2 F' (20q, 16f)
...
.** (type 42)
.*.
30. exchanged rings
B' U' B' L' D B U D2 B U L D' L' U' L2 D (18q)
F U D' L' B2 L U' D F U R2 L2 U' L2 F2 (15f)
31. twisted rings
F D F' D2 L' B' U L D R U L' F' U L U2 (18q, 16f)
32. anaconda, worm
L U B' U' R L' B R' F B' D R D' F' (14q, 14f)
.*.
.*. (type 43)
...
33. six U's type 6
U D' F' U R L' B' U F U D' R' (12q, 12f)
...
.*. (type 44)
...
34. six spot, six O's
U D' R L' F B' U D' (8q, 8f)
mike
From cube-lovers-errors@mc.lcs.mit.edu Mon Sep 1 22:33:57 1997
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Mail-from: From SCHMIDTG@iccgcc.cle.ab.com Mon Sep 1 16:35:03 1997
From: SCHMIDTG@iccgcc.cle.ab.com
Date: Mon, 1 Sep 1997 16:32:10 -0400 (EDT)
To: cube-lovers@ai.mit.edu
Message-Id: <970901163210.20217b13@iccgcc.cle.ab.com>
Subject: Re: Open and Closed Subgroups of G
I'd like to thank Jerry for taking the time to put together his
message discussing basic group theory as it applies to the cube as
well as the basics of Thistlewaite's algorithm. Although I consider
myself somewhat beyond the "layman" level in this area, I'm not
always able to follow the various posts to this group. Besides,
it's also helpful to read a little "refresher" every now and then
to help reinforce and clarify previously digested concepts.
It might also be helpful for someone to cover the basics of cube
parity. Although I think I understand the basic group theoretic
concepts of permutation parity, the asymmetry of the marked faces
of the cube have never quite left me feeling comfortable about
how this concept is applied to the cube. Hofstadter, covers this,
but does not discuss it in enough detail for one to fully grasp
the concept.
Regards,
-- Greg Schmidt
From cube-lovers-errors@mc.lcs.mit.edu Mon Sep 1 23:26:56 1997
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Mail-from: From SCHMIDTG@iccgcc.cle.ab.com Mon Sep 1 16:50:08 1997
From: SCHMIDTG@iccgcc.cle.ab.com
Date: Mon, 1 Sep 1997 16:46:33 -0400 (EDT)
To: cube-lovers@ai.mit.edu
Message-Id: <970901164633.20217b13@iccgcc.cle.ab.com>
Subject: Re[2]: Open and Closed Subgroups of G
Oh, and I forgot to mention...
My ultimate goal of understanding parity would be such that someone could
hand me an arbitrary permutation puzzle and I'd be able to examine it and
determine from the set of legal moves both the parity constraints and also
be able to construct a parity test valid from any given puzzle state.
I find it interesting that the method seems to differ across puzzles.
For example, 15 puzzle parity can be determined by the number of pairwise
exchanges required to solve the puzzle, whereas with the cube, it seems
a more direct approach is possible by examining cubie orientations with
respect to marked cubicles.
Still, I'm somewhat mystified.
Regards,
-- Greg Schmidt
From cube-lovers-errors@mc.lcs.mit.edu Tue Sep 2 11:08:15 1997
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Mail-from: From nbodley@tiac.net Tue Sep 2 09:46:52 1997
Date: Tue, 2 Sep 1997 08:44:30 -0400 (EDT)
From: Nicholas Bodley
To: SCHMIDTG@iccgcc.cle.ab.com
Cc: cube-lovers@ai.mit.edu
Subject: Parity (Was Re: Re[2]: Open and Closed Subgroups of G)
In-Reply-To: <970901164633.20217b13@iccgcc.cle.ab.com>
Message-Id:
If I understand parity, Greg's examination would reveal whether someone
had reassembled a Cube (or other mathematically-related puzzle) into a
state that can't be solved.
|* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath
|* Waltham, Mass. *|* -----------------------------------------------
|* nbodley@tiac.net *|* Waltham is now in the new 781 area code.
|* Amateur musician *|* 617 will be recognized until the end of 1997.
--------------------------------------------------------------------------
From cube-lovers-errors@mc.lcs.mit.edu Wed Sep 3 18:01:12 1997
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Mail-from: From lvt-cfc@servtech.com Wed Sep 3 13:05:50 1997
From: "christopher f. chiesa"
Message-Id: <199709031702.NAA20567@cyber1.servtech.com>
Subject: Re: Open and Closed Subgroups of G
To: cube-lovers@ai.mit.edu
Date: Wed, 3 Sep 1997 13:02:11 -0400 (EDT)
Greg Schmidt (SCHMIDTG@iccgcc.cle.ab.com) mentions discomfort about
how concepts of "parity" are applied to the Cube. I second the
notion! :-)
I assume that by "parity" we mean that which is conserved as the "twist"
of corner cubies or the "flip" of edge cubies. I myself have a HELL of a
time determining a particular corner cubie's precise amount (N/3, N an
integer) of "twist," or a particular edge cubie's precise amount (N/2, N an
integer) of "flip," other than in the case of an observable change in ONLY
that particular cubie -- and moreover, ONLY in its ORIENTATION. Any change
in a cubie's POSITION, relative OR absolute, renders my notions of "twist"
and "flip" rather fuzzy.
F'rinstance, start with a Cube in the "solved" state and perform the sequence
(generator?):
R' D2 R F D2 F' U2 F D2 F' R' D2 R U2
You will find that "FRU has been twisted -1/3 ("one 'notch' CCW"), and BLU
has been twisted +1/3 ("one 'notch' CW")," relative to their previous
orientations (i.e., relative to "solved") -- and that this is easy to assess
largely because the "solved" state of the rest of the Cube makes it very
clear how the corner cubies' orientations have changed (and their positions
have NOT). The sequence/generator would produce the same net effect
(twisting FRU -1/3, and BLU +1/3) when performed on the Cube in ANY state; it
would merely be more difficult for the casual observer to identify against
the background of a "scrambled" Cube state.
But, back to the start-from-"solved" example. If I now make the single turn
B'
I no longer find it so easy to characterize the corner-twist parity state of
the Cube, because (all of) the corner-cubies affected by this particular
Cube-state-change have left their previous positions, leaving me to wonder,
"RELATIVE TO WHAT" their twist is to be assessed. How is it done? What can
now be said about the "twist state" of, say, the former BLU (now BRU) cubie?
What about the former BLD (now BLU) cubie?
My efforts to "reason it out," within the limitations of my group-theory
background (which is now infinitely broader thanks to Jerry Bryan!), lead to
what almost seems a paradox. For what it's worth, I present it for your
discussion, and will be very interested to hear what you Cubemeisters are
able to contribute!
Observe that the orientations of all corners in the F layer remain unchanged
by the B' operation last performed. In particular, the FRU cubie retains its
-1/3 twist relative to (what's left of) the "solved" state. Assuming that
the "twist" of a cubie which "hasn't moved" REMAINS THE SAME, as opposed to
being, say, "implicitly redefined" by the movement of OTHER cubies, I can
still say a few things -- though not as many things as I would like! -- about
the twist-states of the corner-cubies in the "B layer" after that B' face
turn.
Invoking twist-parity-conservation (let's just say "twist-conservation,"
okay?), I assert that "the TOTAL twist of all corner cubies in the B layer
must still be 'some integer plus 1/3,'" so as to "cancel out" the -1/3 twist
remaining on FRU. The B' turn thus imparted "some integer" TOTAL twist,
which is to say, a total of 0 "net" twist, to the corner cubies in the B
layer -- but was it e.g. "0, 0, 0, 0" or "+1/3, +1/3, -1/3, -1/3?" (I
believe all other combinations reduce to these.) Note that this boils down
to asking, "does a face turn, if it twists corner-cubies AT ALL, twist ALL
FOUR the SAME WAY (i.e. apply the same "net twist" to all four), or NOT?"
Is there a definitive answer? A standard assumption? Proof or disproof of
either? It seems there would _have_ to be, in order to have "meaningful"
discussions of "twist" at all.
For a while I thought I could prove that it was the "0, 0, 0, 0" case, but it
turned out that one of my working assumptions was equivalent to STATING that
it was the "0, 0, 0, 0" case. I was only "proving" my own ASSUMPTION. Glad
I didn't post THAT. :-)
Naturally, analogous issues and questions will arise when discussing
edge-cubie "flip" and the conservation thereof. :-)
All in all, I'd be VERY interested in seeing the professional theoretical
dissection of this issue!
...
That's all I have today on the subjects of "twist," "flip," and "parity/
conservation thereof." But before I go, I'll leave you with two more
demented, blue-sky thoughts. Beware; this is what I get for reading Star
Trek novels before bed, and again at breakfast...
1) At the edge of my intuition, beyond my ability to formalize, I fancy
I sense that there might be a way of looking at the Cube, perhaps
through the use of additional spatial dimensions or their mathemati-
cal equivalents, in which the Cube is in some sense "always" in the
"solved" state, or at least in which it is trivially obvious where lies
the "direct path" back TO the "solved" state. I'm visualizing some
sort of extra-spatial "rubber bands," or "strings" (in those higher
spatial dimensions specifically so as to avoid "tangling" issues)
that "trace" the route (or "net" route) taken by each cubie, or arbi-
trary collection of cubies, from its/their position(s)-and-orienta-
tion(s) in the "solved" state, to its/their p(s)-and-o(s) in a "scram-
bled" Cube. In such a perception, one could simply "tug on the
strings" and "pull" the Cube back to "solved." Does this make ANY
kind of sense to ANYBODY else here? I feel as though I can "almost
see it."
2) Is there a notion, has anybody done any work, on Cube states which
are each other's "duals?" I define the "dual" of a Cube state X as
that Cube state reached by performing, on a "solved" Cube, the same
sequence of turns/moves which "solve" Cube state X. In other words,
define a sequence of turns which transforms the Cube from state X
to "solved," then apply that sequence again to the "solved" cube to
arrive at state Y. State Y is then the "dual" of state X. Ques-
tions abound:
- does each state have EXACTLY ONE dual? Or many, depending on
the specific sequence (as we know, there are many) of moves
performed in solving state X ? (My gut feeling is that each
state has exactly one dual. This would seem to be pretty easy
to prove using the group-theory math at the disposal of many
readers here.)
- are there states which are their OWN duals? (Yes, clearly;
the trivial "checkerboard" pattern arising from a single 180-
degree turn of each face, is its own dual)
- a state which is its own dual, is a "two-cycle" with the
"solved" state: perform the generating sequence on either and
get to the other. Are there "three-cycles?" "Four-cycles?"
etc.?
Looking forward to the followups,
Chris Chiesa
lvt-cfc@servtech.com
From cube-lovers-errors@mc.lcs.mit.edu Wed Sep 3 18:42:04 1997
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Mail-from: From jbryan@pstcc.cc.tn.us Wed Sep 3 13:55:04 1997
Date: Wed, 03 Sep 1997 13:51:02 -0400 (Eastern Daylight Time)
From: Jerry Bryan
Subject: Re: Open and Closed Subgroups of G
In-Reply-To: <970901163210.20217b13@iccgcc.cle.ab.com>
To: SCHMIDTG@iccgcc.cle.ab.com
Cc: cube-lovers@ai.mit.edu
Message-Id:
On Mon, 1 Sep 1997 SCHMIDTG@iccgcc.cle.ab.com wrote:
> It might also be helpful for someone to cover the basics of cube
> parity. Although I think I understand the basic group theoretic
> concepts of permutation parity, the asymmetry of the marked faces
> of the cube have never quite left me feeling comfortable about
> how this concept is applied to the cube. Hofstadter, covers this,
> but does not discuss it in enough detail for one to fully grasp
> the concept.
>
I'll take your question as literal, assuming you mean just parity and
not twist and flip, and assuming you know the basic group theoretic
concepts of permutation parity.
Parity of the cube is best described (I think) as applying to whole
cubies rather than to facelets. As such, a quarter turn of any face is
a 4-cycle on the corner cubies and a 4-cycle on the edge cubies. A
4-cycle is odd, which is to say that it can be decomposed into an odd
number of 2-cycles. The "obvious" way to decompose a 4-cycle is into
three 2-cycles. Although decomposition of a 4-cycle into 2-cycles is
not unique, any such decomposition will contain an odd number of
2-cycles.
Start is even for both the edges and the corners (the identity consists
of zero 2-cycles). If you any quarter turn from Start, both edges and
corners become odd. Make another quarter turn, both edges and corners
become even. Make another quarter turn, both edges and corners become
odd. Etc. Edges and corners are either both even or both odd.
In the constructable group, you can have odd corners with even edges or
vice versa. For example, remove two edge cubies from a cube and
exchange them without moving any of the other cubies around. You will
be changing the parity of the edges without changing the parity of the
corners.
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us
Pellissippi State (423) 539-7198
10915 Hardin Valley Road (423) 694-6435 (fax)
P.O. Box 22990
Knoxville, TN 37933-0990
From cube-lovers-errors@mc.lcs.mit.edu Thu Sep 4 17:02:06 1997
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Mail-from: From jbryan@pstcc.cc.tn.us Thu Sep 4 12:54:09 1997
Date: Thu, 04 Sep 1997 12:50:11 -0400 (Eastern Daylight Time)
From: Jerry Bryan
Subject: Re: Open and Closed Subgroups of G
In-Reply-To: <199709031702.NAA20567@cyber1.servtech.com>
To: "christopher f. chiesa"
Cc: cube-lovers@ai.mit.edu
Message-Id:
On Wed, 3 Sep 1997, christopher f. chiesa wrote:
> 2) Is there a notion, has anybody done any work, on Cube states which
> are each other's "duals?" I define the "dual" of a Cube state X as
> that Cube state reached by performing, on a "solved" Cube, the same
> sequence of turns/moves which "solve" Cube state X. In other words,
> define a sequence of turns which transforms the Cube from state X
> to "solved," then apply that sequence again to the "solved" cube to
> arrive at state Y. State Y is then the "dual" of state X. Ques-
> tions abound:
The concept of "dual" which you are describing is standard in group
theory (and be extension, in cube theory). A "dual" is properly called
an inverse. If you have a sequence of turns which creates a position,
the inverse sequence consists of writing the turns in reverse order, and
converting clockwise turns to counterclockwise turns and vice versa. So
the inverse of FRU' is UR'F'. If there are multiple sequences for a
position (and most typically there are), you can do the same thing for
any such sequence.
Also, a position can be described in terms of which cubies have gone
where. For example, you might have something like
flu --> fur
fur --> frd
frd --> fdl
fdl --> flu
(flu is the front-left-up cubie etc. Standard Singmaster notation uses
lower case letters for cubies and upper case letters for the moves
themselves.)
You could get the inverse by reversing the arrows like so.
flu <-- fur
fur <-- frd
frd <-- fdl
fdl <-- flu
More commonly, you would write the inverse by swapping the cubie
designations between the left and right side of the arrows like so.
fur --> flu
frd --> fur
fdl --> frd
flu --> fdl
I don't know what you mean by "any work", but here are some standard
information about inverses. The length of a position X is the same as
the length of its inverse X', where length is the minimum number of
moves to create the position. If X' is the inverse of X, then X is the
inverse of X'. The symmetry of an inverse X' is the same as the
symmetry of a position X (see Symmetry and Local Maxima in the archives
for a discussion of symmetry). A local maximum is a position such that
no matter which move you make, you will be one move closer to Start. It
is not necessarily the case that the inverse of a local maximum is also
a local maximum.
>
> - does each state have EXACTLY ONE dual? Or many, depending on
> the specific sequence (as we know, there are many) of moves
> performed in solving state X ?
Yes, inverses are unique, both for groups in general, and for cubes in
particular.
>
> - are there states which are their OWN duals? (Yes, clearly;
> the trivial "checkerboard" pattern arising from a single 180-
> degree turn of each face, is its own dual)
You have answered your own question. Many positions are their own
inverse. Some of them are much more complicated than the one which you
describe.
>
> - a state which is its own dual, is a "two-cycle" with the
> "solved" state: perform the generating sequence on either and
> get to the other. Are there "three-cycles?" "Four-cycles?"
> etc.?
>
The proper term for the concept you are describing is order. If you
repeat a maneuver n times from Start and return to Start, then the
position is of order n. (Strictly speaking, the order of a position is
the smallest n which will work. Obviously, if n will work then so too
will 2n, 3n, etc.) There are many different orders for which there are
cube positions of that order. One of David Singmaster's early Cubic
Circulars (I don't have the reference handy) had a table of possible
cube orders and how many positions there were of each order.
The term cycle is also very important in group theory (and by extension
in cube theory). Suppose you look at a scrambled cube and determine
that cubie a has gone to cubie b's place, cubie b has gone to cubie c's
place, and cubie c has gone to cubie a's place, then a, b, and c form a
3-cycle. The way I have defined this particular 3-cycle, you could
write it as (a,b,c), as (b,c,a), or as (c,a,b). This so-called cycle
notation is circular, so it does't really matter which you write first.
However, (a,c,b) is a different cycle than (a,b,c). In fact, (a,c,b) is
the inverse of (a,b,c). Just for emphasis, (a,b,c) is not like an
ordered pair (or really an ordered triple in this case). (a,b,c) means
a goes to b, b goes to c, c goes to a.
As an example of a cycle in purely cube terms, the cycle for the example
I gave earlier would be (flu,fur,frd,fdl), so it is a 4-cycle.
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us
Pellissippi State (423) 539-7198
10915 Hardin Valley Road (423) 694-6435 (fax)
P.O. Box 22990
Knoxville, TN 37933-0990
From cube-lovers-errors@mc.lcs.mit.edu Fri Sep 5 21:03:58 1997
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Mail-from: From Hoey@AIC.NRL.Navy.Mil Fri Sep 5 21:08:00 1997
Date: Fri, 5 Sep 1997 21:07:48 -0400
Message-Id: <199709060107.VAA04503@sun30.aic.nrl.navy.mil>
From: Dan Hoey
To: lvt-cfc@servtech.com
Cc: cube-lovers@ai.mit.edu
In-Reply-To: <199709031702.NAA20567@cyber1.servtech.com> (lvt-cfc@servtech.com)
Subject: Re: Open and Closed Subgroups of G (fwd)
Chris Chiesa , among other things, writes
> If I now make the single turn
> B'
> I no longer find it so easy to characterize the corner-twist parity state of
> the Cube, because (all of) the corner-cubies affected by this particular
> Cube-state-change have left their previous positions, leaving me to wonder,
> "RELATIVE TO WHAT" their twist is to be assessed.
At the risk of being repetitious, the answer is, "relative to the home
orientation of the position they find themselves in". You choose a
special facelet for each corner cubie. When the cubie is in its home
position, its twist is the position of its special facelet relative to
the home of the special facelet. When cubie X is in cubie Y's home
position, the twist of cubie X is the position of X's special facelet
relative to the home of Y's special facelet. The edges are done the
same way, except mod 2.
Cube-lovers can find this in Vanderschel's article (6 Aug 1980) and
the extension by Saxe (3 September 1980). I mentioned (23 September
1982) that the choice of special facelets is arbitrary, and that a
conservation of twist occurs for a set of pieces of any puzzle that
1. have an Abelian orientation group, and
2. are moved in untwisted cycles by the generators.
This is true even if not all the cycles have the same length. For
instance, we could have a Rubik's cube in which generators move
corners in permutations like (FTR,FRD,FDL,FLT)(BRT,BTL,BLD), and twist
would be preserved. The key is that for each piece, the minimum power
of the generator that returns that piece to its home position must
also return it to its home orientation.
I'm quite uncertain about what orientation constraints can arise in
puzzles with non-Abelian orientation groups. For instance, the
hypercorners of a Rubik's tesseract have the symmetry group A4, and
any orientation is achievable up to a constraint imposed by an Abelian
quotient of A4 of type 3 (See 22 Oct 1982). Does every group have a
unique maximal Abelian quotient? Is that the only orientation
constraint that can occur?
Dan Hoey
Hoey@AIC.NRL.Navy.Mil
[ Moderator's Note: Cube-lovers will be down Saturday and Sunday due to
major electrical work at MIT. ]
From cube-lovers-errors@mc.lcs.mit.edu Mon Sep 8 09:47:22 1997
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Mail-from: From kociemba@hrz1.hrz.th-darmstadt.de Sun Sep 7 17:51:08 1997
Message-Id: <3411D734.6471@hrz1.hrz.th-darmstadt.de>
Date: Sun, 07 Sep 1997 00:20:36 +0200
From: Herbert Kociemba
To: cube-lovers@ai.mit.edu
Subject: Number of maneuvers with n face turns
The number of maneuvers with 1, 2, 3,.. face turns for Rubik's cube are
of course well known and are 18, 243, 3240... But I did not see a closed
formula for these numbers before, so maybe you find the following
formula interesting:
Let r:= sqrt(6), then you have with n face turns
P(n) = [(3+r)*(6+3r)^n + (3-r)*(6-3r)^n]/4
maneuvers. Because the second part in brackets is much smaller than the
first, asymptotically you have
(3+r)*(6+3r)^n /4 maneuvers.
Even for small n, this approximation is very good. So for n=3 you get
3240.33 instead of 3240. The asymptotic branching factor P(n+1)/P(n) is
therefore (6+3r), which is about 13.348469 .
Herbert
From cube-lovers-errors@mc.lcs.mit.edu Tue Sep 9 11:01:44 1997
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Mail-from: From reid@math.brown.edu Tue Sep 9 00:17:33 1997
Message-Id: <199709090413.AAA00748@life.ai.mit.edu>
Date: Tue, 9 Sep 1997 00:20:27 -0400
From: michael reid
To: cube-lovers@ai.mit.edu
Subject: maximal abelian quotients
dan asks
> Does every group have a
> unique maximal Abelian quotient?
yes. let G be a group. it's not difficult to show that
1) the commutator subgroup G' is normal,
2) the quotient group G / G' is abelian, and
3) if G --> A is a homomorphism to any abelian group A ,
then G' is in the kernel, so there is a unique homomorphism
G / G' --> A such that the original homomorphism is the composite
G --> G / G' --> A .
this last one is kind of technical, but in the special case where
A = G / N for some normal subgroup N , it says that if G / N is
abelian, then N contains the commutator subgroup. thus, G / G'
is the maximal abelian quotient of G .
the quotient G / G' is sometimes written G^ab (the "abelianization" of G).
as you might guess, this is an important construction in group theory,
and it's one of the reasons why commutator subgroups are important.
mike
From cube-lovers-errors@mc.lcs.mit.edu Tue Sep 9 14:56:02 1997
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Mail-from: From jbryan@pstcc.cc.tn.us Tue Sep 9 11:06:36 1997
Date: Tue, 09 Sep 1997 11:02:32 -0400 (EDT)
From: Jerry Bryan
Subject: Re: Open and Closed Subgroups of G
In-Reply-To: <199709060107.VAA04503@sun30.aic.nrl.navy.mil>
To: Cube-Lovers
Cc: lvt-cfc@servtech.com
Message-Id:
On Fri, 5 Sep 1997, Dan Hoey wrote:
> Chris Chiesa , among other things, writes
>
> > If I now make the single turn
>
> > B'
>
> > I no longer find it so easy to characterize the corner-twist
> > parity state of the Cube, because (all of) the corner-cubies
> > affected by this particular Cube-state-change have left their
> > previous positions, leaving me to wonder, "RELATIVE TO WHAT" their
> > twist is to be assessed.
>
> At the risk of being repetitious, the answer is, "relative to the home
> orientation of the position they find themselves in". You choose a
> special facelet for each corner cubie. When the cubie is in its home
> position, its twist is the position of its special facelet relative to
> the home of the special facelet. When cubie X is in cubie Y's home
> position, the twist of cubie X is the position of X's special facelet
> relative to the home of Y's special facelet. The edges are done the
> same way, except mod 2.
Dan's response (plus his references in the Cube-Lovers archives) pretty
well covers it. I would just like to add a couple of points.
1. There is a reference in the archives to a way of demonstrating
conservation of twist without first establishing a frame of
reference, but I can't find the reference. The best I can
recall, the same technique did not work for edges. But I prefer
the frame of reference technique anyway because it is closely
tied to some of the more usual ways of representing the cube in a
computer.
2. For example, number the corner facelets from 1 to 24. Each
facelet has two companion facelets which are bound to it on
the same cubie. By knowing where one of the three facelets
of a cubie is in a computer program, you automatically know
where the other two facelets are, so you only have to store
one of the three facelets. The one that you store can be the
"special" facelet that Dan described for the purposes of
determining conservation of twist.
The collection of eight "special" facelets for the corners
have been described in the archives as constituting a
supplement for the group, but I have yet to find a discussion
group supplements in any group theory book.
As Dan says, your choice of "special" facelet is totally
arbitrary for each cubie, but most typically you choose
the Front and Back facelets, or the Right and Left facelets,
or something equally well organized.
3. For another example, number the corner cubies from 1 to 8, and
for each of the cubies describe the twist with a number from 0
to 2. This is essentially a wreath product representation of
the cube. The numbers from 0 to 2 which describe the twist
can be used to describe whether a cubie is twisted when it is
not home, and can therefore be used to prove conservation of
twist.
Without knowing any more than I do about supplements, it seems
very likely that it should be easy to represent any group
which can be representated as a supplement as a wreath product
and vice versa. The isomorphism seems obvious. I wonder if
anybody out there can shed any light on this issue?
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us
Pellissippi State (423) 539-7198
10915 Hardin Valley Road (423) 694-6435 (fax)
P.O. Box 22990
Knoxville, TN 37933-0990
From cube-lovers-errors@mc.lcs.mit.edu Fri Sep 12 17:53:00 1997
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Mail-from: From jbryan@pstcc.cc.tn.us Fri Sep 12 17:08:04 1997
Date: Fri, 12 Sep 1997 17:07:47 -0400 (Eastern Daylight Time)
From: Jerry Bryan
Subject: Re: isoglyphs
In-Reply-To: <199708182216.SAA00604@sun30.aic.nrl.navy.mil>
To: cube-lovers@ai.mit.edu
Reply-To: Jerry Bryan
Message-Id:
On Mon, 18 Aug 1997, Dan Hoey wrote:
> A "chiral isoglyph" is one in which the handedness of the glyph is
> taken into account in testing for isoglyphy,* so that the glyph
> appears only in one variety.
>
> Mike used "achiral" for an isoglyph that fails to be a chiral
> isoglyph, though I would tend to use "non-chiral". I would rather use
> "achiral" for a situation that lacked chirality, as in an isoglyph of
> a mirror-symmetric glyph.
Let me see what I can do to muddy these waters.
It seems to me that we might ought to consider the chirality of an
isoglyph as being a different issue than the chirality of a glyph. I
think the two are clearly related, but I am not sure that the one
necessarily derives from the other.
As to a glyph, it seems to me that a glyph is chiral only if conjugating
the position by each of the four reflections of the square yields a
different set of positions than does conjugating the position by each of
the four rotations of the square. Hence, you can have a glyph which
occurs in right-handed or left-handed forms, or one that doesn't. This
is the simple part.
I think the situation with isoglyphs is a little more complicated. For
example, form an isoglyph using both the right-handed and the
left-handed forms of a chiral glyph. You might have 6 right-handed
glyphs and 0 left-handed glyphs, 5 right-handed glyphs and 1 left-handed
glyph, etc. If there are unequal numbers of right-handed and
left-handed glyphs, then it seems natural to define the handedness of
the isoglyph as being that of the dominate glyph. But what if there are
three right-handed glyphs and three left-handed glyphs?
Up to symmetry, there are only two ways to partition the six faces of a
cube into two sets of three faces. For example, the F, U, and B faces
can be of the same chirality, or the F, U, and R faces can be of the
same chirality (or any conjugates of these choice of faces). In the
first case, the cube is partitioned like a universal joint, or maybe
like a cubic baseball. Such a position seems to me to lack chirality.
In the second case, three faces with the same chirality cluster around a
common corner. Again, such a position seems to me to lack chirality.
So an isoglyph which lacks chirality can contain chiral glyphs.
On the other hand, even on an isoglyph consisting of three right-handed
and three left-handed glyphs, you still might be able to find a
distinguishing characteristic of the right hand part that was different
from the left-handed part. For example, the glyph boundaries which were
internal to the right-handed part of the isoglyph might be continuous
whereas the glyph boundaries which were internal to the left-handed part
of the isoglyph might not be continuous. Or for another example, the
rotations of the three right-handed faces relative to each other might
be different than the rotations of the left-handed faces relative to
each other.
(By the way, I have not verified that any of these positions I have
described are actually in G. I guess I am thinking in terms of the
constructible group of the facelets -- conceptually, peeling all the
facelets off and reattaching them.)
On the other hand, two glyphs which lack chirality when placed side by
side can be chiral. For example,
XOOXXX (the base glyph is XXX
XXXOXO OXO
XOOOXO OXO )
I really haven't thought through the implications of using six glyphs
instead of two, but it seems to me quite likely that an isoglyph could
be constructed using six glyphs which lack chirality and which are the
same pattern, and where the we could attribute chirality to the isoglyph
as a whole.
I have thought about this in terms of Herbert's Cube Explorer 1.5
program. The pattern editor has a check box for continuous. If you
don't check the box, the program finds both continuous and
non-continuous isoglyphs. If you do check the box, it finds only
continuous ones. So I have considered what would happen if the program
had a check box for chiral. What should it do? The obvious thing would
be that in normal operation, it would consider conjugates of both
rotations and reflections of the square when building an isoglyph from a
glyph, but that if the chiral box were checked it would consider only
conjugates of rotations of the square. But is that sufficient to
satisfy our various definitions of chiral, achiral, and/or non-chiral?
I'm not sure. Maybe Dan or Mike would be kind enough to clarify further
their thoughts on this issue.
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us
Pellissippi State (423) 539-7198
10915 Hardin Valley Road (423) 694-6435 (fax)
P.O. Box 22990
Knoxville, TN 37933-0990
From cube-lovers-errors@mc.lcs.mit.edu Sun Sep 14 22:54:52 1997
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Mail-from: From reid@math.brown.edu Sat Sep 13 21:32:35 1997
Message-Id: <199709140132.VAA09760@life.ai.mit.edu>
Date: Sat, 13 Sep 1997 21:33:59 -0400
From: michael reid
To: cube-lovers@ai.mit.edu
Subject: optimal solver is available
for those who are interested in my optimal cube solver, you can now
get it from the web page
http://www.math.brown.edu/~reid/rubik/optimal_solver.html
i've reduced the size of my transformation tables, so now i think there's
a reasonable chance that it will run within 80Mb of RAM.
enjoy the program. if you make any new exciting discoveries, please share
them with the entire mailing list.
mike
From cube-lovers-errors@mc.lcs.mit.edu Mon Sep 29 13:08:14 1997
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Mail-from: From reid@math.brown.edu Sun Sep 28 14:45:54 1997
Message-Id: <199709281845.OAA08068@life.ai.mit.edu>
Date: Sun, 28 Sep 1997 14:46:39 -0400
From: michael reid
To: cube-lovers@ai.mit.edu
Subject: idea for smaller optimal solver
a number of people have told me that they don't have 80Mb of RAM on their
computers, so that my optimal solver won't work on their machine. here's
an idea for an optimal solver that uses much less memory; it should fit
within 16Mb, or 20Mb at the most. of course, it's a space/time tradeoff,
but perhaps will still be fairly good.
in my current program, i use distances to the subgroup
H = __
as my "heuristic" function. there is another subgroup, H' , which
contains H as a subgroup of index 8. H' is the subgroup of all
elements of H composed with all (valid) flips of U-D slice edges.
another way to describe H' is the subgroup of all elements where
the U face has only the colors U and D, and the same for the D face.
from this latter description, we see that if H1' , H2' and H3'
are the three orientations of this subgroup, then their intersection
is the subgroup of elements that "look like" they're in the square
group. this is the same target subgroup that my current program has.
the subgroup H' also has 16 symmetries. using this to reduce the
size of the pattern database, and storing each entry with 4 bits,
it should take about 8.5Mb. my current program also has about 8.5Mb
of transformation tables (but 3Mb of these are not used while searching).
the transformation tables will probably be slightly smaller (certainly
no larger), so it seems plausible that this could run with 16Mb of RAM.
what about running time? in his paper, rich korf hypothesizes that
the number of nodes generated should be roughly proportional to the
inverse of the size of the pattern databases. this suggests that
using the smaller tables above would result in about 8 times as many
nodes as my current program. this isn't bad, especially given that
the branching factor (6 + 3 * sqrt(6) = 13.348469 for face turns,
9.3736596 for quarter turns) is larger than this. so this approach
would be within 1 turn of my current program.
i don't foresee having enough spare time anytime soon to program this,
so i'll just post it here and maybe someone who is interested will
program this.
mike
From cube-lovers-errors@mc.lcs.mit.edu Tue Sep 30 12:12:19 1997
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Mail-from: From C.McCaig@Queens-Belfast.AC.UK Tue Sep 30 05:39:44 1997
From: C.McCaig@queens-belfast.ac.uk
Date: Tue, 30 Sep 1997 10:27:55 GMT
To: cube-lovers@ai.mit.edu
Message-Id: <009BB113.FAA1EEF6.44@a1.qub.ac.uk>
Subject: 4x4x4 solution
i recently borrowed a friends 4x4x4, and i know the basic method for
solving it. ie get the 6 centres, pair up all the edges, and then
solve for the normal cube. however, about half the time i end up
with a single edge pair inverted and cant figure out a move for
reorientating the single edge pair. usually i break a few pairs
and try and reorientate them this way, but this seems rather longwinded...
does anyone have a move for this?. for example, say the green edge
is on the blue face, and the blue edge is on the green face...
thanks.
clive
---
clive mccaig
queens university belfast
northern ireland
From cube-lovers-errors@mc.lcs.mit.edu Tue Sep 30 17:44:19 1997
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Mail-from: From Cube-Lovers-Request@ai.mit.edu Tue Sep 30 15:39:10 1997
Date: Tue, 30 Sep 1997 17:43:57 -0400
Message-Id: <30Sep1997.174357.Cube-Lovers@AI.MIT.EDU>
From: Cube Lovers Moderator
Sender: Cube-Lovers-Request@AI.MIT.EDU
To: Cube-Lovers@AI.MIT.EDU
Subject: 4x4x4 solution -- [Digest v23 #159]
Cube-Lovers Digest Tue, 30 Sep 1997 Volume 23 : Issue 159
Today's Topic:
4x4x4 solution
[ I have gathered together several similar messages on a single topic,
putting them in digest format. It would be nice to get an explicit
process for this problem, though. --Moderator. ]
----------------------------------------------------------------------
Date: Tue, 30 Sep 1997 13:39:46 -0400 (EDT)
From: der Mouse
To: C.McCaig@queens-belfast.ac.uk
Cc: cube-lovers@ai.mit.edu
Subject: Re: 4x4x4 solution
> i recently borrowed a friends 4x4x4, and i know the basic method for
> solving it. [...] however, about half the time i end up with a
> single edge pair inverted and cant figure out a move for
> reorientating the single edge pair.
Make a single 90-degree inner-slice turn, then solve as before. This
introduces an odd permutation on the edge pairs, which gets you back
into easily solvable space. (It's usually easiest if you make sure
that the two swapped edge cubies are part of the slice turn, by placing
on the same slice beforehand if necessary.)
I'm not sure quite what the parity constraint here is. There is some
kind of even-parity constraint on the edge cubies, it appears, with a
linked constraint on the face centres, but it's not as simple as the
parity of the edge and face permutations being both even or both odd,
because the single slice turn introduces two nonoverlapping 4-cycles on
the face centre cubies - which is, overall, an even permutation on
them.
I do notice, though, that a slice turn produces a 4-cycle on the edges
and two 4-cycles on the face centres; a face turn produces a 4-cycle on
the face centres and two 4-cycles on the edges (and a 4-cycle on the
corners, which may or may not be relevant). I wonder if there's a
multiple-of-three constraint lurking.
Doubtless some group theorist has long ago worked out exactly what the
constraints are, but I haven't heard. (I tried to work through a
group-theory text recently, got stalled along about the time it got to
cosets, quotient groups, normal subgroups, etc.)
der Mouse
mouse@rodents.montreal.qc.ca
7D C8 61 52 5D E7 2D 39 4E F1 31 3E E8 B3 27 4B
------------------------------
Date: Tue, 30 Sep 1997 14:11:28 -0400 (EDT)
From: Allan Wechsler
To: C.McCaig@queens-belfast.ac.uk
Cc: cube-lovers@ai.mit.edu
Subject: 4x4x4 solution
[C. McCaig:]
i recently borrowed a friends 4x4x4, and i know the basic method for
solving it. ie get the 6 centres, pair up all the edges, and then
solve for the normal cube. however, about half the time i end up
with a single edge pair inverted and cant figure out a move for
reorientating the single edge pair. usually i break a few pairs
and try and reorientate them this way, but this seems rather longwinded...
does anyone have a move for this?. for example, say the green edge
is on the blue face, and the blue edge is on the green face...
What's happened here is that you've got those two edge-cubies
_exchanged_. Here's how it works. Look at any face. You will see
eight edge stickers, arranged around the face like eight square
dancers (or Irish set dancers, if you prefer). Now I hope you are
familiar with one or the other of these kinds of folk-dancing, because
otherwise what I am going to say won't make sense. Those eight decals
are either men or women, and no matter how they dance around the cube,
they will never change sex. Every edge-cubie on the 444 has one
permanently male and one permanently female sticker. If I haven't
clarified things, at least I've spiced them up a bit.
House around to home.
- -A
------------------------------
Date: Tue, 30 Sep 1997 14:18:17 -0400 (Eastern Daylight Time)
From: Jerry Bryan
Subject: Re: 4x4x4 solution
To: C.McCaig@Queens-Belfast.AC.UK
Cc: cube-lovers@ai.mit.edu
On Tue, 30 Sep 1997 C.McCaig@Queens-Belfast.AC.UK wrote:
> i recently borrowed a friends 4x4x4, and i know the basic method for
> solving it. ie get the 6 centres, pair up all the edges, and then
> solve for the normal cube. however, about half the time i end up
> with a single edge pair inverted and cant figure out a move for
> reorientating the single edge pair. usually i break a few pairs
> and try and reorientate them this way, but this seems rather longwinded...
> does anyone have a move for this?. for example, say the green edge
> is on the blue face, and the blue edge is on the green face...
>
Your problem is one of parity. You have two edges cubies swapped (this
swap is visible) and two face center (centre) cubies of the same color
swapped (this swap is invisible). You have to have an even number of
swaps in the total cube. If you want an even number in the edges (and
you do), then you also have to have an even number in the face centers,
even if swaps in the face centers are invisible.
There is probably a more elegant solution, but the following will work.
If you encounter the situation you describe, make any middle slice
quarter turn. This will disturb the centers. The centers will now have
an even numbers of swaps. Solve the centers again without simply undoing
the middle slice you just made. The parity of the edges will then be
ok. (I'm assuming that your solution for the face centers will maintain
their parity after you correct it as described.)
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us
Pellissippi State (423) 539-7198
10915 Hardin Valley Road (423) 694-6435 (fax)
P.O. Box 22990
Knoxville, TN 37933-0990
------------------------------
End of Cube-Lovers Digest
*************************
From cube-lovers-errors@mc.lcs.mit.edu Tue Sep 30 18:10:34 1997
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Mail-from: From roger.broadie@iclweb.com Tue Sep 30 18:05:38 1997
From: roger.broadie@iclweb.com (Roger Broadie)
To:
Subject: Re: 4x4x4 solution
Date: Tue, 30 Sep 1997 23:02:48 +0100
Message-Id: <19970930230037.AAA21244@home>
C.McCaig@queens-belfast.ac.uk wrote:
> ...I recently borrowed a friends 4x4x4, and I ... can't figure out a
> move for reorientating the single edge pair....
It is possible to solve the problem with a sequence based on a quarter
turn of a central slice, since that, like a swap of two edge pieces,
involves an odd-parity cycle of the edge pieces. Thus
r2 U2 r U2 r2
(where r is the turn of the inner slice next to R in the direction
parallel to R)
puts a 4-cycle of edges onto the top face, but leaves you with the task
of restoring the centres.
It was the desire to find something less cumbersome that first lead me
to investigate the archives of this list, and there the answer was:
Date: Fri, 20 Oct 95 12:46:32 -0400 (EDT)
From: Georges Helm
Subject: Re: Old question about 2 adj edges
how to flip 2 adj. edges (and nothing else) in 4x4x4 cube?
r^2 U^2 r l' U^2 r' U^2 r U^2 r l U^2 l' U^2 r U^2 l r^2 U^2
Georges
geohelm@pt.lu
It does indeed contain an odd number of turns of the central slices to
give the desired parity.
Roger Broadie
From cube-lovers-errors@mc.lcs.mit.edu Wed Oct 1 11:53:15 1997
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Mail-from: From dokon@MIT.EDU Tue Sep 30 19:30:29 1997
Message-Id: <3.0.32.19970930192820.006ce8ac@po9.mit.edu>
Date: Tue, 30 Sep 1997 19:28:21 -0400
To: cube-lovers@ai.mit.edu
From: Dennis Okon
Subject: God's Number
I just found out that Keith Randall for the theory group of LCS (Lab for
Computer Science) at MIT gave a talk Monday about God's number for the
rubik's cube. He upped the lower bound 24 and gave "evidence" that it is
24. I don't know what moves he was counting (e.g. slice, quarter).
Unfortunately, I missed it. Does anyone have any information on this?
I'll see what I can find out.
-Dennis Okon
dokon@mit.edu
From cube-lovers-errors@mc.lcs.mit.edu Wed Oct 1 13:18:18 1997
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Mail-from: From ERCO@compuserve.com Wed Oct 1 01:56:54 1997
Date: Wed, 1 Oct 1997 01:52:24 -0400
From: Edwin Saesen
Subject: Re: 4x4x4 solution -- [Digest v23 #159]
Sender: Edwin Saesen
To: CUBE
Message-Id: <199710010152_MC2-225C-6120@compuserve.com>
jbryan@pstcc.cc.tn.us wrote:
>Your problem is one of parity. You have two edges cubies swapped
>(this swap is visible) and two face center (centre) cubies of the
>same color swapped (this swap is invisible). You have to have an
>even number of swaps in the total cube. If you want an even number
>in the edges (and you do), then you also have to have an even number
>in the face centers, even if swaps in the face centers are invisible.
I've had this problem as well. If I understand you correctly, this
problem simply doesn't occur anymore as soon as you number (or mark in
any other way) the center pieces which
a) makes solving the cube a bit more difficult
b) makes sure that you'll always get back to the original
configuration of center pieces.
I've had a similar problem on my 5x5x5 as well, and I assume that
marking the nine center pieces might solve the problem as well.
On my 4x4x4 I also had a problem of having two pairs of edges
exchanged which simply can't happen on a 3x3x3. By experimenting with
3x3x3 moves I found a 24move solution to this, and I wonder if that's
also sort of automatically solved by marking center pieces.
Can anyone confirm this?
Michael Ehrt
---------------------------------------------
ERCO@compuserve.com
From cube-lovers-errors@mc.lcs.mit.edu Wed Oct 1 13:55:36 1997
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Mail-from: From darinh@ldr.com Wed Oct 1 13:22:36 1997
Message-Id: <34328707.3792@ldr.com>
Date: Wed, 01 Oct 1997 10:23:24 -0700
From: Darin Haines
Organization: Litho Development & Research
To: Cube
Subject: Piece for a Rubik's Revenge
Hi Everyone,
Does anyone know of someone wanting to sell a BROKEN Rubik's Revenge?
or maybe a center piece from the same?
Did anyone else have problems with the center pieces breaking on their
RR? or am I the only one?
My RR has been sitting useless on the shelf since '84. Hey, I was young
and careless. ;-)
-Darin
[Moderator's note: I'll be away from cube-lovers from 2 Oct to 5 Oct.
Messages received during that time will be distributed on the 6th. ]
From cube-lovers-errors@mc.lcs.mit.edu Wed Oct 1 15:49:18 1997
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Mail-from: From bagleyd@americas.sun.sed.monmouth.army.mil Wed Oct 1 14:41:30 1997
From: bagleyd@americas.sun.sed.monmouth.army.mil (David Bagley x21081)
Message-Id: <199710011842.OAA21977@asia.sed.monmouth.army.mil>
Subject: Piece for Alexander's Star
To: Cube-Lovers@ai.mit.edu
Date: Wed, 1 Oct 1997 14:42:15 -0400 (EDT)
In-Reply-To: <34328707.3792@ldr.com> from "Darin Haines" at Oct 1, 97 10:23:24 am
>
> Hi Everyone,
>
> Does anyone know of someone wanting to sell a BROKEN Rubik's Revenge?
> or maybe a center piece from the same?
>
That reminds me:
If anyone needs a piece or 2 for the Alexander's Star let me know.
They seem to break pretty easily IMHO. Please specify colors.
I have a center piece too.
Mine broke a while back and I have since gotten another one.
--
Cheers,
/X\ David A. Bagley
(( X bagleyd@bigfoot.com http://wauug.erols.com/~bagleyd/
\X/ xlockmore ftp://wauug.erols.com/pub/X-Windows/xlockmore/index.html
From cube-lovers-errors@mc.lcs.mit.edu Wed Oct 1 16:57:31 1997
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Mail-from: From Cube-Lovers-Request@ai.mit.edu Wed Oct 1 16:53:46 1997
Date: Wed, 1 Oct 1997 16:53:46 -0400 (EDT)
Message-Id: <01Oct1997.165346.Cube-Lovers@AI.MIT.EDU>
From: Cube Lovers Moderator
To: Cube-Lovers@AI.MIT.EDU
Subject: 4x4x4 solution -- [Digest v23 #165]
Cube-Lovers Digest Wed, 1 Oct 1997 Volume 23 : Issue 165
Today's Topic:
4x4x4 solution
----------------------------------------------------------------------
Date: Wed, 1 Oct 1997 08:20:34 -0400 (EDT)
From: Assoc Prof W David Joyner
To: C.McCaig@queens-belfast.ac.uk
Cc: cube-lovers@ai.mit.edu
Subject: Re: 4x4x4 solution
On Tue, 30 Sep 1997 C.McCaig@queens-belfast.ac.uk wrote:
> i recently borrowed a friends 4x4x4, and i know the basic method for
> solving it. ie get the 6 centres, pair up all the edges, and then
> solve for the normal cube. however, about half the time i end up
> with a single edge pair inverted and cant figure out a move for
> reorientating the single edge pair. usually i break a few pairs
> and try and reorientate them this way, but this seems rather longwinded...
> does anyone have a move for this?. for example, say the green edge
> is on the blue face, and the blue edge is on the green face...
The idea is on the www page
http://www.nadn.navy.mil/MathDept/wdj/solve4.txt
Try
L2^2*D1^2*U2*F1^3*U2^3*F1*D1^2*L2^2*L1*U1*L1^3*U2^3*L1*U1^3*L1^3
(due to Jeff Adams). - David Joyner
------------------------------
Date: Wed, 1 Oct 1997 14:13:50 -0400 (EDT)
From: Nichael Cramer
To: Edwin Saesen
Cc: CUBE
Subject: Re: 4x4x4 solution -- [Digest v23 #159]
On Wed, 1 Oct 1997, Edwin Saesen wrote:
> On my 4x4x4 I also had a problem of having two pairs of edges
> exchanged which simply can't happen on a 3x3x3.
I find it most convienent to think of this situation as the following:
One of the "center-slices" containing one of the "swapped" edge pieces is
rotated by 90 degrees.
(This is roughly analogous to the the 3X case where the whole cube is
solved except for two corners and two edge pieces being --respectively--
swapped. The problem is that the unfinished face is 90dg out of phase.)
Rotate that center-slice by 90 degrees and re-solve from there.
This is surely not the most efficient (i.e. shortest) solution; but it is
conceptually straight forward.
Nichael Cramer
work: ncramer@bbn.com
home: nichael@sover.net
http://www.sover.net/~nichael/
------------------------------
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Date: Wed, 1 Oct 1997 16:23:14 -0400
From: Jim Mahoney
To: ERCO@compuserve.com
Cc: CUBE-LOVERS@ai.mit.edu
Subject: Re: 4x4x4 solution -- [Digest v23 #159]
>Your problem is one of parity. You have two edges cubies swapped
>(this swap is visible) and two face center (centre) cubies of the
>same color swapped (this swap is invisible). You have to have an
Edwin> I've had this problem as well.... If I understand you
Edwin> correctly, this problem simply doesn't occur anymore as
Edwin> soon as you number (or mark in any other way) the center
Edwin> pieces which a) makes solving the cube a bit more difficult
Edwin> b) makes sure that you'll always get back to the original
Edwin> configuration of center pieces.
This isn't quite true, at least not on the 4x4x4.
While it is true that parity is the question at hand, and also that on
the 4x4x4 cube a quarter of a central slice performs an odd permutation
on the edges which is otherwise "invisible", it is *not* true that
marking the centers will help. The reason is that a quarter turn on a
center slice of the 4x4x4 performs a cyclic rearrangement of 4 edges -
an odd permutation - while at the same time rearranges *two* sets of 4
central pieces - an even permutation of the centers. Thus parity does
not prohibit swapping two edges while leaving the centers untouched.
Moreover, in fact there are move sequences which will exchange two
edges without disturbing the position of any other piece, corner or
center - though I don't have any on hand which are short. If there's
interest, though, I can produce a move sequence to exchange two 4x4x4
edges while leaving all corners and centers in their original
positions.
A cross-section looks like this. A quarter turn cycles the four E's,
the four C1's, and the four C2's. This is an odd permutation of the
E's but an even permutation of the C's. (All the C's are corners, and
can be put into each other's positions with a combination of face and
center turns.)
E C1 C2 E
C2 C1
C1 C2
E C2 C1 E
A full account of parity and possible 4x4x4 moves gives
4x4x4
type , how many , parity after: 1/4 face turn , 1/4 center turn
- ---------------------------------------------------------------------
corners | 8 | odd | even (untouched)
edges | 24=2x(12 edges) | even (8 move) | odd
centers | 24=4x(6 faces) | odd | even (8 move)
Thus to solve a 4x4x4 cube you must have made both (1) an even
total number of moves on the faces (to restore the corners and centers to
even parity), as well as (2) an even total number of moves on the center
slices, to restore the edges to even parity.
The parity constraints on the 5x5x5 are a bit different. In that case
there are two types of edges (the one in the middle of an edge vs the
ones next to the corners) and three types of centers. Each has its
own parity change under each different slice. A bit of playing around
shows that any central slice move which cycles 4 edges must also cycle
several kinds of centers. At least one of those center cycles is odd.
Therefore on the 5x5x5 you cannot exchange a pair of edges without
also exchanging two centers somewhere. So marking where the centers
go will help on the 5x5x5.
Regards,
Jim Mahoney mahoney@marlboro.edu
Physics & Astronomy
Marlboro College, Marlboro, VT 05344
------------------------------
End of Cube-Lovers Digest
*************************
From cube-lovers-errors@mc.lcs.mit.edu Wed Oct 1 17:46:05 1997
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Mail-from: From SCHMIDTG@iccgcc.cle.ab.com Wed Oct 1 16:48:58 1997
From: SCHMIDTG@iccgcc.cle.ab.com
Date: Wed, 1 Oct 1997 16:48:44 -0400 (EDT)
To: cube-lovers@ai.mit.edu
Cc: ljl@basmark.com
Message-Id: <971001164844.2023493c@iccgcc.cle.ab.com>
Subject: New cube program available
I've recently finished my implementation of Kociemba's algorithm and it
is now available from the cube-lovers ftp site at:
ftp.ai.mit.edu /pub/cube-lovers/contrib/kcube1_0.zip
The .zip files contains a README.TXT file, commented C++ source, and
an executable program that runs on Win95/NT. Here's a brief description
of the program that appears in the README file within the "contrib"
directory:
File: kcube1_0.zip
Author: Greg Schmidt
Description:
A cube solver that implements Kociemba's algorithm. This program was
written for the express purpose of understanding the algorithm in
sufficient detail for me to implement it. The source code is included
and commented with the hope of providing others with a similar
understanding.
I welcome feedback concerning any aspects of this program. Many thanks to
Dik Winter and especially Herbert Kociemba for answering some of my
detailed questions as well as allowing me to use their ideas and offer
them to cube-lovers in the form of this program.
Regards,
-- Greg
From cube-lovers-errors@mc.lcs.mit.edu Wed Oct 1 19:01:21 1997
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Mail-from: From joemcg3@snowcrest.net Wed Oct 1 18:42:12 1997
Message-Id: <3432D0DB.4B19@snowcrest.net>
Date: Wed, 01 Oct 1997 15:38:19 -0700
From: Joe McGarity
Reply-To: joemcg3@snowcrest.net
To: "Mailing List, Rubik's Cube"
Subject: My Revenge is Complete
How strange that I both have a broken Rubik's Revenge and need a piece
to an Alexander's Star. What are the odds?
I haven't looked at it for quite some time, but I think my Revenge is
complete. The problem is the ball in the center. One of the corners
is broken (if you have seen a dissasmbled RR it makes sense for a ball
to have corners) and has resisted all attempts at being glued. So it
may be that my broken cube will not be of any help to Darin Haines, but
the rest of the cubies are intact if anyone needs any of them. Which
brings up that I have a fairly large collection of broken or otherwise
incomplete puzzles. I suspect that this is true for many of us. I would
be more than willing to do some trading if anyone has any particular
needs. Let me know.
Joe McGarity
From cube-lovers-errors@mc.lcs.mit.edu Wed Oct 1 20:02:37 1997
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Mail-from: From randall@theory.lcs.mit.edu Wed Oct 1 19:49:18 1997
Date: Wed, 1 Oct 1997 19:46:08 -0400
Message-Id: <199710012346.TAA06162@hemp>
From: Keith H Randall
To: reid@math.brown.edu
Cc: cube-lovers@ai.mit.edu
In-Reply-To: <199710012120.AA25636@theory.lcs.mit.edu> (message from michael
reid on Wed, 1 Oct 1997 17:19:02 -0400)
Subject: Re: God's Number
Don Dailey, Aske Plaat, and myself have a program that will do a
complete 22-ply search in about 24 hours on an 8 processor Sun
machine. The program measures distance in the QT (quarter-turn)
metric.
I've run some experiments on random cubes, summarized as follows:
112 random odd cubes:
20 depth 19
92 depth 21
57 random even cubes:
41 depth 20
16 depth 22
>From this random sample, it seems as if less than 1% of cubes are
depth 23, let alone more than depth 24. In fact, the only depth 23
cubes I know of so far are the twelve cubes 1 move away from the
superflip. This fact gives some evidence that God's number is
probably 24.
By the way, below are solutions and depths for all of the symmetric
cubes enumerated by Hoey and Saxe in their message of Sun, 14 Dec 80.
These are obvious cubes to try because they are local maxima, and they
are all depth 22 or less except for the superflip. Only one
representative from each of the 26 conjugacy classes is given. All
solutions were obtained from the program, except for the superflip
solution which is absconded from a post from Reid on Tue, 10 Jan 95.
All depths are exact minimal depths, i.e. no shorter solutions exist.
M-symmetric cubes
0 solved
--
12 pons asinorum
F F B B L L R R U U D D
24 superflip
R' U U B L' F U' B D F U D' L D D F' R B' D F' U' B' U D'
20 pons asinorum * superflip
F' U' B' R' F R L' D' R L' U D' L' U D' F R B U F
T-symmetric cubes
22 girdleflip
F F U F F B' U R' L B U F D' F F B D' R L' B' D' F
19 girdleswap
F U F R U' L' U' B U' B' R' F' R' L' F' R L L F'
21 girdleflip * girdleswap
F U' L U F' U' B B D B U B' D' R D' R' B' R' D R B'
22 girdleflip * pons asinorum
F F U L F L' D' R L' U' L L U U R F' B D' F' U R' D'
17 girdleswap * pons asinorum
F R F B R' F' B' L D D F F D D R' L' F'
21 girdleflip * girdleswap * pons asinorum
F R' L B R U' R R U' D F' R F' B L B R' F B' U' L'
20 girdleflip * superflip
F U U F' R' U' L F' D F B' L U' L U' F' L U D' F
21 girdleswap * superflip
F R F B U D' F' B R R U F B D' R L D' F' B' U F
21 girdleflip * girdleswap * superflip
F U D B' R' F' D' R' U R' L' B R F U F D B D L' B'
20 girdleflip * pons asinorum * superflip
F F B R' F U' B' R' L D L U' R' U' D F L B' D F
21 girdleswap * pons asinorum * superflip
F U U B D' L' U F F B R' U R B U D' L B U D' L
21 girdleflip * girdleswap * pons asinorum * superflip
F B U F' U' F R B' R' F' U R' U F B U' F' B' U R U'
H-symmetric cubes
22 plummer
F F R B' U L U R F L U' L L B R' D F D B L F D'
16 six-H
F F R R F B' R R L L F' B R R B B
20 plummer * six-H
F F U F' R' B' D' F' R U D L B' U' F' L' B' U' F F
20 plummer^2 * six-H
F F U F R B U F R' U' D' L' B D F L B U' F F
20 plummer^2 * pons asinorum
F R U F D' B B L F L' F' L F R R D' L U B L
20 plummer^2 * superflip
F B U F R L' U' D' L U' D L B L' B' U F D L U
18 six-H * superflip
F R' U D D F' B R F R' L D' F' R' L U L F'
22 plummer * six-H * superflip
F U D F' R L F U D R R L L B' U D F' R L B' R' L
22 plummer^2 * six-H * superflip
F B U F' B' D R' L' U F F B B R' L' D' F' B' D R' L' D'
22 plummer * pons asinorum * superflip
F B R' U' D' R' F' B' R U' D' F F B B L U' D' R' F' B' L'
reference for cube names:
pons asinorum
W B W
B W B
W B W
O R O G Y G R O R Y G Y
R O R Y G Y O R O G Y G
O R O G Y G R O R Y G Y
B W B
W B W
B W B
superflip
W Y W
O W R
W G W
O W O G W G R W R Y W Y
Y O G O G R G R Y R Y O
O B O G B G R B R Y B Y
B G B
O B R
B Y B
plummer
Y W Y
W W W
G W G
W O W O G R W R W R Y O
O O O G G G R R R Y Y Y
B O B O G R B R B R Y O
G B G
B B B
Y B Y
six-H
W W W
B W B
W W W
O O O G Y G R R R Y G Y
R O R G G G O R O Y Y Y
O O O G Y G R R R Y G Y
B B B
W B W
B B B
girdle flip (about ULF-DRB axis)
W Y W
W W R
W W W
O O O G G G R W R Y W Y
Y O O G G R G R R Y Y O
O B O G B G R R R Y Y Y
B G B
O B B
B B B
girdle swap (about ULF-DRB axis)
R B B
W W B
W W Y
B O O G G B O O G O G G
R O O G G Y O R R Y Y G
R R Y R Y Y W R R Y Y W
W W O
W B B
G B B
-Keith
From cube-lovers-errors@mc.lcs.mit.edu Wed Oct 1 20:49:17 1997
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Mail-from: From roger.broadie@iclweb.com Wed Oct 1 19:12:26 1997
From: roger.broadie@iclweb.com (Roger Broadie)
To:
Subject: Re: 4x4x4 solution
Date: Thu, 2 Oct 1997 00:09:47 +0100
Message-Id: <19971002000735.AAA23683@home>
I'm tempted to try a little more analysis of the parity constraints on
the 4x4x4 cube, though no doubt it's all been done before. As der
Mouse said,
A slice turn produces a 4-cycle on the edges and two 4-cycles on the
face centres; a face turn produces a 4-cycle on the face centres and
two 4-cycles on the edges (and a 4-cycle on the corners, which may or
may not be relevant).
I think it is very relevant. We can set the effects out as follows:
Turn Piece Cycle(s) Parity
------ ------- -------- -------
Slice edge 1x4 odd
centre 2x4 even
Face edge 2x4 even
centre 1x4 odd
corner 1x4 odd
The consequence is that the parity of the centre pieces depends
entirely on the number of face turns - any slice turns do not affect
the parity of these pieces since the changes they introduce will be of
even parity. For face turns, the changes to the parity of the corner
pieces and the centre pieces are the same. Hence if the corner pieces
are in place, the centres will be in an even permutation, and that will
not be changed even if the edge pieces are in an odd permutation, which
was the essence of Clive McCaig's original question. Nor will that be
changed by any turn of a central slice to bring them back to an even
permutation.
I the corners are correct (which I guess is the normal situation when
the problem with the swapped edge pieces shows up) then, though I say
so with some hesitation, I do not think Jerry Bryan is right in saying
that the pair of swapped edge pieces will be matched by a pair of
swapped centre pieces. For example, the process I quoted switches edge
pieces, and though it has no visible effect on the centre pieces, it
does in fact change the positions of the centre pieces on the front
face (if I have correctly identified the results of a bit of hasty work
with little Post-it stickers). However, the whole block of four
rotates through 180 degrees, which is two 2-cycles and thus of even
parity. Edwin Saesen could mark the centre pieces, get them back to
their original position and still find the edge pieces swapped, but
that will not prevent his correcting the edge pieces, and then, if he
wants to, correcting the centre pieces with even-parity processes.
Luckily, for the 4x4x4, we do not have to worry about twists for the
edge pieces or the centre pieces, since that is fixed geometrically for
each position they can occupy. When an edge piece is in its home
position it must be the right way round. When it moves to its
next-door position it must flip. I imagine this is the point behind
Allan Wechsler's charming square-dancing analogy. The centre pieces
always present the same corner to the central intersection of the face.
Roger Broadie
From cube-lovers-errors@mc.lcs.mit.edu Mon Oct 6 19:55:23 1997
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Mail-from: From Goyra@iol.ie Wed Oct 1 20:07:28 1997
Message-Id: <199710020007.BAA10545@GPO.iol.ie>
From: "Goyra (David Byrden)"
To:
Subject: For all cube programmers
Date: Thu, 2 Oct 1997 01:02:23 +0100
When writing a program to manipulate
the Cube, you're interested in your algorithm.
The output usually looks like RLULRURL
because you won't waste time programming
any graphics.
I will shortly release a freeware
software component that displays a standard
Rubik's Cube. You can incorporate it into
your software and manipulate the cube
directly. See your cube solutions executed in
front of your eyes.
For an idea of what this component
will look like, take a Java browser to my
pages at
http://www.iol.ie/~goyra/Rubik.html
The component will be a Java Bean,
meaning you can use it in Java, and also
in any Activex environment such as Visual
C++ or Visual Basic.
Anyone with suggestions about how
the programmatic interface to the component
should look, please mail me.
David
From cube-lovers-errors@mc.lcs.mit.edu Mon Oct 6 21:04:33 1997
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Mail-from: From Cube-Lovers-Request@ai.mit.edu Mon Oct 6 21:02:21 1997
Date: Mon, 6 Oct 1997 21:02:21 -0400 (EDT)
Message-Id: <06Oct1997.210221.Cube-Lovers@AI.MIT.EDU>
From: Cube Lovers Moderator
To: Cube-Lovers@AI.MIT.EDU
Subject: 4x4x4 solution -- [Digest v23 #170]
Cube-Lovers Digest Mon, 6 Oct 1997 Volume 23 : Issue 170
Today's Topic:
4x4x4 solution
----------------------------------------------------------------------
Date: Wed, 1 Oct 1997 23:26:34 -0400 (EDT)
From: Nichael Cramer
To: Roger Broadie
Cc: Cube-Lovers@ai.mit.edu
Subject: Re: 4x4x4 solution
Message-Id:
Roger Broadie wrote:
> A slice turn produces a 4-cycle on the edges and two 4-cycles on the
> face centres; a face turn produces a 4-cycle on the face centres and
> two 4-cycles on the edges (and a 4-cycle on the corners, which may or
> may not be relevant).
>
> I think it is very relevant. We can set the effects out as follows:
>
> Turn Piece Cycle(s) Parity
> ------ ------- -------- -------
>
> Slice edge 1x4 odd
> centre 2x4 even
>
> Face edge 2x4 even
> centre 1x4 odd
> corner 1x4 odd
>
> The consequence is that the parity of the centre pieces depends
> entirely on the number of face turns - any slice turns do not affect
> the parity of these pieces since the changes they introduce will be of
> even parity. For face turns, the changes to the parity of the corner
> pieces and the centre pieces are the same. Hence if the corner pieces
> are in place, the centres will be in an even permutation, and that will
> not be changed even if the edge pieces are in an odd permutation, which
> was the essence of Clive McCaig's original question. Nor will that be
> changed by any turn of a central slice to bring them back to an even
> permutation.
As one of the folks who advocated rotating a center slice, let me explain
my (admittedly non-optimal) process for getting out of this fix and perhaps
you can explain where my reasoning is wrong.
1] Imagine a 4X which is completely solved except for two flipped (i.e.
swapped) edge-pieces.
2] For simplicity's sake --and without loss of generality--, assume the 2
flipped/swapped pieces are adjacent and in the top front location. So the
top of the cube will look like this:
X X X X
X X X X
X X X X
X 1 2 X
(Here the numbers are meant to indicate only where the cubies are located,
having nothing to do with their colors.)
3] I now rotate one of the center slices (say, the one on the right, i.e.
the one containing the cubie "2") 90dg away from me.
4] The top of the cube now looks like:
X X 2 X
X X O X
X X O X
X 1 3 X
5] I can now perform the 3-cycle 1->3->2 (i.e. without affecting any of the
rest of the cube).
The top of the cube now looks like:
X X 3 X
X X O X
X X O X
X 2 1 X
6] In particular, note that "2" and "1" are now in their correct positions
(and, of course, necessarily in their proper "flip" orientation).
7] Moreover, note that I now have exactly three edge cubes in the wrong
place (i.e. "3" from above and the other two edge cubes which were
misplaced during my original 90dg rotation of the center slice).
I can now perform a 3-cycle on these edges pieces (similar to the one used
in step 5 above) again without affecting any of the other locations on the
cube.
8] My cube now has all the edge pieces in their correct location.
9] I now have only to "fix" the 8 central-face cubes which were misplaced
during my initial 90dg twist. I can now do this is short order.
QED[?]
Nichael Cramer
work: ncramer@bbn.com
home: nichael@sover.net
http://www.sover.net/~nichael/
------------------------------
From: roger.broadie@iclweb.com (Roger Broadie)
To: "Nichael Cramer"
Cc:
Subject: Re: 4x4x4 solution
Date: Thu, 2 Oct 1997 23:12:39 +0100
> From: Nichael Cramer
> To: Roger Broadie
> Cc: Cube-Lovers@ai.mit.edu
> Subject: Re: 4x4x4 solution
> Date: 2 October 1997 4:26
>
> As one of the folks who advocated rotating a center slice, let me
> explain my (admittedly non-optimal) process for getting out of this
> fix and perhaps you can explain where my reasoning is wrong.
>[followed by a procedure in which a quarter turn of a centre slice is
followed, first, by a 3-cycle of edges on the top to restore the two
swapped pieces, second, by a 3-cycle of edges to restore the other
displaced edges, and, third, by restoring the displaced centres]
I absolutely agree with your reasoning. A quarter turn of a central
slice must be at the heart of any procedure to perform an edge swap,
because it is the only way to change the parity of the edges. That was
what I said in my first post on 1 October 1997.
In my second post I was trying to look at the effect of that quarter
turn of the central slice on the centre pieces, and show that, as they
had been subjected to an even permutation by reason of the centre-slice
turn, the centre pieces could not have undergone an invisible swap of a
single pair of centre pieces. Having made a single quarter turn of the
central slice, all the other edge and centre pieces can be restored
with processes of even parity, like your two 3-cycles.
Roger Broadie
------------------------------
End of Cube-Lovers Digest
*************************
From cube-lovers-errors@mc.lcs.mit.edu Mon Oct 6 22:28:22 1997
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Mail-from: From Cube-Lovers-Request@ai.mit.edu Mon Oct 6 22:27:32 1997
Date: Mon, 6 Oct 1997 22:27:32 -0400 (EDT)
Message-Id: <06Oct1997.222732.Cube-Lovers@AI.MIT.EDU>
From: Cube Lovers Moderator
To: Cube-Lovers@AI.MIT.EDU
Subject: 4x4x4 solution -- [Digest v23 #171]
Cube-Lovers Digest Mon, 6 Oct 1997 Volume 23 : Issue 171
Today's Topic: Pieces of broken cubes:
Rubik's Revenge (Clarified)
My Revenge is Complete
Piece for a Rubik's Revenge
Piece for Alexander's Star
----------------------------------------------------------------------
Date: Thu, 02 Oct 1997 08:01:03 -0700
From: Darin Haines
To: Cube
Subject: Rubik's Revenge (Clarified)
I guess my terminology was incorrect. The parts I need are actually the
center cubies of which there are 4 on each side (for a total of 24). It
sounds like Joe McGarity's broken RR will help me out just fine (as will
a couple of other responses I've received).
I got to looking last night and found that I actually need 3 (not just
1) of these center cubies.
- -Darin
------------------------------
To: cube-lovers@ai.mit.edu
From: "Bryan Main"
Subject: Re: My Revenge is Complete
Date: Thu, 02 Oct 1997 14:32:15 Eastern Daylight Time
At 03:38 PM 10/1/97 -0700, you wrote:
>I haven't looked at it for quite some time, but I think my Revenge is
>complete.
How stable are the 4x4x4 and 5x5x5? I was thinking on getting one but they
cost quite a lot of money and was wondering how easy it is to break them.
Also what kind of paint should I use to paint my cubes as I have 4 normal
ones and would like to make different patterns on them to make them more
intersting. Also any patterns would be helpful.
bryan
__________________________________________________________________
Bryan Main
Cartographic Specialist
http://caddscan.com
------------------------------
Date: Thu, 2 Oct 1997 22:42:11 -0400 (EDT)
From: Nicholas Bodley
To: Darin Haines
Cc: Cube
Subject: Re: Piece for a Rubik's Revenge
On Wed, 1 Oct 1997, Darin Haines wrote:
{Snips}
}Did anyone else have problems with the center pieces breaking on their
}RR? or am I the only one?
These pieces >are< rather fragile, as I remember.
|* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath
|* Waltham, Mass. *|* -----------------------------------------------
|* nbodley@tiac.net *|* Waltham is now in the new 781 area code.
|* Amateur musician *|* 617 will be recognized until 1 Dec. 1997.
------------------------------
Date: Sun, 5 Oct 1997 18:40:17 -0400 (EDT)
From: Nicholas Bodley
To: David Bagley x21081
Cc: Cube-Lovers@ai.mit.edu
Subject: Re: Piece for Alexander's Star
They do break easily. I haven't had mine out of storage for some time, but
I well remember that it needed conscious care when manipulating; nothing
like a properly-lubricated deluxe Ideal 3^3 (the one with plastic color
tiles, and changes to the shapes of the pieces that tend to make it
self-align).
|* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath
|* Waltham, Mass. *|* -----------------------------------------------
|* nbodley@tiac.net *|* Waltham is now in the new 781 area code.
|* Amateur musician *|* 617 will be recognized until 1 Dec. 1997.
------------------------------
End of Cube-Lovers Digest
*************************
From cube-lovers-errors@mc.lcs.mit.edu Mon Oct 6 23:26:59 1997
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Mail-from: From Cube-Lovers-Request@ai.mit.edu Mon Oct 6 23:26:19 1997
Date: Mon, 6 Oct 1997 23:26:19 -0400 (EDT)
Message-Id: <06Oct1997.232619.Cube-Lovers@AI.MIT.EDU>
From: Cube Lovers Moderator
To: Cube-Lovers@AI.MIT.EDU
Subject: God's number -- [Digest v23 #172]
Cube-Lovers Digest Mon, 6 Oct 1997 Volume 23 : Issue 172
Today's Topic:
God's Number
----------------------------------------------------------------------
Date: Thu, 02 Oct 1997 17:04:33 -0400 (Eastern Daylight Time)
From: Jerry Bryan
Subject: Re: God's Number
To: Keith H Randall
Cc: reid@math.brown.edu, cube-lovers@ai.mit.edu
Message-Id:
On Wed, 1 Oct 1997, Keith H Randall wrote:
> Don Dailey, Aske Plaat, and myself have a program that will do a
> complete 22-ply search in about 24 hours on an 8 processor Sun
> machine. The program measures distance in the QT (quarter-turn)
> metric.
>
> I've run some experiments on random cubes, summarized as follows:
>
> 112 random odd cubes:
> 20 depth 19
> 92 depth 21
>
> 57 random even cubes:
> 41 depth 20
> 16 depth 22
Wow. I am impressed with how much data you have. For the case of
random cubes and guaranteed optimal solutions, I believe this is the
most data which has been posted to Cube-Lovers.
It would be nice to examine enough cases to raise the probability that a
few positions of length 17q would show up for odd cubes and of length
18q for even cubes. At this distance from Start, the branching factor
for one level is about 9.3, so the branching factor for two levels
(e.g., between level 17 and level 19) would be about 85 or so. So you
are just at the edge of the sample size where you would expect the
shorter lengths to show up.
Notwithstanding that, I decided to play with the numbers to see if I
could make any reasonable projection about the overall distribution of
lengths in the quarter-turn metric. Here is what I have come up with.
Consider the 19q case. Your results suggest that about 17.8% of odd
positions, and hence about 8.6% or 8.7% of all positions are exactly 19q
from Start. (The sample size does not support an estimate of that
precision, of course, but let's continue anyway). It's easy to
calculate that no more than about 8.4% of positions can be 19q from
Start. From this, I would conclude two things. First, your results
seem right on, well within the bounds of sampling error. Second, your
results suggest that it is very unlikely that the branching factor drops
below about 9.3 until you pass 19q from Start. Using the best available
known results, plus using your results as an estimate, plus some other
guessing, I would propose that the actual search tree for the q-turn
case looks something like the following.
Distance Number Branching Cumulative
from of Factor Number of
Start Positions Positions
0 1 1
1 12 12.000 13
2 114 9.500 127
3 1068 9.368 1195
4 10011 9.374 11206
5 93840 9.374 105046
6 878880 9.366 983926
7 8221632 9.355 9205558
8 76843595 9.347 86049153
9 717789576 9.341 803838729
10 6701836858 9.337 7505675587
11 62549615248 9.333 70055290835
12 5.838E+11 9.333 6.538E+11
13 5.449E+12 9.333 6.102E+12
14 5.085E+13 9.333 5.696E+13
15 4.746E+14 9.333 5.316E+14
16 4.430E+15 9.333 4.961E+15
17 4.134E+16 9.333 4.631E+16
18 3.859E+17 9.333 4.322E+17
19 3.601E+18 9.333 4.034E+18
20 1.546E+19 4.294 1.950E+19
21 1.657E+19 1.071 3.606E+19
22 6.035E+18 0.364 4.210E+19
23 12 0.000 4.210E+19
24 1 0.083 4.210E+19
Notice that my table does not quite reach |G|, so there are probably a
few more positions than this at 20q, 21q, and 22q from Start (there
can't be more any closer to Start than that). Also, the branching factor
probably does not remain constant at 9.333 all the way out to 19q from
Start; it probably declines slightly, maybe to 9.300 or so. Finally,
the distribution is probably bimodal, with modes at 20q and 21q (it
almost has to be bimodal because of odd/even parity considerations).
(By the way, I am making no claim whatsover that the diameter of the
cube group is 24q. This is only an educated guess based on the evidence
at hand. In fact I tend to doubt it. I think the branching factor in
the chart just drops off too sharply at levels 21q, 22q, and 23q for
the chart to be real.)
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us
Pellissippi State (423) 539-7198
10915 Hardin Valley Road (423) 694-6435 (fax)
P.O. Box 22990
Knoxville, TN 37933-0990
------------------------------
Date: Sun, 5 Oct 1997 18:54:32 -0400
From: michael reid
To: cube-lovers@ai.mit.edu, randall@theory.lcs.mit.edu
Subject: Re: God's Number
keith randall writes
> Don Dailey, Aske Plaat, and myself have a program that will do a
> complete 22-ply search in about 24 hours on an 8 processor Sun
> machine. The program measures distance in the QT (quarter-turn)
> metric.
wow, that's quite a bit faster than my optimal solver! how about searches
through other depths (20q, 21q, 23q, ... )? does the run time depend upon
the input position? could you describe your searching algorithm? i'm
sure that this would be of interest to many people on the cube-lovers
mailing list.
> By the way, below are solutions and depths for all of the symmetric
> cubes enumerated by Hoey and Saxe in their message of Sun, 14 Dec 80.
i already posted data for these positions, but it's always nice to have
confirmation. however, ...
> 22 girdleflip * pons asinorum
> F F U L F L' D' R L' U' L L U U R F' B D' F' U R' D'
this is solvable in 18q:
) 3. U R U' F D R L' B' L' F R F B' U' L' D B' D' (18q, 18f)
although i gave it in a different orientation.
> 22 plummer * six-H * superflip
> F U D F' R L F U D R R L L B' U D F' R L B' R' L
there's a slight typo here; the last twist should be L' .
mike
------------------------------
End of Cube-Lovers Digest
*************************
From cube-lovers-errors@mc.lcs.mit.edu Tue Oct 7 13:06:59 1997
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Mail-from: From nichael@sover.net Tue Oct 7 00:45:56 1997
Message-Id:
In-Reply-To: <06Oct1997.222732.Cube-Lovers@AI.MIT.EDU>
Date: Mon, 6 Oct 1997 23:03:54 -0400
To: Cube-Lovers@ai.mit.edu
From: Nichael Lynn Cramer
Subject: Re: 4x4x4 solution -- [Digest v23 #171]
Cc: bmain@caddscan.com
[ Moderator's note--The subject is misleading, because I erroneously
titled Digest v23 #171 as "4x4x4 solutions". It was actually about
"Pieces of broken cubes"--the discussion of breakability and the idea
of trading pieces of cubes. I regret the error. ]
>To: cube-lovers@ai.mit.edu
>From: "Bryan Main"
>Subject: Re: My Revenge is Complete
>How stable are the 4x4x4 and 5x5x5? I was thinking on getting one but they
>cost quite a lot of money and was wondering how easy it is to break them.
5Xs are pretty stable; each side has a fixed center piece (i.e. like a 3X).
I've had three and never had any problem with any of them.
4Xs are another ballgame altogether. Since they don't have a fixed center,
they depend on an internal configuration, consisting of a cluster of four
plates, to hold the faces on.
Each of these plates is held on with a screw and this adjustment is
_critical_. Too tight and it can be all but impossible to twist the
faces; too loose and the cube tends to dissolve in your hands.
I've owned four; one was fine, one was OK/usable, one was too stiff to use
and one couldn't be kept together. So, the "usability" rate was approx
1/3. (OTOH I picked them all up for $2/ea at a ToysRU clearance...)
Nichael
nichael@sover.net 6.501
http://www.sover.net/~nichael/ -- the ln of the Beast
From cube-lovers-errors@mc.lcs.mit.edu Tue Oct 7 17:04:52 1997
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Mail-from: From jbryan@pstcc.cc.tn.us Tue Oct 7 16:59:58 1997
Date: Tue, 07 Oct 1997 16:59:25 -0400 (Eastern Daylight Time)
From: Jerry Bryan
Subject: Maximality Analysis Through 11q
To: cube-lovers@ai.mit.edu
Message-Id:
Not too long ago, I reported that my Shamir program had completed
searching through 11q from Start, that the results did confirm my
previous results using tape spinning programs, that no local maxima were
found 11q from Start, and that otherwise nothing new was found. I have
come to realize that there is a small bit of new information. I really
should post the maximality analysis in its entirety, because the whole
row 11q from Start is new. The row 11q from Start does include the
failure to find any new local maxima. As always, the local maxima are
in the right-most column, where all 12 moves go closer to Start.
Maximalility Analysis
In Terms of Patterns (M-conjugacy classes)
Number of Moves which go Closer to Start
1 1 1
0 1 2 3 4 5 6 7 8 9 0 1 2
|x|
0 1 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 0 0 0 0 0 0 0 0 0 0 0
2 0 2 3 0 0 0 0 0 0 0 0 0 0
3 0 20 4 1 0 0 0 0 0 0 0 0 0
4 0 182 34 2 1 0 0 0 0 0 0 0 0
5 0 1677 280 20 1 0 0 0 0 0 0 0 0
6 0 15642 2561 184 8 0 0 0 0 0 0 0 0
7 0 145974 23773 1721 61 0 0 0 0 0 0 0 0
8 0 1362579 222235 16241 663 1 3 0 3 0 0 0 0
9 0 12719643 2077549 153026 5954 74 15 2 3 0 0 0 0
10 0 118711701 19418503 1438825 58862 925 318 11 37 0 8 0 4
11 0 1107594690 181433604 13517370 576891 11843 3442 251 321 10 21 2 0
Maximalility Analysis
In Terms of Positions
Number of Moves which go Closer to Start
0 1 2 3 4 5 6 7
|x|
0 1 0 0 0 0 0 0 0
1 0 12 0 0 0 0 0 0
2 0 96 18 0 0 0 0 0
3 0 912 144 12 0 0 0 0
4 0 8544 1368 96 3 0 0 0
5 0 80088 12816 912 24 0 0 0
6 0 749376 120612 8640 252 0 0 0
7 0 7001712 1135104 82152 2664 0 0 0
8 0 65391504 10645824 777936 28200 48 56 0
9 0 610499652 99666528 7338720 280800 3048 624 96
10 0 5698027296 931905180 69049264 2796978 43800 12336 528
11 0 53164171632 8708296416 648777868 27618360 563880 159024 11904
1 1 1
8 9 0 1 2
|x|
0 0 0 0 0 0
1 0 0 0 0 0
2 0 0 0 0 0
3 0 0 0 0 0
4 0 0 0 0 0
5 0 0 0 0 0
6 0 0 0 0 0
7 0 0 0 0 0
8 27 0 0 0 0
9 108 0 0 0 0
10 1296 0 138 0 42
11 14856 408 828 72 0
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us
Pellissippi State (423) 539-7198
10915 Hardin Valley Road (423) 694-6435 (fax)
P.O. Box 22990
Knoxville, TN 37933-0990
From cube-lovers-errors@mc.lcs.mit.edu Wed Oct 8 12:48:49 1997
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Mail-from: From dzander@solaria.sol.net Tue Oct 7 19:21:35 1997
From: Douglas Zander
Message-Id: <199710072320.SAA09034@solaria.sol.net>
Subject: broken rubik's cube: help!
To: cube-lovers@ai.mit.edu (cube)
Date: Tue, 7 Oct 97 18:20:57 CDT
Hello,
I wonder if someone can suggest a way to fix my 3x3x3 cube. The screw
that holds a center cubie to the spindle has stripped out of the spindle.
I thought of just super-glueing it back in; would this work? Also, I
wonder if there is to be any tension (compression) on the spring inside
the center cubie when the screw is set in? I'm afraid that I will have to
open up the center cubie and use a driver to screw the cube together
again. I don't want to pry open my center cubie.
Thanks for any suggestions.
--
Douglas Zander |
dzander@solaria.sol.net |
Milwaukee, Wisconsin, USA |
From cube-lovers-errors@mc.lcs.mit.edu Tue Oct 14 12:51:26 1997
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Mail-from: From jbryan@pstcc.cc.tn.us Mon Oct 13 16:19:54 1997
Date: Mon, 13 Oct 1997 16:18:30 -0400 (Eastern Daylight Time)
From: Jerry Bryan
Subject: Re: God's Number
In-Reply-To: <3.0.32.19970930192820.006ce8ac@po9.mit.edu>
To: Dennis Okon
Cc: cube-lovers@ai.mit.edu
Message-Id:
On Tue, 30 Sep 1997, Dennis Okon wrote:
> I just found out that Keith Randall for the theory group of LCS (Lab for
> Computer Science) at MIT gave a talk Monday about God's number for the
> rubik's cube. He upped the lower bound 24 and gave "evidence" that it is
> 24. I don't know what moves he was counting (e.g. slice, quarter).
> Unfortunately, I missed it. Does anyone have any information on this?
> I'll see what I can find out.
Was there ever any more information on this?
The lower bound for the diameter of the cube group was raised to 24q on
19 February 1995. I would be very surprised if Keith Randall presented
a position requiring 24f. I don't know of any published results in
metrics which include both slice and face turns.
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us
Pellissippi State (423) 539-7198
10915 Hardin Valley Road (423) 694-6435 (fax)
P.O. Box 22990
Knoxville, TN 37933-0990
From cube-lovers-errors@mc.lcs.mit.edu Tue Oct 14 18:44:16 1997
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Mail-from: From fb91@dial.pipex.com Tue Oct 14 08:00:33 1997
Message-Id: <199710141200.IAA10374@life.ai.mit.edu>
From: "Richard Armitage"
To:
Subject: VRML puzzles/newsletter
Date: Tue, 14 Oct 1997 12:59:59 +0100
We are shortly to create a full VRML site of cubes, and other similar
puzzles a la Rubiks and spacecubes. It will contain both free and for sale
items and will evolve as demand requests from people like you!!
I am going to be publishing a monthly newsletter from November 1997,
covering 3D puzzles (real and virtual) and SpaceCubes news. You can sign
up from the first page of our website or by sending an e-mail to
info@spacecubes.com with the subject newsletter. You will receive a
SpaceCubes info standard letter for now until we set up all the right
autoresponses but I wiil happily deal with all feedback.
Thankyou and looking forward to giving you good challenges
Richard
Richard Armitage (SpaceCubes Marketing)
tel:44 191 281 6011 US fax 2125048016
autorespond:
or
From cube-lovers-errors@mc.lcs.mit.edu Thu Oct 23 12:00:22 1997
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Mail-from: From whuang@ugcs.caltech.edu Thu Oct 23 07:05:58 1997
To: Cube-Lovers@AI.MIT.Edu
From: whuang@ugcs.caltech.edu (Wei-Hwa Huang)
Subject: Magic Make-a-cube
Date: 23 Oct 1997 11:05:18 GMT
Organization: California Institute of Technology, Pasadena
Message-Id: <62nb1e$q4a@gap.cco.caltech.edu>
I have just acquired what appear to be the components to Rubik's Magic
Make-a-cube. Unfortunately, two color paper pieces are missing. Can
anyone tell me what the color arrangements are, and in what order? Thanks
for anything you can dig up.
--
Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/
---------------------------------------------------------------------------
"[Lucy's eyes] look like little round dots of India ink..." -- Charlie Brown
From cube-lovers-errors@mc.lcs.mit.edu Fri Oct 31 11:43:29 1997
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Mail-from: From MO374@cnsvax.albany.edu Wed Oct 29 11:06:48 1997
Date: Wed, 29 Oct 1997 11:02:53 -0500 (EST)
From: Mary Osielski
Subject: Where to buy one???
To: cube-lovers@ai.mit.edu
Message-Id: <01IPDKFAISLE90NIU9@cnsvax.albany.edu>
I'm trying to buy a regular, standard, run-of-the-mill Rubik's cube
which I now realize is not so easy. Can you please direct me to a
source? Are they no longer produced? I got your address from the
mountains of material on the Internet about Rubik. Is there a store,
a phone number, a person from whom I can buy one. I'm in Albany, NY
but mail-order is fine. Thanks in advance for the help!
Mary Osielski
mo374@cnsvax.albany.edu
From cube-lovers-errors@mc.lcs.mit.edu Fri Oct 31 12:09:35 1997
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Mail-from: From mouse@Rodents.Montreal.QC.CA Thu Oct 30 12:30:33 1997
Date: Thu, 30 Oct 1997 12:29:32 -0500 (EST)
From: der Mouse
Message-Id: <199710301729.MAA08700@Twig.Rodents.Montreal.QC.CA>
To: cube-lovers@ai.mit.edu
Subject: Re: A* versus IDA*
It's a little off-topic and rather old (June 1st) anyway, so I'll make
this quick:
> [...discussion of FreeCell and "Baker's game"...]
Could someone interested in these contact me? I'd like to learn more
about Baker's game, whatever that is, and discuss some empirical
results with with Seahaven.
der Mouse
mouse@rodents.montreal.qc.ca
7D C8 61 52 5D E7 2D 39 4E F1 31 3E E8 B3 27 4B
From cube-lovers-errors@mc.lcs.mit.edu Fri Oct 31 12:45:42 1997
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Mail-from: From mouse@Rodents.Montreal.QC.CA Thu Oct 30 19:12:43 1997
Date: Thu, 30 Oct 1997 19:11:51 -0500 (EST)
From: der Mouse
Message-Id: <199710310011.TAA10960@Twig.Rodents.Montreal.QC.CA>
Mime-Version: 1.0
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Content-Transfer-Encoding: 8bit
To: cube-lovers@ai.mit.edu
Subject: Re: Categorization of cube solving programs
This is a response to a pretty old message:
> Date: Thu, 5 Jun 1997 22:56:56 -0400 (EDT)
However, I kept the message around, which usually means I never did
anything with it. If I already did, my apologies to the list for
duplication.
> Since I'm interested in such things, I came up with the following
> categories of cube solving programs in general order of increasing
> sophistication:
> Class 1: Simply provide a simulation of the cube and allow the
> user to manipulate the cube model [...]. Often these
> programs have very nice 3D graphics.
> Class 2: A program which solves the cube by implementing a
> canned algorithm (or 'book procedure'). [...]
> Class 3: A program that when given a specific instance of the
> cube, attempts to 'discover' or learn a sequence which
> will solve that particular instance. [eg, Kociemba]
> Class 4: A program which attempts to discover an ALGORITHM to
> solve ALL randomized cubes. [...] Korf wrote a
> program to do this in the mid 1980s. [Such programs
> generally produce Class-2-ish solutions.] I believe
> Korf's program is the only program ever achieved that
> can be placed in this category.
I wish to speak to the last sentence of the Class 4 description. Back
in my larval stage (mid-'80s), someone at a lab I worked for build a
Class 4 program in Franz Lisp. It wasn't fast, but that was probably
because it had nothing more than a VAX-11/780 to run on. (I remember
it particularly as it was one of the most impressive pieces of hot-spot
optimization I ever did; replacing about 20 lines of Lisp with about 20
lines of assembly got a speedup of between two and three orders of
magnitude overall.)
I have no idea whether the program still exists in any form. I do
believe I can still reach its author, if anyone would like me to
inquire.
der Mouse
mouse@rodents.montreal.qc.ca
7D C8 61 52 5D E7 2D 39 4E F1 31 3E E8 B3 27 4B
From cube-lovers-errors@mc.lcs.mit.edu Fri Oct 31 21:19:56 1997
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Mail-from: From mouse@Rodents.Montreal.QC.CA Thu Oct 30 19:28:57 1997
Date: Thu, 30 Oct 1997 19:28:16 -0500 (EST)
From: der Mouse
Message-Id: <199710310028.TAA11074@Twig.Rodents.Montreal.QC.CA>
To: cube-lovers@ai.mit.edu
Subject: Re: 5x5x5 Stuctural Integrtity
> Where are you able to find 5x5x5 cubes that don't instantly fall
> apart?
I've owned only one 5-Cube and have had no mechanical problem with it
at all. I bought it in mid-December 1993, but unfortunately I don't
know where it came from. I probably got it at a retail toy/game store
here in the city called Valet de Coeur ("Jack of Hearts" in French),
but (a) am not sure of even that by now (though I have trouble
imagining where else might have had it) and (b) I have no idea where it
was made or what distributor they got it from.
> The orange stickers seem to have a habit of fleeing the cube in
> terror. (It's always the orange ones on any cube that fall off
> first. Has anyone else noticed this?)
I sure have, with my 5-Cube. Three of the 25 have come off, and one
has been completely lost (the other two are attached to the cube with a
piece of masking tape, pending my doing something more permanent). I
may do to it what I did to one of my 3-Cubes recently: take all the
stickers off and use plastic-model paint to color the cubies. (I
actually may do this to just the orange face, since that's the only
problematic one.)
der Mouse
mouse@rodents.montreal.qc.ca
7D C8 61 52 5D E7 2D 39 4E F1 31 3E E8 B3 27 4B
From cube-lovers-errors@mc.lcs.mit.edu Fri Oct 31 22:06:26 1997
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Mail-from: From jferro@knave.ece.cmu.edu Fri Oct 31 13:50:54 1997
Date: Fri, 31 Oct 1997 13:50:10 -0500
From: "Jonathan R. Ferro"
Message-Id: <199710311850.NAA26736@knave.ece.cmu.edu>
Organization: Electrical and Computer Engineering, CMU
To: cube-lovers@ai.mit.edu
In-Reply-To: <01IPDKFAISLE90NIU9@cnsvax.albany.edu> (message from Mary Osielski on Wed, 29 Oct 1997 11:02:53 -0500 (EST))
Subject: Re: Where to buy one???
"Mary" == Mary Osielski writes:
Mary> I'm trying to buy a regular, standard, run-of-the-mill Rubik's
Mary> cube which I now realize is not so easy. Can you please direct me
Mary> to a source? Are they no longer produced?
There has been a new run (I'm not sure if it's by Ideal or not), and I
saw two on the shelf under the Lego Brand Construction Blocks (tm) (Note
to self: kill the lawyers) at K-Mart just last week.
-- Jon
From cube-lovers-errors@mc.lcs.mit.edu Sat Nov 1 22:17:34 1997
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Mail-from: From SCHMIDTG@iccgcc.cle.ab.com Sat Nov 1 20:01:52 1997
From: SCHMIDTG@iccgcc.cle.ab.com
Date: Sat, 1 Nov 1997 20:01:05 -0500 (EST)
To: cube-lovers@ai.mit.edu
Message-Id: <971101200105.20201302@iccgcc.cle.ab.com>
Subject: Re: Categorization of cube solving programs
"der Mouse" wrote:
>This is a response to a pretty old message:
>> Since I'm interested in such things, I came up with the following
>> categories of cube solving programs in general order of increasing
>> sophistication:
>[...Class 1 through Class 2...]
> Class 3: A program that when given a specific instance of the
> cube, attempts to 'discover' or learn a sequence which
> will solve that particular instance. [eg, Kociemba]
> Class 4: A program which attempts to discover an ALGORITHM to
> solve ALL randomized cubes. [...] Korf wrote a
> program to do this in the mid 1980s. [Such programs
> generally produce Class-2-ish solutions.] I believe
> Korf's program is the only program ever achieved that
> can be placed in this category.
In retrospect, Class 4 programs are not necessarily more sophisticated
than Class 3 programs especially when one considers that the latter
should be be able to produce a macro-table solution by solving for
a sufficient set of specific sequences. Perhaps, I'm overly fascinated
by a learning program which, in essence, outputs a solving program but
I don't want to discount the fact that there are some very interesting
and sophisticated Class 3 programs out there.
Richard Korf points out a suggestion by Jon Bently that the learning
program can be be interleaved with the solving program, as co-routines,
and only running the learning program when a new macro is needed to
solve a particular problem instance. Thus, the specific entries
required in the macro-table do not have to be planned out in advance.
>I wish to speak to the last sentence of the Class 4 description. Back
>in my larval stage (mid-'80s), someone at a lab I worked for build a
>Class 4 program in Franz Lisp. It wasn't fast, but that was probably
>because it had nothing more than a VAX-11/780 to run on. (I remember
>it particularly as it was one of the most impressive pieces of hot-spot
>optimization I ever did; replacing about 20 lines of Lisp with about 20
>lines of assembly got a speedup of between two and three orders of
>magnitude overall.)
>I have no idea whether the program still exists in any form. I do
>believe I can still reach its author, if anyone would like me to
>inquire.
It would be interesting to compare the approach of this program to
Korf's learning program. If the program is still available I suggest
it would make a quite excellent addition to the cube lovers archive.
Regards,
-- Greg Schmidt
From cube-lovers-errors@mc.lcs.mit.edu Mon Nov 3 12:42:36 1997
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Mail-from: From mouse@Rodents.Montreal.QC.CA Sun Nov 2 06:52:25 1997
Date: Sun, 2 Nov 1997 06:51:36 -0500 (EST)
From: der Mouse
Message-Id: <199711021151.GAA27954@Twig.Rodents.Montreal.QC.CA>
To: cube-lovers@ai.mit.edu
Subject: Re: Categorization of cube solving programs
>> Class 3: A program that when given a specific instance of the
>> cube, attempts to [solve it] [eg, Kociemba]
>> Class 4: A program which attempts to [find an algorithm to solve
>> arbitrary cubes].
> In retrospect, Class 4 programs are not necessarily more
> sophisticated than Class 3 programs especially when one considers
> that the latter should be be able to produce a macro-table solution
> by solving for a sufficient set of specific sequences.
Sure...but who picks the specific instances for them?
> Richard Korf points out a suggestion by Jon Bently that the learning
> program can be be interleaved with the solving program, as
> co-routines, and only running the learning program when a new macro
> is needed to solve a particular problem instance.
This means that the solving program has to imagine macros, try to
choose a useful one, determine whether it's actually possible (you
gotta keep the program from trying to produce, for example, a single
edge flipper). You also have to decide when it's worth trying for a
macro and when it's better to just hit the (sub)problem with brute
force. I would expect all these problems to be quite hard.
>> I wish to speak to the last sentence of the Class 4 description.
>> Back in my larval stage (mid-'80s), someone at a lab I worked for
>> build a Class 4 program in Franz Lisp. [...]
>> I have no idea whether the program still exists in any form. I do
>> believe I can still reach its author, if anyone would like me to
>> inquire.
> It would be interesting to compare the approach of this program to
> Korf's learning program. If the program is still available I suggest
> it would make a quite excellent addition to the cube lovers archive.
I'll send off a missive to the author.
der Mouse
mouse@rodents.montreal.qc.ca
7D C8 61 52 5D E7 2D 39 4E F1 31 3E E8 B3 27 4B
From cube-lovers-errors@mc.lcs.mit.edu Mon Nov 3 13:18:15 1997
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Mail-from: From nbodley@tiac.net Sun Nov 2 20:32:15 1997
Date: Sun, 2 Nov 1997 20:31:16 -0500 (EST)
From: Nicholas Bodley
To: der Mouse
Cc: cube-lovers@ai.mit.edu
Subject: 5^3 orange stickers
In-Reply-To: <199710310028.TAA11074@Twig.Rodents.Montreal.QC.CA>
Message-Id:
I think these might be made of plastic instead of paper, and they seem to
have a different adhesive. I thought mine were loose because I had tried
several lubricants on my cube, and the lube. had interacted with the
adhesive; apparently not. Someone who's smart with solvents might be able
to remove all the adhesive, and reattach them with a better adhesive.
CA ("Krazy Glue"; cyanoacrylate) might be good, as might plastic-model
cement. However, one should be careful; a 5^3 is not something to
mistreat!
My regards to all,
|* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath
|* Waltham, Mass. *|* -----------------------------------------------
|* nbodley@tiac.net *|* Waltham is now in the new 781 area code.
|* Amateur musician *|* 617 will be recognized until 1 Dec. 1997.
--------------------------------------------------------------------------
From cube-lovers-errors@mc.lcs.mit.edu Mon Nov 3 14:03:20 1997
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Mail-from: From bmain@caddscan.com Mon Nov 3 10:09:08 1997
To: cube-lovers@ai.mit.edu
From: "Bryan Main"
Subject: Re: Where to buy one???
Date: Mon, 03 Nov 1997 10:07:11 EST
Message-Id: <19971103100711.0054df7a.in@caddscan.com>
At 01:50 PM 10/31/97 -0500, you wrote:
>"Mary" == Mary Osielski writes:
>Mary> I'm trying to buy a regular, standard, run-of-the-mill Rubik's
>Mary> cube which I now realize is not so easy. Can you please direct me
>Mary> to a source? Are they no longer produced?
>
>There has been a new run (I'm not sure if it's by Ideal or not), and I
>saw two on the shelf under the Lego Brand Construction Blocks (tm) (Note
>to self: kill the lawyers) at K-Mart just last week.
The new ones, at least the ones that I've gotten in the last year or less,
are made by Oddz-on (sp?). I think that they still make them but I haven't
looked in a few months. I called them a few months ago to see if they had
plans to make a 4x4x4 but they said no. Also they did make 2x2x2's for
awhile but I don't think they do anymore, plus the 2's were hard to rotate
and fell apart eaisly.
bryan
__________________________________________________________________
Bryan Main
Cartographic Specialist
http://caddscan.com
CADDScan Engineering Inc.
NOAA Site Number: 301-713-0388 X 110
From cube-lovers-errors@mc.lcs.mit.edu Mon Nov 3 19:04:36 1997
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Date: Mon, 3 Nov 1997 13:48:19 -0700 (MST)
From: cube-lovers-request@ai.mit.edu
To: cube-lovers@ai.mit.edu
Reply-To: Paul Hart
Subject: Auction on Rubik's Revenge (4x4x4) cubes
Paul Hart has announced he has 6 unopened Rubik's
Revenge cubes for sale to the highest bidder. The Cube-lovers list
will not include details of the offer; I am passing this information
on only because a number of persons on this list have asked about
finding Rubik's Revenge cubes, apparently without success. Contact
hart@iserver.com for any further information. As always, beware of
fraud.
Dan Hoey, Interim moderator
Cube-Lovers-Request@AI.MIT.Edu
From cube-lovers-errors@mc.lcs.mit.edu Mon Nov 3 19:40:01 1997
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Mail-from: From dzander@solaria.sol.net Mon Nov 3 18:52:31 1997
From: Douglas Zander
Message-Id: <199711032351.RAA14876@solaria.sol.net>
Subject: Re: Where to buy one???
To: bmain@caddscan.com (Bryan Main)
Date: Mon, 3 Nov 97 17:51:42 CST
Cc: cube-lovers@ai.mit.edu (cube)
In-Reply-To: <19971103100711.0054df7a.in@caddscan.com> from "Bryan Main" at Nov 3, 97 10:07:11 am
Can you comment how good the new cubes from Oddz-on rotate? Are they
smooth and slick like the original Rubik's Cubes were or hard to turn like
the knock-offs were?
--
Douglas Zander |
dzander@solaria.sol.net |
Milwaukee, Wisconsin, USA |
From cube-lovers-errors@mc.lcs.mit.edu Tue Nov 4 14:24:17 1997
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Mail-from: From SCHMIDTG@iccgcc.cle.ab.com Tue Nov 4 01:38:07 1997
From: SCHMIDTG@iccgcc.cle.ab.com
Date: Tue, 4 Nov 1997 1:37:40 -0500 (EST)
To: cube-lovers@ai.mit.edu
Message-Id: <971104013740.202034d6@iccgcc.cle.ab.com>
Subject: Re: Categorization of cube solving programs
"der Mouse" wrote:
>>> Class 3: A program that when given a specific instance of the
>>> cube, attempts to [solve it] [eg, Kociemba]
>>> Class 4: A program which attempts to [find an algorithm to solve
>>> arbitrary cubes].
>
>> In retrospect, Class 4 programs are not necessarily more
>> sophisticated than Class 3 programs especially when one considers
>> that the latter should be be able to produce a macro-table solution
>> by solving for a sufficient set of specific sequences.
>
>Sure...but who picks the specific instances for them?
See below...
>> Richard Korf points out a suggestion by Jon Bently that the learning
>> program can be be interleaved with the solving program, as
>> co-routines, and only running the learning program when a new macro
>> is needed to solve a particular problem instance.
>
>This means that the solving program has to imagine macros, try to
>choose a useful one, determine whether it's actually possible (you
>gotta keep the program from trying to produce, for example, a single
>edge flipper). You also have to decide when it's worth trying for a
>macro and when it's better to just hit the (sub)problem with brute
>force. I would expect all these problems to be quite hard.
Although I haven't verified this with Richard Korf, I think there
is a very simple approach to this. Consider each cubie to have one
of two states, either "fixed" or "don't care". Initially, all cubies
are in the "don't care" state. If a cubie state is "don't care" then
that means we disregard it's position (i.e. location and orientation)
in the target state for a particular macro.
Number all 20 of the corner and edge cubies. Now perform the following
"Pidgin C" algorithm:
Mark all cubies[1 through 20] as "don't care" in current_cube_state
for (i = 1 to 20)
{
target_state = cubies 1 through i in proper home cubicle position
and marked as "fixed", all other cubies are in a "don't care" state
Construct a unique macro index =
f(IN = current_cubie_position[i], IN = desired_cubie_position[i])
if (the macro at "index" doesn't exist)
{
Class_3_Solve(IN = current_cube_state, IN = target_cube_state, OUT = macro)
add the new "macro" to the macro table at "index"
}
Apply the macro to our current_cube_state
Mark cubie[i] as "fixed" in current_cube_state
}
Note: Class_3_solve must be able to accept an initial and goal state
augmented with the "fixed" and "don't care" markings and should honor
the constraints implied by them. To put it another way, if a cubicle
is marked as "don't care" then a valid target state allows this
cubie to be placed in any other cubicle not currently occupied by a
"fixed" cubie. Not really a big deal for any search procedure as we
are simply relaxing the goal state condition to a partial match rather
than requiring an exact match.
So we start out by solving for one cubie only and ignore the effect
this has on the remaining 19 cubies. We continue doing this, each
time successively fixing another cubie and ignoring the rest, until
all cubies are finally in place. For any valid cube configuration,
we are always guaranteed to find a macro that can solve this subproblem.
Actually, we will never fully iterate to all 20 cubies since it
is impossible to move just a single cubie. For example, the very last
subproblem for cubie #19 might be an edge flip. Once we've discovered
and applied the appropriate macro for this particular edge flip
we will have also flipped the #20 cubie and placed the cube in its
solved configuration.
Initially, the macros are very easy to find since most cubies
can be relocated. At the very end, we can only move very few
cubies, and the macros are more difficult. But a class 3 program
can solve any cube and thus can find even the most difficult
macros (e.g. an edge flip).
Eventually, once we've solved enough cube instances, our macro table
will be complete and all future cubes can be solved via macro table
lookup without the aid of the solving portion of the program.
Regards,
-- Greg
From cube-lovers-errors@mc.lcs.mit.edu Thu Nov 6 18:53:55 1997
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Mail-from: From bmain@caddscan.com Thu Nov 6 10:15:24 1997
To: Richard E Korf
From: "Bryan Main"
Subject: Re: Where to buy one???
Cc: cube-lovers@ai.mit.edu
Date: Thu, 06 Nov 1997 10:13:37 Eastern Standard Time
Message-Id: <19971106101337.000d3578.in@caddscan.com>
At 11:26 AM 11/5/97 -0800, you wrote:
>Douglas,
> I bought an Oddz-on Cube the other day, and although I don't have a
> large basis for comparison, it seems to work pretty well.
> -rich
This got sent to me and I think that it was for the list so I'm forwarding
it. On the same note I have three of these cubes and they work well but the
stickers become old fast. They begin to come off around the edges and the
protective cover sometimes comes off.
__________________________________________________________________
Bryan Main
Cartographic Specialist
http://caddscan.com
From cube-lovers-errors@mc.lcs.mit.edu Thu Nov 6 19:32:59 1997
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Mail-from: From tenie1@juno.com Thu Nov 6 18:13:12 1997
To: Cube-Lovers@ai.mit.edu
Subject: Better way to flip a middle edge?
Message-Id: <19971106.151149.11046.0.tenie1@juno.com>
From: tenie1@juno.com (Tenie Remmel)
Date: Thu, 06 Nov 1997 18:10:26 EST
Is there a short way to flip a middle edge cubie without disturbing the
top layer or the other middle edges? I mean, better than replacing it
with one from the bottom and then putting it back in the correct
orientation, which takes 15 moves.
--Tenie Remmel (tenie1@juno.com)
From cube-lovers-errors@mc.lcs.mit.edu Fri Nov 7 13:52:09 1997
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Mail-from: From cubeman@idirect.com Thu Nov 6 23:15:29 1997
Message-Id: <34629664.2B5E@idirect.com>
Date: Thu, 06 Nov 1997 23:17:40 -0500
From: Mark Longridge
To: cube lovers
Subject: Availablility of Rubik's Cube
I've followed the thread about the availibility of Rubik's Cubes.
Ideal Toy is once again manufacturing Rubik's Cube for the mass market.
New packaging (with the correct number of permutations) has been
purchased by myself in a Canadian Toys R Us store just recently.
I know nothing of the Oddz-On cubes, but the new Ideal Toy cubes are
wonderful. The new cubes sport a new logo and brighter colours, but
they use the same colour arrangement as the Ideal Cubes of old.
There is also an official site for Rubik's Cube at http://www.rubiks.com
Unfortunately they are not answering their mail and are attracting a
mostly younger crowd. They are also using Karl Hornell's rubik's cube
java applet (sporting the incorrect colour arrangement I might add)
without giving any mention of Karl's name. It is an exact byte for
byte copy. Although Karl does give out the Rubik's Cube java applet
as freeware, I think he deserves credit from the Ideal web site.
As for my own web site (which does sport Karl's java applet with the
correct standard colouration, and also BEFUDDLER support!) I intend
to record the entire chronology of all the cube contests from every
country, including all the records from the World Championships.
My Rubik's Cube web page is currently http://web.idirect.com/~cubeman
If anyone has any information about the cube contests I have missed,
please email me. Thanks!
-> Mark Longridge <-
The Cubeman of the Internet
From cube-lovers-errors@mc.lcs.mit.edu Fri Nov 7 14:22:06 1997
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Mail-from: From C.McCaig@Queens-Belfast.AC.UK Fri Nov 7 05:33:32 1997
From: C.McCaig@queens-belfast.ac.uk
Date: Fri, 07 Nov 1997 10:28:08 GMT
To: cube-lovers@ai.mit.edu
Message-Id: <009BCEF0.4DC8B4D9.41@a1.qub.ac.uk>
Subject: Re: Where to buy one???
i've noticed that here in northern ireland, there are a couple of
places selling cubes. one is a standard copy of the rubik's cube,
and the other is called "magic cube" which has holographic stickers
on it, and the cubies are much squarer making it very difficult to
take apart.. it comes with a locking key which allows you to remove
one of the faces.. the turning mechanism is _really_ loose, too
loose in fact, but mine hasnt fallen apart.
as an aside, i have an original cube, that my grandmother bought me
16 or 17 years ago, and it's still got all it's original stickers!
clive
---
Clive McCaig
Dept. Applied Mathematics
Queens University Belfast
Northern Ireland
From cube-lovers-errors@mc.lcs.mit.edu Fri Nov 7 14:46:43 1997
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Mail-from: From davidbarr@iname.com Fri Nov 7 02:09:32 1997
Sender: davidb@davidb.concentric.net
Message-Id: <3462BE64.3F20493A@iname.com>
Date: Thu, 06 Nov 1997 23:08:20 -0800
From: David Barr
Organization: Medweb
To: Tenie Remmel , Cube-Lovers@ai.mit.edu
Subject: Re: Better way to flip a middle edge?
References: <19971106.151149.11046.0.tenie1@juno.com>
Tenie Remmel wrote:
> Is there a short way to flip a middle edge cubie without disturbing the
> top layer or the other middle edges? I mean, better than replacing it
> with one from the bottom and then putting it back in the correct
> orientation, which takes 15 moves.
>
> --Tenie Remmel (tenie1@juno.com)
Take a look at http://ssie.binghamton.edu/~jirif/Mike/middle.html. I
think this is the sequence you want:
2) R2 D2 F' R2 F D2 R D' R
This sequence flips the cubie on the front-right edge without disturbing
the upper face or the other middle edges.
--
mailto:davidbarr@iname.com
http://www.concentric.net/~Davebarr/
From cube-lovers-errors@mc.lcs.mit.edu Fri Nov 7 18:43:53 1997
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Mail-from: From Michael.Swart@switchview.com Fri Nov 7 09:33:48 1997
Message-Id: <199711071431.JAA05030@support.switchview.com>
From: "Michael Swart"
To: , "Tenie Remmel"
Subject: Re: Better way to flip a middle edge?
Date: Fri, 7 Nov 1997 09:26:36 -0500
> Is there a short way to flip a middle edge cubie without disturbing the
> top layer or the other middle edges? I mean, better than replacing it
> with one from the bottom and then putting it back in the correct
> orientation, which takes 15 moves.
F' L' R F L' R T' B2 T L R' F L R' D2 F'
18 q turns 16 h turns. Guess it wasn't any better but you may notice
that this sequence also leaves the bottom intact except for one flipped
cubie at the back down edge.
Michael Swart
From cube-lovers-errors@mc.lcs.mit.edu Sun Nov 9 14:40:21 1997
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Mail-from: From charlied@erols.com Sun Nov 9 01:06:22 1997
Message-Id:
Date: Sun, 9 Nov 1997 01:05:27 -0500
To: Cube-Lovers@ai.mit.edu
From: Charlie Dickman
Subject: A 4 Dimensional Rubik's Cube
About 18 months ago I sent this group an email about a Mac based program I
have written that simulates a 4 dimensional (3x3x3x3) Rubik's Cube based on
an unpublished paper by Harry Kamack and Tom Keene.
Some of you who were interested in the paper that describes the model and
the program had difficulty with the copies I sent you and, I suspect, were
unable to read it after you received it. Someone suggested that I translate
the document into HTML and this email is to let you know that I have done
that and will send either a ZIP or STUFFIT archive of the document to
anyone interested.
I know that maybe I should get a web site and put the paper there but I'm
not up for designing a web page or maintaining it. If you would like a copy
of the document and would also like to put it on your web site, let me know
that too.
The HTML version of the document consists of 36 fairly small GIFs that
illustrate the words. The STUFFIT archive is 328K and the ZIP file is 320K.
The documentation for the Mac based ZIP program claims that the file can be
successfully unZIPped on non-Mac platforms. The STUFFIT archive is
self-extracting if you're Mac enabled.
Send me an email if you are interested in either the program or the HTML
document or both. If you just want the document, tell me which format you
want. If you ask for the program I will assume you have a Mac and will send
everything in a STUFFIT sea.
Regards to all...
Charlie Dickman
charlied@erols.com
From cube-lovers-errors@mc.lcs.mit.edu Sun Nov 9 15:59:22 1997
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Mail-from: From whuang@ugcs.caltech.edu Sun Nov 9 12:26:11 1997
To: cube-lovers@ai.mit.edu
From: whuang@ugcs.caltech.edu (Wei-Hwa Huang)
Subject: Re: Where to buy one???
Date: 9 Nov 1997 17:25:11 GMT
Organization: California Institute of Technology, Pasadena
Message-Id: <644rln$45r@gap.cco.caltech.edu>
References:
C.McCaig@queens-belfast.ac.uk writes:
>and the other is called "magic cube" which has holographic stickers
>on it, and the cubies are much squarer making it very difficult to
>take apart.. it comes with a locking key which allows you to remove
>one of the faces.. the turning mechanism is _really_ loose, too
>loose in fact, but mine hasnt fallen apart.
Ah... this is a recent Taiwanese invention.
--
Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/
---------------------------------------------------------------------------
"[Lucy's eyes] look like little round dots of India ink..." -- Charlie Brown
From cube-lovers-errors@mc.lcs.mit.edu Mon Nov 10 10:58:12 1997
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Mail-from: From whuang@ugcs.caltech.edu Sun Nov 9 12:36:08 1997
To: cube-lovers@ai.mit.edu
From: whuang@ugcs.caltech.edu (Wei-Hwa Huang)
Subject: Re: Better way to flip a middle edge?
Date: 9 Nov 1997 17:35:21 GMT
Organization: California Institute of Technology, Pasadena
Message-Id: <644s8p$4er@gap.cco.caltech.edu>
References:
"Michael Swart" writes:
>> Is there a short way to flip a middle edge cubie without disturbing the
>> top layer or the other middle edges? I mean, better than replacing it
>> with one from the bottom and then putting it back in the correct
>> orientation, which takes 15 moves.
>F' L' R F L' R T' B2 T L R' F L R' D2 F'
>18 q turns 16 h turns. Guess it wasn't any better but you may notice
>that this sequence also leaves the bottom intact except for one flipped
>cubie at the back down edge.
This can be slightly improved:
D R' F D' R' L B D' R B' D R L' F'
14 quarter turns; does exactly the same thing.
--
Wei-Hwa Huang, whuang@ugcs.caltech.edu, http://www.ugcs.caltech.edu/~whuang/
---------------------------------------------------------------------------
"[Lucy's eyes] look like little round dots of India ink..." -- Charlie Brown
From cube-lovers-errors@mc.lcs.mit.edu Mon Nov 10 11:32:49 1997
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Mail-from: From cubeman@idirect.com Sun Nov 9 23:05:52 1997
Message-Id: <34668899.248A@idirect.com>
Date: Sun, 09 Nov 1997 23:07:53 -0500
From: Mark Longridge
To: cube lovers
Cc: joyner.david@mathnt1.sma.usna.navy.mil
Subject: Megaminx, the 10-spot and GAP
First of all, the STANDARD colour arrangement used by Ideal Toy is as
follows:
UP = White
DOWN = Blue
FRONT = Yellow
BACK = Green
LEFT = Red
RIGHT = Orange
All the official Ideal Toy 2x2x2, 3x3x3 & 4x4x4 cubes used this
arrangement.
Even my 5x5x5 cube is the same.
Secondly, I have at last resolved the 10-spot pattern for the megaminx
in GAP. I created the process m1a which is the sequence of operators
to generate the 10-spot. I had no C_U operator, so it was more
difficult than I thought it would be.
To see all the gory details surf to the following URLs
(These are all GAP text files)
http://web.idirect.com/~cubeman/dodeca.txt describes the megaminx
http://web.idirect.com/~cubeman/megaop.txt describes operators
http://web.idirect.com/~cubeman/spot.txt generates the 10-spot
Note that after executing spot.txt (which loads the other necessary
files) in gives the order of process m1a correctly as 5.
This generator uses all of the megaminx operators except the top and
bottom faces, so it is a pretty good test of the correctness of the
all of dodeca.txt, megaop.txt, and spot.txt
I believe this is the first simulation of the megaminx generating
the 10-spot although Dr. David Joyner is very close! His work is
more graphically interesting (using Maple to generate 3d pics
of the megaminx) but his operators to rotate the whole megaminx
are cooked. However, we have both verified that processes
m2, m3 and m3a are correct and have been graphed correctly using
Maple.
-> Mark <-
From cube-lovers-errors@mc.lcs.mit.edu Tue Nov 11 20:07:56 1997
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Mail-from: From tim@mail.htp.net Tue Nov 11 00:07:26 1997
From: tim@mail.htp.net (Tim Mirabile)
To: cube lovers
Subject: Re: Megaminx, the 10-spot and GAP
Date: Tue, 11 Nov 1997 05:06:29 GMT
Organization: http://www.webcom.com/timm/
Message-Id: <3467e58b.881450@mail.htp.net>
References: <34668899.248A@idirect.com>
In-Reply-To: <34668899.248A@idirect.com>
On Sun, 09 Nov 1997 23:07:53 -0500, Mark Longridge wrote:
>First of all, the STANDARD colour arrangement used by Ideal Toy is as
>follows:
>
>UP = White
>DOWN = Blue
>FRONT = Yellow
>BACK = Green
>LEFT = Red
>RIGHT = Orange
>...
I remember having one if the early Ideal cubes (at least I think it
was), and green was opposite blue.
Recently I bought one of those "odds-on" cubes, and within a week I
wore the plastic coating off the faces, so I decided to peel all the
stickers off and paint it using model paint. I decided to keep the
most similar colors opposite each other (blue-green, red-orange,
yellow-white). I find this arrangement makes things easier when
cubing under dim lighting. :)
--
Long Island chess -> http://www.webcom.com/timm/ TimM on ICC and A-FICS
The opinions of my employers are not necessarily mine and vice versa.
From cube-lovers-errors@mc.lcs.mit.edu Tue Nov 11 20:43:51 1997
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Mail-from: From Anders.Larsson@hvi.uu.se Tue Nov 11 02:25:24 1997
Message-Id: <346807C1.ACFE4D68@hvi.uu.se>
Date: Tue, 11 Nov 1997 08:22:41 +0100
From: Anders Larsson
To: cube lovers
Subject: Colour arrangements (Was: Re: Megaminx, the 10-spot and GAP)
References: <34668899.248A@idirect.com>
Mark Longridge wrote:
> First of all, the STANDARD colour arrangement used by Ideal Toy is as
> follows:
>
> UP = White
> DOWN = Blue
> FRONT = Yellow
> BACK = Green
> LEFT = Red
> RIGHT = Orange
In front of me I hold a cube from one of the first batches from Hungary
with the following colour arrangement:
Up = white
Down = yellow
Front = blue
Back = green
Left = red
Right = orange
Does anybody know the history why this colour arrangement was changed?
BTW: Even if Ideal Toys has their own local standard, it doesn't change
the original ("correct") colour arrangement.
/Anders
--
Anders Larsson, PhD
Institute of High Voltage Research Tel.: +46 (0)18 532702
Uppsala University Fax.: +46 (0)18 502619
Husbyborg E-mail: Anders.Larsson@hvi.uu.se
S-752 28 Uppsala, Sweden
http://www.hvi.uu.se/IFH/staff/Anders/Anders.html
From cube-lovers-errors@mc.lcs.mit.edu Wed Nov 12 21:40:03 1997
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Mail-from: From ck1@home.com Tue Nov 11 22:26:29 1997
From: "Chris and Kori Pelley"
To:
Subject: Colors and other variations between brands
Date: Tue, 11 Nov 1997 22:26:17 -0500
Message-Id: <01bcef1a$bcbe2f60$da460318@CC623255-A.srst1.fl.home.com>
Most of the early "clone" cubes had the Blue/Green arrangement instead of
the Blue/White. Most Ideal cubes seemed to have the Blue/White. There were
exceptions, though...
I remember there were several factories where Ideal had their cubes made.
Some factories were better than others in terms of their quality. My
favorites were the ones that said "Made in Korea" on a little peel-off gold
sticker. Back in those days I would refer to "my Korean cube." Believe it
or not, these all had the Blue/Green arrangement but they were genuine Ideal
cubes! Their cubes were also the smoothest. I still have one of them that
is in near perfect condition. It was the cube I used in the competitions.
The other factories included Japan and Hong Kong. The Japanese cubes seemed
more prevalent and I still have at least three of those-- all featuring the
Blue/White arrangement.
The earliest Rubik's Cube I ever saw had strange colors-- grey instead of
white and the shades of green and blue were very different from later cubes.
I don't think it was an Ideal cube.
The Blue/White arrangement definitely won out as Ideal's "standard"
arrangement since their 4x4x4 Revenge and 2x2x2 Pocket Cubes featured the
identical coloring. Some Ideal 3x3x3 cubes were Blue/White but
"non-standard" because the Yellow/Green would be reversed (mirror image).
Who knows why these variations existed-- probably something as simple as
some factory tech switching the sticker feeds accidentally?
The new "Rubik's Cubes" made by Oddz-On are not all that great, in my
opinion. They look shiny and great in the box, but after mild use the
stickers get ruined. The Square-1 puzzles suffer the same fate. Also their
turning mechanism is nowhere near the quality of the "Korean cubes." Their
2x2x2 "Mini-Cube" as it is now called also lacks in quality compared to the
old Ideal Pocket Cubes. Still, it warms my heart to see them back in toy
stores again!
Much better are the "Magic Cube" clones that appeared last year. I have
purchased several of these (only $3.99 at Walgreen's!) and they turn very
smoothly. The holographic stickers are different, but they don't wear out
like the Oddz-On cubes. Also, mine feature the Blue/White arrangement!
I recently saw a post that Ideal is now making cubes again. This seems
strange since I thought they went out of business, but I could be wrong.
Anybody know the real scoop?
Finally, Square-1 seems to have made a reappearance. I thought they only
made one batch of these, but maybe they've made another lately?
Chris Pelley
ck1@home.com
http://members.home.net/ck1
From cube-lovers-errors@mc.lcs.mit.edu Wed Nov 12 22:10:31 1997
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Mail-from: From chrono@ibm.net Wed Nov 12 00:37:48 1997
Message-Id: <34694088.5E3AC959@ibm.net>
Date: Tue, 11 Nov 1997 21:37:12 -0800
From: "Jin 'Time Traveler' Kim"
Organization: The Fourth Dimension
To: cube lovers
Subject: Rubik's Cube Color arrangements
References: <34668899.248A@idirect.com> <346807C1.ACFE4D68@hvi.uu.se>
I have serveral cubes spanning over a decade and a half in front of me.
Here's the quick color arrangements:
Rubik's Cube (circa 1982)
Up = White
Down = Yellow
Front = Blue
Back = Green
Left = Orange
Right = Red
I've owned this cube for what feels like forever. I'm not even sure how I
even found it again, because I dug it out of some old junk after not having
a puzzle for about 5 years. This is slightly different from the one you
described as being from Hungary. I suspect mine is from there too, so
maybe production values weren't as high as they could be.
Rubik's Cube "4th Dimension" (Golden Toys, circa 1988) - Poor quality in my
opinion, as the stickers are paper with a clear plastic laminate, but
despite only being taken out of the box only 4 Times ever, the plastic
laminate is already peeling in spots.
Rubik's Mini Cube (OddzOn, circa 1996)
Rubik's Cube (OddzOn, circa 1997)
All of the above have the same color arrangement as what you described
below as being the "Ideal" solution, which I believe isn't the best. I
still think that the opposite pairing of red/orange, white/yellow, and
blue/green makes for the best balanced color combination. Not to mention
it's also the best quality with plastic stickers instead of paper.
Anders Larsson wrote:
> Mark Longridge wrote:
>
> > First of all, the STANDARD colour arrangement used by Ideal Toy is as
> > follows:
> >
> > UP = White
> > DOWN = Blue
> > FRONT = Yellow
> > BACK = Green
> > LEFT = Red
> > RIGHT = Orange
>
> In front of me I hold a cube from one of the first batches from Hungary
> with the following colour arrangement:
>
> Up = white
> Down = yellow
> Front = blue
> Back = green
> Left = red
> Right = orange
>
> Does anybody know the history why this colour arrangement was changed?
>
> BTW: Even if Ideal Toys has their own local standard, it doesn't change
> the original ("correct") colour arrangement.
--
Jin "Time Traveler" Kim
chrono@ibm.net
VGL Costa Mesa
http://www.geocities.com/timessquare/alley/9895
http://www.slamsite.com
From cube-lovers-errors@mc.lcs.mit.edu Wed Nov 12 22:54:25 1997
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Mail-from: From richard_morton@icom-solutions.com Wed Nov 12 04:56:12 1997
Date: Wed, 12 Nov 1997 17:31:37 GMT
From: David Singmaster
To: Anders.Larsson@hvi.uu.se
Cc: cube-lovers@ai.mit.edu
Message-Id: <009BD319.4B14FD66.61@ice.sbu.ac.uk>
Subject: RE: Colour arrangements (Was: Re: Megaminx, the 10-spot and GAP)
The colour arrangement on the early Hungarian cubes was quite random!!
I even have two examples where two faces have the same colour!! It was not
until about 1980 that the idea of having a standardised colour pattern was
adopted and the most common was to have the opposite faces differ by yellow.
That is the opposite faces were White - Yellow, Blue - Green, Red - Orange.
Rubik went to some effort to select six colours that would be maximally
distinct, but I think the yellow, red and orange tended to be too close in the
sense that either the orange was too close to the red or too close to the
yellow!
However, this does not completely determine the colour pattern. Just
as with a die, there are two possible arrangements. Conway and Guy etc.
observed that Blue, Orange and Yellow meet at a corner and they can occur
clockwise or counterclockwise, spelling BOY or YOB. Some people have
expressly asked me for one form rather than the other!
An early anecdote, from about 1979. A friend's son was trying to help
another friend solve his cube over the telephone. This is a pretty formidable
task at the best of times, but their two cubes had different colour patterns,
so the son was making statements like: turn the red face, that's blue on your
cube, ....
DAVID SINGMASTER, Professor of Mathematics and Metagrobologist
School of Computing, Information Systems and Mathematics
Southbank University, London, SE1 0AA, UK.
Tel: 0171-815 7411; fax: 0171-815 7499;
email: zingmast or David.Singmaster @sbu.ac.uk
From cube-lovers-errors@mc.lcs.mit.edu Thu Nov 13 13:15:59 1997
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Mail-from: From richard_morton@icom-solutions.com Wed Nov 12 04:56:12 1997
Message-Id: <199711120955.EAA01537@life.ai.mit.edu>
Date: Wed, 12 Nov 1997 04:55:28 EST
From: "Richard M Morton"
To: cube-lovers@ai.mit.edu
Subject: Cube Colours
Mark Longridge wrote:
> First of all, the STANDARD colour arrangement used by Ideal Toy is as
> follows:
>
> UP = White
> DOWN = Blue
> FRONT = Yellow
> BACK = Green
> LEFT = Red
> RIGHT = Orange
Is the orientation of the above fixed in some way or is it arbitrary ?
My second cube (can't remember what happened to the first one) is a later
edition (not sure if it is Ideal) with the same arrangement to above
except, the orientation is different (UP is either RED or ORANGE) The
reason I say this is that the LEFT,RIGHT,DOWN and FRONT faces have
symbols printed in the centre as follows :
YELLOW - signature of Erno Rubik
WHITE - Rubik's CUBE tm
GREEN - C*4**4 (actually uses superscript 4 for the power)
BLUE - silhouette (of Erno Rubik)
The symbols are designed to make the cube harder to solve - the challenge
is to solve the cube with the centre cubes all in the correct
orientation. I recall that there are sequences of moves that rotate pairs
of centre cubes.
This cube is definitely a lot stiffer than my original cube but the
novelty of speed cubing has worn off anyway.
Richard Morton
(If my employers views are not necessarily those of my own, why am I
still working here ?)
Icom Solutions
http://www.icom-solutions.com/offprods/default.htm
From cube-lovers-errors@mc.lcs.mit.edu Thu Nov 13 13:57:08 1997
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Mail-from: From Geoffroy.VanLerberghe@ping.be Wed Nov 12 15:37:45 1997
Message-Id: <346A13C7.7C38@ping.be>
Date: Wed, 12 Nov 1997 21:38:32 +0100
From: Geoffroy Van Lerberghe
To: Cube-Lovers
Subject: Cubes in London
In one month I am going to London for a few days and I would like to
know where I can buy brainteasers there (mainly Rubik's cubes and
related puzzles).
Could you help me and send me all the information you have?
In Brussels, Belgium, you can (sometimes) find Magic Dodecahedron,
Pyraminx, Skewb and "555" cube at
Dedale
Galerie du Cinquantenaire
Avenue de Tervuren 32
1040 Brussels
Thank you for your help.
Geoffroy
From cube-lovers-errors@mc.lcs.mit.edu Thu Nov 13 14:26:35 1997
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Mail-from: From tenie1@juno.com Thu Nov 13 13:02:44 1997
To: Cube-Lovers@ai.mit.edu
Date: Thu, 13 Nov 1997 10:01:41 -0800
Subject: 6x6x6 cube design
Message-Id: <19971113.100159.5094.0.tenie1@juno.com>
From: tenie1@juno.com (Tenie Remmel)
I am attempting to design a 6x6x6 cube. My idea to make it structurally
sound is to attach both the center cubies and the middle edge cubies to
a ball in the center. Then all other pieces are wedged behind those. I
think that extending from the 5x5x5 design the same way the 4x4x4 was
extended from the 3x3x3 design would be way too flimsy, mainly because
the centers would have to be attached via long, thin struts which are
apt to break easily unless made out of metal, which would make the thing
way too heavy. The width of the cubies probably could not be more than
14 or 15 mm; if they were larger, the cube would be quite big and so it
would be difficult to manipulate.
Unfortunately the ball would be quite complicated, with six or even nine
tracks in it instead of just three as in the 4x4x4 cube. It might have
to be made of metal instead of plastic (it shouldn't be too heavy if it
is hollow). Also the 152 pieces will be a real pain to put together...
Of course, even if it can be built, does anyone know how to solve it?
Here is a rather crude diagram of a cross section through the center of
the cube. Actually it is just a quarter of a cube.
------------------------------------------------------------------------
aaaaaaaaaaaaaaaaaaaabbbbbbbbbbbbbbbbbbbbcccccccccccccccccccc
aaaaaaaaaaaaaaaaaaaabbbbbbbbbbbbbbbbbbbbcccccccccccccccccccc
aaaaaaaaaaaaaabbbbbbbbbbbbbbbbbbbbbbbbbbcccccccccccccccccccc
aaaa......aaaabbbbbbbbbbbbbbbbbbbbbbbbbbbbbbcccccccccccccccc
aaaa..............bbbbbbbbbbbbbbbbbbbbbbbbbbcccccccccccccccc
aaaaaaaa................bbbbbbbbbbbbbbbbbbbbcccccccccccccccc
aaaaaaaa....................bbbbbbbbbbbbbbbbcccccccccccccccc
................................bbbbbbbbbbcccccccccccccccccc
..................................bbbbbbccccccbbbbbbbbcccccc
....................................bbccccccbbbbbbbbbbcccccc
..............................cccc..ccccccbbbbbbbbbbbbbbbbbb
..............................ccccccccccbbbbbbbbbbbbbbbbbbbb
................................cccccc....bbbbbbbbbbbbbbbbbb
..................................cccccc....bbbbbbbbbbbbbbbb
....................................cccc......bbbbbbbbbbbbbb
..............................................bbbbbbbbbbbbbb
................................................bbbbbbbbbbbb
................................................bbbbbbbbbbbb
..................................................bbbbbbbbbb
..................................................bbbbbbbbbb
..................................................bbbbbbaaaa
....................................................bbbbaaaa
....................................................bbbbaaaa
....................................................aaaaaaaa
....................................................aaaaaaaa
......................................................aaaaaa
..............................................aaaa....aaaaaa
..............................................aaaa....aaaaaa
..............................................aaaaaaaaaaaaaa
..............................................aaaaaaaaaaaaaa
------------------------------------------------------------------------
BTW, Does anyone have experience with TurboCAD? Can it be used to
design this type of thing? It sure would be easier to use a computer
program than to use graph paper.
I believe that the 6x6x6 is the largest mechanically possible, because
with the 7x7x7 and higher cubes, the corner cubies aren't attached to
anything at all! Is this correct?
Also what is the mechanism for a 2x2x2 cube? Could it be extended to
make a more stable 4x4x4 and/or 6x6x6 cubes...
And how about a GigaMinx, a 5x5 version of the MegaMinx magic pentagonal
dodecahedron, with five pieces on each edge, 31 pieces on each face
(5 corners, 11 edges and 11 central pieces), 242 pieces total. I would
draw a diagram if it wasn't so hard to make a pentagon out of chars...
--Tenie Remmel (tenie1@juno.com)
[ Moderator's note: The purported impossibility of a Rubik's 7^3 has been
discussed and refuted repeatedly on this list, and several mechanisms
have been proposed for it; see the archives. It is not true that the
corner cubies "aren't attached to anything". Each corner will be attached
to at least two edge cubies, though not always the same two edge cubies.
You should also look in the archives to find descriptions of the 2^3, some
as recently as 28 July. Unfortunately, I haven't been able to understand
it. I'd like to see a clear description, as I haven't got a 2^3 handy to
try myself. -Dan ]
From cube-lovers-errors@mc.lcs.mit.edu Fri Nov 14 10:37:37 1997
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Mail-from: From cubeman@idirect.com Thu Nov 13 21:03:37 1997
Date: Thu, 13 Nov 1997 19:06:53 -0500 (EST)
From: Mark Longridge
To: Richard M Morton
Cc: cube-lovers@ai.mit.edu
Subject: Re: Cube Colours
In-Reply-To: <199711120955.EAA01537@life.ai.mit.edu>
Message-Id:
On Wed, 12 Nov 1997, Richard M Morton wrote:
> Mark Longridge wrote:
>
> > First of all, the STANDARD colour arrangement used by Ideal Toy is as
> > follows:
> >
> > UP = White
> > DOWN = Blue
> > FRONT = Yellow
> > BACK = Green
> > LEFT = Red
> > RIGHT = Orange
>
> ...
Ok folks, one last bit of info about the cube colour controversy
The colouring "standard" I was referring to was used by Canadian and
the USA cube contests. Having said that there were probably contests
where people brought there own cubes, and that would make it potpourri.
Moreover, this was stipulated in the rules of the contest. I still have
the form. The only difference between differ by yellow and the standard
Ideal cube was the transposition of yellow and blue.
There isn't really a standard orientation, save for the orientation
I use in my own cube programs. All the Ideal cubes I have conform
to White/Blue, Yellow/Green, Red/Orange for Top/Down, Front/BACK,
Left/Right.
So I suppose it is open to interpretation. I thought David Singmaster
might mention what colour arrangement was used in the World Championship.
So a case may be made for both "Differ by Yellow" and Ideal Contest
Colours.
Would someone like to pick one?? :-)
-> Mark <-
The Colourist
From cube-lovers-errors@mc.lcs.mit.edu Fri Nov 14 11:12:41 1997
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Mail-from: From chrono@ibm.net Thu Nov 13 22:03:44 1997
Message-Id: <346BBF7F.ADFDE0D4@ibm.net>
Date: Thu, 13 Nov 1997 19:03:27 -0800
From: "Jin 'Time Traveler' Kim"
Organization: The Fourth Dimension
To: Cube-Lovers@ai.mit.edu
Subject: Re: 6x6x6 cube design
References: <19971113.100159.5094.0.tenie1@juno.com>
Tenie Remmel wrote:
> I am attempting to design a 6x6x6 cube. My idea to make it structurally
> sound is to attach both the center cubies and the middle edge cubies to
> a ball in the center. Then all other pieces are wedged behind those. I
> think that extending from the 5x5x5 design the same way the 4x4x4 was
> extended from the 3x3x3 design would be way too flimsy, mainly because
> the centers would have to be attached via long, thin struts which are
> apt to break easily unless made out of metal, which would make the thing
> way too heavy. The width of the cubies probably could not be more than
> 14 or 15 mm; if they were larger, the cube would be quite big and so it
> would be difficult to manipulate.
> Of course, even if it can be built, does anyone know how to solve it?
If it can be built and scrambled, it can be solved. In fact, it could
make for a very interesting puzzle since it could behave identically
to a 3x3x3 if one wanted it to, just like a 4x4x4 can be manipulated
like a 2x2x2. Heck, the 6x6x6 could also behave like a 2x2x2... One
puzzle could take the place of two others. Sort of a "mix and match"
difficulty setting. Regardless, I suspect that many would applaud the
ingenuity of a 6x6x6 if it was executed elegantly and worked well,
like the 5x5x5.
> I believe that the 6x6x6 is the largest mechanically possible, because
> with the 7x7x7 and higher cubes, the corner cubies aren't attached to
> anything at all! Is this correct?
The moderator of the mailing list stated that a 7x7x7 cube could be
built, but I counter that it would require "cubes" of dissimilar size
or some kind of groove type scheme, which actually isn't quite in the
spirit of a cube. Even a 6x6x6 would require some careful engineering
since the corner cubes just barely overlap.
> Also what is the mechanism for a 2x2x2 cube? Could it be extended to
> make a more stable 4x4x4 and/or 6x6x6 cubes...
The mechanism of the 2x2x2 is similar to the 4x4x4, which makes both
of them rather stiff.
> And how about a GigaMinx, a 5x5 version of the MegaMinx magic pentagonal
> dodecahedron, with five pieces on each edge, 31 pieces on each face
> (5 corners, 11 edges and 11 central pieces), 242 pieces total. I would
> draw a diagram if it wasn't so hard to make a pentagon out of chars...
I'm sure supersets of many existing puzzles have been considered. I
myself spent some hours contemplating and drafting the possibility of
a pyraminx to the next level. I called it Tut's Curse as a sort of
'project' name, despite the fact that Tut was never buried in a
pyramid. Maybe that's why I never completed the project. Oh well.
The best laid plans of mice and men...
--
Jin "Time Traveler" Kim
chrono@ibm.net
VGL Costa Mesa
http://www.geocities.com/timessquare/alley/9895
http://www.slamsite.com
From cube-lovers-errors@mc.lcs.mit.edu Fri Nov 14 11:46:24 1997
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Mail-from: From cube-lovers-request@life.ai.mit.edu Fri Nov 14 10:27:59 1997
Date: Fri, 14 Nov 1997 15:25:06 GMT
From: David Singmaster
To: tenie1@juno.com
Cc: cube-lovers@ai.mit.edu
Message-Id: <009BD499.F2FD74E5.202@ice.sbu.ac.uk>
Subject: RE: 6x6x6 cube design
First, regarding the 6^3 and 7^3. As noted, when you get to
these sizes, the connection of the corners while turning becomes
problematic. For the 6^3, the overlap is about 15% of the edge length
of the cubie, probably too small to be practicable. One can imagine
some clever mechanism to hold onto the corners, but it would be tricky
and I've never seen one clearly described.
However, if you think about it, there's no reason for all the
levels of the cube to be the same size. That it, the parallel cutting
planes of the entire cube do not have to be equally spaced. One can
thus have the corner cubies be very large with much smaller centre
cubies. The edge cubies will be cuboids, rather than cubes. Using
this idea, one can make arbitrarily large cubes, but the interior
pieces become impossible to manipulate.
Now let me try my hand at describing three versions of the 2^3.
I'll start with the simplest which was sent to me from Japan
about 1980. This had a steel sphere in the middle and each cubie had
a magnet in it. Although the sphere and the cubies were carefully
machined, when one moved it quickly, a piece would catch against
another piece and lift off and then fall off. Not very successful.
The second version was patented by Ishige in Japan about 1977?
and several versions were made. I received a batch of seven with
different colouring patterns made by a German sports firm - three or
four had broken just in the post! This version has a central sphere
and six of what I call 'umbrellas' sticking out toward each face
centre. Each of the pieces has a notch around the part that rest
against the inner sphere. The umbrellas catch into these notches.
One can also think of the cubies as having their own umbrellas, but of
triangular form and concave. This is the same mechanism used in the
Impossiball.
The third version is the most common and is shown in Rubik's
Hungarian patent, but is hard to interpret as I've never had the text
translated. Basically, his 2^3 is a 3^3 with the edge and centre
pieces concealed. I gather from earlier messages that there were
several versions of this, but I only recall one, but I only ever took
a few apart.
At the very centre was a cube. On each face was a square rod
extending almost to the face center. The ends of these had a +
groove. Between the rods were pieces in the form of a quadrant with a
groove on the outer, curved, edge. When all these pieces are in
place, each of the midplanes of the cube is seen to contain a circle
with a groove on its outer edge. The corner pieces are basically
hollow, but each interior face is a layer ending in a quarter-circular
curve, which fits as a tongue into the groove just mentioned. Where
two of these meet, at the interior edge of the piece, a section is cut
away to allow the piece to slide past the projections of the end of
the square rods.
In theory, one might be able to avoid the quadrant pieces, but
I think they give the structure stability.
A more serious problem is that the inner, concealed, pieces
can get out of synch with the visible pieces, The early patent of
Gustafson left gaps so one could see the inner pieces and move them.
The method used by Rubik and in some similar puzzles is to fix one
corner piece to the inner structure by some method. Rubik's 2^3 did
this by making some of the rod ends solid rather than grooved (or
perhaps they were fixed to the central cube so they couldn't rotate).
One could also not notch one of the corner pieces. Whatever one does,
it must have the effect of preventing one corner from moving in
relation to the inner structure. I seem to recall that the 4^3 uses
this idea also.
Don't know how much this helps, but that's the best I can do
off-hand.
DAVID SINGMASTER, Professor of Mathematics and Metagrobologist
School of Computing, Information Systems and Mathematics
Southbank University, London, SE1 0AA, UK.
Tel: 0171-815 7411; fax: 0171-815 7499;
email: zingmast or David.Singmaster @sbu.ac.uk
From cube-lovers-errors@mc.lcs.mit.edu Sat Nov 15 22:37:51 1997
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Mail-from: From cube-lovers-request@life.ai.mit.edu Sat Nov 15 14:55:00 1997
Date: Sat, 15 Nov 1997 14:54:07 -0500 (EST)
From: Nicholas Bodley
Reply-To: Nicholas Bodley
To: David Singmaster
Cc: tenie1@juno.com, cube-lovers@ai.mit.edu
Subject: RE: 6x6x6 cube design; also notes about the 2^3 innards. (Fairly long)
In-Reply-To: <009BD499.F2FD74E5.202@ice.sbu.ac.uk>
Message-Id:
There's a short mention in passing in Douglas Hofstadter's (second
major?) book (Metamagical Themas?) to the effect that a physical
prototype exists for the 6^3, and a paper design for the 7^3. This was
ca. 1982, iirc.
On Fri, 14 Nov 1997, David Singmaster wrote:
{Snips}
} Now let me try my hand at describing three versions of the 2^3.
} The third version is the most common and is shown in Rubik's
}Hungarian patent, but is hard to interpret as I've never had the text
}translated. Basically, his 2^3 is a 3^3 with the edge and centre
}pieces concealed.
The ones I had were quite difficult to take apart and reassemble; if
they weren't made of a strong, resilient engineering plastic, they would
not have been possible to make, I would say.
} At the very centre was a cube.
In mine, this cube was almost tiny; perhaps 15% (along an edge) of the
size of a cubie as seen from the outside.
} On each face was a square rod
}extending almost to the face center.
In mine, just about sure that three adjacent faces of this inner cube
each had thin cylindrical rods extending toward the face centers. These
were surrounded by square rods of the same width as the other three
which were part of the center cube.
The thin cylindrical rods served as pivots for the square rods of the
same width. When you rotated one half of the Cube, these pivots allowed
one half to rotate with respect to the other without prying anything
apart.
The three fixed square rods, which are extensions and "part of" the
center cube, stayed fixed within their half of the Cube when the other
half was rotated, much as the ball inside a 4^3 stays fixed.
} The ends of these [rods -nb] had a +
}groove. Between the rods were pieces in the form of a quadrant with a
}groove on the outer, curved, edge. When all these pieces are in
}place, each of the midplanes of the cube is seen to contain a circle
}with a groove on its outer edge.
The aforementioned rods are required to keep the quadrants from moving
inward and therefore out of engagement with the inner, "cut-away" edges
of the cubies. If that were to happen, the Cube would fall apart.
(Please see the next paragraph.) (When I tried to describe the innards
of a 2^3 a while back, I called these quadrants "clips". My hat's off to
Mr. Singmaster for his fluency!)
} The corner pieces are basically
}hollow, but each interior face is a layer ending in a quarter-circular
}curve, which fits as a tongue into the groove just mentioned. Where
}two of these meet, at the interior edge of the piece, a section is cut
}away to allow the piece to slide past the projections of the end of
}the square rods.
} In theory, one might be able to avoid the quadrant pieces, but
}I think they give the structure stability.
With all due respect, without them, the Cube would instantly fall
apart! They are essential.
} A more serious problem is that the inner, concealed, pieces
}can get out of synch with the visible pieces.
The natural tendency is to squeeze the cubies of each half together
when maneuvering.
Because the thin square rods molded along with the center cube are
"attached" to adjacent faces of that cube, the other three faces of that
cube carry the swiveling rods. No matter how you pick up the Cube, one
half will contain a fixed rod. Squeezing the cubies together around that
rod will make the center cube stay aligned with those four cubies that
are squeezing one of its rods. (Actually, the cubies squeeze the
quadrants, and the quadrants squeeze the rods.)
Keeping that center cube aligned also means it will keep aligned the
four rods that have their axes in the current shear plane. These rods
will then keep the quadrants aligned with the half of the cubie that is
squeezing the fixed rod.
The four quadrants in the swiveling half will squeeze the hollow,
swiveling rod, which will rotate around the thin cylindrical [rod] that
extends from that face of the center cube.
I'm indebted to Mr. Singmaster for his clarifying description.
This mechanism seems to be a real challenge to describe solely in
words! Here, a few images equal many kB of ASCII...
}DAVID SINGMASTER, Professor of Mathematics and Metagrobologist
} email:
}zingmast or David.Singmaster @sbu.ac.uk
* * *
Here's another go, for those who have the patience:
Imagine that each cubie is hollow. (They really are.) Imagine that they
are separated from each other in 3-D space by moderate and equal
distances, but still not tilted with respect to each other. In other
words, there's a large gap between any two.
Now, imagine a spherical rotary cutter, spinning in the center of the
3-D array of 8 cubies. Move the cubies toward the cutter, along radii of
the cutter passing through their outermost corners. Don't tilt or
rotate, just translate radially inward toward the cutter along a
[45-degree] axis.
Let the cutter machine a curved outline in each of the three inner
faces. (The diameter of the cutter is maybe 80% of the edge of a
complete Cube.) Make the cutter disappear, and you have a spherical
cutaway inside the whole cluster of eight. (This is real, in essence.)
The cubies are hollow, and they really have this curved "cut" in each
of their concealed inside faces. Of course, this was molded in, not
machined by a cutter.
Now, you need something to hold the cubies together.
If you've seen a radar corner reflector used by small boat owners,
think of one made of three intersecting, mutually-orthogonal circles.
They intersect at a common, center point. Make this corner reflector
tiny, maybe 3,5 (3.5) cm (?) in diameter.
Cut this apart into eight quadrants. Make them thick, if they aren't.
Make a rectangular groove in each curved edge. Remove some material from
the straight edges; line up the curved edges with a circle (same size as
the original structure before you cut it apart) on your workbench).
Space them equally apart. The gaps form a cross (or an "X", if you like
45-degree angles). The rods will go into those gaps.
OK: These are now positioned the way they will be in one of the three
shear planes in a Cube.
[The radius of the corner reflector is somewhat bigger than that of the
ball cutter.)
Thinking back to the corner reflector, if you replace all 12 quadrants
where they used to be in 3-D space, with gaps between them as they were
on the bench, that is how they are positioned in an aligned Cube.
To start assembling the Cube, you take four cubies, lay them down next
to each other (touching) with colors properly aligned, but with their
inside surfaces facing upward. Pick up four quadrants, placing the
grooves you made (in the curved edges) onto the curved "cutaways" in the
adjacent inside edges of the hollow cubies, where the cubies touch. As
long as these quadrants don't move toward each other, they will keep the
cubies together.
This, in two dimensions, is what holds the Cube together. The next four
quadrants fit into the remaining cutouts. They lie flat, and form a
circle, the way they did when you laid them down on the workbench.
To keep the quadrants away from the center, you now insert the center
cube and its rods. However, assembling the remaining four cubies (and
their four quadrants) to what you have so far, is, in the real world, a
major struggle. It involves some worrisome distortions of the pieces!
This "geometric interference" is also what makes it so hard to
disassemble.
Wonder how these are assembled at the factory?
My best regards to all,
|* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath
|* Waltham, Mass. *|* -----------------------------------------------
|* nbodley@tiac.net *|* 'T was the night before Xmas, and all through
|* Amateur musician *|* the coffeehouse, not a creature was stirring.
--------------------------------------------------------------------------
From cube-lovers-errors@mc.lcs.mit.edu Sat Nov 15 23:21:58 1997
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Mail-from: From roger.broadie@iclweb.com Thu Nov 13 19:09:04 1997
From: roger.broadie@iclweb.com (Roger Broadie)
To: "Geoffroy Van Lerberghe" ,
"Cube-Lovers"
Subject: Re: Cubes in London, plus OddzOn, clones and colours
Date: Fri, 14 Nov 1997 00:07:33 -0000
Message-Id: <19971114000519.AAA4398@home>
London's most famous toy-shop is Hamley's in Regent St. On the fourth
floor they have a wall of OddzOn cubes at 39.00 pounds each, together
with snakes and magics.
I say OddzOn because that is the name given in the copyright notice at the
back of the instruction booklet, although in this country that name appears
nowhere else. The packaging (a rectangular cardboard casing with a clear
central panel holding the cube at an angle) gives the distributors as
Toybrokers Ltd. This cube sometimes appears in the British Toys 'R' Us,
and my family bought one in Jenners, the big Edinburgh department store,
this summer. I wouldn't rate it as highly as the Ideal cubes.
I had a quick look for clones in the sort of shop in London that I have
seen them in in the past, but found none. I bought a couple of Taiwanese
clones in Dublin a few weeks ago - they came in a cube-sized cardboard box
with a picture of a cube on the front with two yellow centre pieces, five
green edge pieces and five red corner pieces. I did not complain that the
cube inside did not match this picture. I tried both the sample on display
and one of the cubes which I later bought. They turned quite well. I did
not try the other one until I got it home, when to my annoyance I found it
much stiffer. The colours are rather dull, but yellow and white are
opposite, which I prefer, because then the colours of the opposite faces
seem to have a sensible connection helping recognition of a piece that is
in the opposite face to its home face. They cost 35.00 Irish pounds each .
Among various other puzzle, Hamleys also has those from Meffert, including
the skewb and an annoying dodecahedron - the colours are duplicated at
opposite poles. So does Toys 'R' Us. What I have not been able to find is
the 5x5x5 that is shown on the Meffert packaging.
The OddzOn cube is the one that is associated with the www.rubiks.com site,
which reproduces the instruction booklet and uses the same logo in chubby
capitals. Rubik himself is clearly involved - he is quoted on the site.
I had concluded that Ideal Toys (the US company) had gone out of business,
having failed to find any reference to it currently. There is a British
company Ideal Toys (UK) Limited, but that is a subsidiary of Triumph Adler
AG. There was a company The Ideal Toy Company Limited, but that was
dissolved, as was CBS Toys Limited, which may have been connected.
OddzOn Products Inc appears to be a subsidiary of Hasbro Inc.
I have a suspicion that Ideal may have deliberately adopted the
yellow-opposite-green configuration to create a new colour arrangement that
would help them expunge clones by relying on their trade dress.
Roger Broadie
From cube-lovers-errors@mc.lcs.mit.edu Sun Nov 16 14:26:45 1997
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Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Nov 16 06:12:15 1997
From: roger.broadie@iclweb.com (Roger Broadie)
To: "Cube-Lovers"
Subject: Re: Cubes in London
Date: Sun, 16 Nov 1997 11:07:06 -0000
Message-Id: <19971116110836.AAA18652@home>
I apparently wrote
> .. at 39.00 pounds ..
Before people get the wrong idea about the price of cubes on this side of
the Atlantic, I'd better say that the OddZon cube was 9 British pounds in
Hamleys and the Dublin clone was 5 Irish pounds. I used the pound symbol
and the conversion - I suspect both machine and human - went awry.
I can add to my slightly meandering note on the Dublin clone that the
central spider has now bust.
Life is full of new hazards.
Roger Broadie
[ Sorry, you're a victim of moderator error. While replacing the
pound symbol with the word "pounds", and I left in an extra 3.
Thanks for the information. --Dan ]
From cube-lovers-errors@mc.lcs.mit.edu Sun Nov 16 14:57:58 1997
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Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Nov 16 07:24:33 1997
Message-Id: <199711161045.KAA30105@GPO.iol.ie>
From: "Goyra (David Byrden)"
To: Cube-Lovers@ai.mit.edu
Subject: Re: Cubes in London, plus OddzOn, clones and colours
Date: Sun, 16 Nov 1997 10:43:38 -0000
> From: Roger Broadie
> What I have not been able to find is
> the 5x5x5 that is shown on the Meffert packaging.
Try writing to Dr. Christophe Banelow
An Der Wabeck 37
D-58456 Witten
Germany
tel: 49 2302 71147
fax: 49 2302 77001
I have his catalogue here and he lists the 5^3, Skewb,
Dedecahedron, Pyraminx, Octahedron, Magic Jewel, among
others.
David
From cube-lovers-errors@mc.lcs.mit.edu Mon Nov 17 00:07:22 1997
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Mail-from: From cube-lovers-request@life.ai.mit.edu Sun Nov 16 17:31:22 1997
Date: Sun, 16 Nov 1997 17:30:08 -0500 (EST)
From: Nicholas Bodley
To: "Goyra (David Byrden)"
Cc: Cube-Lovers@ai.mit.edu
Subject: Re: Cubes in London, plus OddzOn, clones and colours
In-Reply-To: <199711161045.KAA30105@GPO.iol.ie>
Message-Id:
On Sun, 16 Nov 1997, Goyra (David Byrden) wrote:
{Snips}
}
} Try writing to Dr. Christophe Banelow
} An Der Wabeck 37
} D-58456 Witten
} Germany
} tel: 49 2302 71147
} fax: 49 2302 77001
I hope I'm not being rude to point out minor typos; his name should be
"Christoph Bandelow". It's an easy slip to make.
Regards,
|* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath
|* Waltham, Mass. *|* -----------------------------------------------
|* nbodley@tiac.net *|* 'T was the night before Xmas, and all through
|* Amateur musician *|* the coffeehouse, not a creature was stirring.
--------------------------------------------------------------------------
From cube-lovers-errors@mc.lcs.mit.edu Mon Nov 17 21:35:42 1997
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Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Nov 17 04:26:22 1997
Message-Id: <34700D89.24426818@ibm.net>
Date: Mon, 17 Nov 1997 01:25:29 -0800
From: "Jin 'Time Traveler' Kim"
Organization: The Fourth Dimension
To: Cube-Lovers@ai.mit.edu
Subject: Color schemes revisited
References: <5dlt1c$baq@gap.cco.caltech.edu>
An interesting thing to note regarding the color patterns on cubes...
on the Rubik's Cube home page (http://www.rubiks.com) a picture is
displayed showing Dr. Rubik himself holding a mixed cube in his hand.
On a whim I decided to figure out what the color scheme of the cube
was.
If you wish to figure out yourself without being told, or if you just
want to try to refute my guess (I'm no stranger to being wrong) then
don't read the "answer" to the puzzle that's below my .sig.
Otherwise, if you can't be bothered with minor trivialities like this
one (it's really not that difficult to figure out the colors anyway)
then read on.
--
Jin "Time Traveler" Kim
chrono@ibm.net
VGL Costa Mesa
http://www.geocities.com/timessquare/alley/9895
http://www.slamsite.com
I determined that the color scheme of the cube held in Erno Rubik's hand is:
Front: Red
Back: Orange
Left: Green
Right: Blue
Top: White
Bottom: Yellow
From cube-lovers-errors@mc.lcs.mit.edu Mon Nov 17 22:05:10 1997
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Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Nov 17 10:56:56 1997
Date: Mon, 17 Nov 1997 15:53:23 GMT
From: David Singmaster
To: chrono@ibm.net
Cc: cube-lovers@ai.mit.edu
Message-Id: <009BD6F9.65F6A7C1.425@ice.sbu.ac.uk>
Subject: Re: 6x6x6 cube design
I'm sure that this has been mentioned before, but the 6^3
etc. actually introduce no further complications than present on the
4^3 (and 5^3). There are just more types of center pieces, but they
all behave in much the same way. In my message on notation and
solution of the 4^3, I gave a method of producing a 3-cycle of center
pieces and it can be used for each class of centre pieces - the puzzle
doesn't get any more interesting, just longer!
The 6^3 introduces a slightly interesting feature
theoretically in that the center pieces break up into more classes
than one might initially expect because the piece at the (1,2)
location is not in the same class as the piece at the (2,1) location.
(Taking a corner as (0,0).)
DAVID SINGMASTER, Professor of Mathematics and Metagrobologist
School of Computing, Information Systems and Mathematics
Southbank University, London, SE1 0AA, UK.
Tel: 0171-815 7411; fax: 0171-815 7499;
email: zingmast or David.Singmaster @sbu.ac.uk
From cube-lovers-errors@mc.lcs.mit.edu Mon Nov 17 22:44:50 1997
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Mail-from: From cube-lovers-request@life.ai.mit.edu Mon Nov 17 10:58:38 1997
Date: Mon, 17 Nov 1997 15:57:19 GMT
From: David Singmaster
To: cubeman@idirect.com
Cc: cube-lovers@ai.mit.edu
Message-Id: <009BD6F9.F2D7ACBC.209@ice.sbu.ac.uk>
Subject: Re: Cube Colours
According to what I wrote in my Cubic Circular 3/4 of Summer 1982, the
cubes used in the World Championship were of the +/- yellow BOY pattern.
DAVID SINGMASTER, Professor of Mathematics and Metagrobologist
School of Computing, Information Systems and Mathematics
Southbank University, London, SE1 0AA, UK.
Tel: 0171-815 7411; fax: 0171-815 7499;
email: zingmast or David.Singmaster @sbu.ac.uk
From cube-lovers-errors@mc.lcs.mit.edu Tue Nov 18 23:50:41 1997
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Mail-from: From cube-lovers-request@life.ai.mit.edu Tue Nov 18 23:43:40 1997
To: Cube-Lovers@ai.mit.edu
Date: Tue, 18 Nov 1997 20:42:55 -0800
Subject: Rubiks Revenge moves
Message-Id: <19971118.204255.7126.1.tenie1@juno.com>
From: tenie1@juno.com (Tenie Remmel)
Is there an easy way to cycle three adjacent top edges on the
Rubiks Revenge? I can't find one shorter than 62 moves, but if
there was a short one I could simplify my solution greatly.
. b c . . a b .
a . . . => c . . .
. . . . . . . .
. . . . . . . .
Hopefully it won't mess up the corners, but it's ok if it does.
I'd also like to see some short moves for the following 3-cycles:
. * * . . . * . . . . . . . * .
. . . . * . . * * . . * . . . *
* . . . . . . . . . . * * . . .
. . . . . . . . . . . . . . . .
Is there a good source anywhere for moves, pretty patterns, etc. for
the Rubiks Revenge? It's quite difficult to find information about
it. Also is there an automatic move generating program for the higher
order cubes like 'Cube Explorer' is for the 3x3x3?
--Tenie Remmel (tjr19@juno.com)
From cube-lovers-errors@mc.lcs.mit.edu Thu Nov 20 11:46:33 1997
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Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Nov 19 08:29:34 1997
From: bagleyd@americas.sun.sed.monmouth.army.mil (David Bagley x21081)
Message-Id: <199711191329.IAA26271@java.sed.monmouth.army.mil>
Subject: Re: A 4 Dimensional Rubik's Cube
To: charlied@erols.com (Charlie Dickman), Cube-Lovers@ai.mit.edu
Date: Wed, 19 Nov 1997 08:29:09 -0500 (EST)
In-Reply-To: from "Charlie Dickman" at Nov 17, 97 08:01:03 pm
Hi All
I added Charlie Dickman's Tesseract (A 4 Dimensional Rubik's Cube) to my
web pages ( http://www.tux.org/~bagleyd/ ). Its in two parts, the docs
(mind twisting stuff) and the Mac Program.
Charlie Dickman: If you make any updates I'll be happy to update the pages.
By the way, I recently reorganized my web pages.... same old junk but its
presented better. :)
--
Cheers,
/X\ David A. Bagley
(( X bagleyd@bigfoot.com http://www.tux.org/~bagleyd/
\X/ xlockmore and more ftp://ftp.tux.org/pub/people/david-bagley
From cube-lovers-errors@mc.lcs.mit.edu Thu Nov 20 12:17:58 1997
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Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Nov 19 16:10:23 1997
Sender: davidb@davidb.concentric.net
Message-Id: <34735538.113A5129@iname.com>
Date: Wed, 19 Nov 1997 13:08:08 -0800
From: David Barr
Organization: Medweb
To: Tenie Remmel , Cube-Lovers
Subject: Re: Rubiks Revenge moves
References: <19971118.204255.7126.1.tenie1@juno.com>
Tenie Remmel wrote:
>
> Is there an easy way to cycle three adjacent top edges on the
> Rubiks Revenge? I can't find one shorter than 62 moves, but if
> there was a short one I could simplify my solution greatly.
>
> . b c . . a b .
> a . . . => c . . .
> . . . . . . . .
> . . . . . . . .
I hold the cube so the bottom looks like this:
. . a .
. . . b
. . . c
. . . .
and do this sequence:
F' b2 L2 / R' D r' D' R D r D' / L2 b2 F
Capital letters are outer slices. Small letters are inner slices.
The slashes are just to show the different parts of the sequence. The
middle part, if performed alone, will cycle three edges. The first
part of the sequence positions the cubies we want to move into the
positions of the cubies that are cycled by the middle sequence. The
last part of the sequence simply reverses the first part.
Left view of cube:
. . . .
. . . a
. . . .
. c b .
b2 R2 / L D' l D L' D' l' D / R2 b2
Bottom view of cube:
. . a .
c . . b
. . . .
. . . .
F' / R' D r' D' R D r D' / F
Bottom view of cube:
. . . .
a . . c
. . . b
. . . .
b2 U' F / R' D r' D' R D r D' / F' U b2
Bottom view of cube:
. . a .
. . . b
c . . .
. . . .
F' b' L2 / R' D r' D' R D r D' / L2 b F
Here are some other three cycles you may find useful:
R' D l D' R D l' D'
R' D L D' R D L' D'
r' D l D' r D l' D'
--
mailto:davidbarr@iname.com
http://www.concentric.net/~Davebarr/
From cube-lovers-errors@mc.lcs.mit.edu Thu Nov 20 12:50:09 1997
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Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Nov 20 12:06:06 1997
Date: Thu, 20 Nov 1997 12:05:48 -0500
Message-Id: <20Nov1997.115617.Hoey@AIC.NRL.Navy.Mil>
From: Dan Hoey
Sender: Cube-Lovers-Request@ai.mit.edu
To: cube-lovers@ai.mit.edu
Subject: Auction on Rubik's Revenge (4x4x4) cubes (REPOST)
Reply-To: Paul Hart
whuang@ugcs.caltech.edu (Wei-Hwa Huang) has passed on a Usenet
announcement of Paul Hart's auction of 6 unopened Rubik's Revenges, as
mentioned previously in Cube-Lovers. The auction ends November 22.
For details on the offer read http://www.enol.com/~hart, check a
Usenet search engine, or inquire by e-mail to Paul Hart
.
- Dan Hoey
Interim Cube-Lovers-Request operator
From cube-lovers-errors@mc.lcs.mit.edu Thu Nov 20 13:16:42 1997
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Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Nov 19 09:32:05 1997
Date: Wed, 19 Nov 1997 14:30:34 GMT
From: David Singmaster
To: chrono@ibm.net
Cc: cube-lovers@ai.mit.edu
Message-Id: <009BD880.292EDB5C.279@ice.sbu.ac.uk>
Subject: RE: Color schemes revisited
In 1979(?) when I had my company David Singmaster Ltd which dealt in
Cubes and cube-related items, we had a tee-shirt designed showing a jumbled
cube with the caption Rubik's Cube Cures Sanity. Only one person ever wrote
in pointing out that the cube was impossible! From the colouring of various
visible pieces, one could tell that the white face was adjacent to all five
other colours!!
DAVID SINGMASTER, Professor of Mathematics and Metagrobologist
School of Computing, Information Systems and Mathematics
Southbank University, London, SE1 0AA, UK.
Tel: 0171-815 7411; fax: 0171-815 7499;
email: zingmast or David.Singmaster @sbu.ac.uk
From cube-lovers-errors@mc.lcs.mit.edu Thu Nov 20 13:47:50 1997
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Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Nov 19 08:55:52 1997
Message-Id:
In-Reply-To: <19971118.204255.7126.1.tenie1@juno.com>
Date: Wed, 19 Nov 1997 08:56:30 -0500
To: Tenie Remmel , Cube-Lovers@ai.mit.edu
From: Nichael Lynn Cramer
Subject: Re: Rubiks Revenge moves
Tenie Remmel wrote:
>Is there an easy way to cycle three adjacent top edges on the
>Rubiks Revenge? I can't find one shorter than 62 moves, but if
>there was a short one I could simplify my solution greatly.
>
>. b c . . a b .
>a . . . => c . . .
>. . . . . . . .
>. . . . . . . .
>
>Hopefully it won't mess up the corners, but it's ok if it does.
The way I approach this is to begin with the following simple 3-cycle for
edge cubies (note that this cycles only the cubies and leave the rest of
the cube unaltered):
1] Imagine the involved cubies in the following configuration:
Top face : . . c . Left face: . . . .
. . . . a . . .
. . . . . . . .
. b . . . . . .
2] Perform the following sequence:
- Rotate Front Face by 1/4 turn clockwise.
- Rotate the slice just below the Top Layer by 180 dgs.
- Rotate the Front Face by 1/4 turn counter-clockwise.
- Rotate the Top Face by 180 dg.
- Rotate Front Face by 1/4 turn clockwise.
- Rotate the slice just below the Top Layer by 180 dgs.
- Rotate the Front Face by 1/4 turn counter-clockwise.
- Rotate the Top Face by 180 dg.
This will result in:
Top face : . . b . Left face: . . . .
. . . . c . . .
. . . . . . . .
. a . . . . . .
with all other cubies in their original locations.
3] Once this step is mastered, it is now only a question of moving the
cubies that you want to swap into the approriate location for this operator
to do its work.
For example, in your example above this can be accomplished by (this
assumes that the figure you have drawn above is your Top Face):
- Rotating the Left-most two slices 1/4 turn clockwise (i.e. towards you)
- Rotating the Top Face 1/4 turn counter-clockwise.
If you now rotate the entire cube by 90dgs clockwise, you will see your
three cubies are now in the proper location to use the above operator.
(When you're done with the operator, repeat the steps just above in the
reverse order to finish.)
>I'd also like to see some short moves for the following 3-cycles:
>
>. * * . . . * . . . . . . . * .
>. . . . * . . * * . . * . . . *
>* . . . . . . . . . . * * . . .
>. . . . . . . . . . . . . . . .
These are just variations on the above. They will be left an exercise for
the reader. ;-)
Hope this helps.
Nichael
Nichael
nichael@sover.net deep autumn my neighbor what does she do
http://www.sover.net/~nichael/ --Basho
From cube-lovers-errors@mc.lcs.mit.edu Thu Nov 20 20:37:44 1997
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Mail-from: From cube-lovers-request@life.ai.mit.edu Thu Nov 20 13:51:23 1997
Sender: davidb@davidb.concentric.net
Message-Id: <3474866B.D1136871@iname.com>
Date: Thu, 20 Nov 1997 10:50:19 -0800
From: David Barr
Organization: Medweb
To: Cube-Lovers
Subject: Re: Rubiks Revenge moves
References: <19971118.204255.7126.1.tenie1@juno.com> <34735538.113A5129@iname.com>
David Barr wrote:
> I hold the cube so the bottom looks like this:
>
> . . a .
> . . . b
> . . . c
> . . . .
>
> and do this sequence:
>
> F' b2 L2 / R' D r' D' R D r D' / L2 b2 F
Actually, you can save a couple moves by doing
d2 L' R' D r' D' R D r D' L d2
but the pieces affected will be on the right side instead of the bottom.
--
mailto:davidbarr@iname.com
http://www.concentric.net/~Davebarr/
From cube-lovers-errors@mc.lcs.mit.edu Thu Nov 20 20:54:43 1997
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Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Nov 19 17:50:07 1997
Sender: mahoney@marlboro.edu
Message-Id: <34736B2E.F9F4962@marlboro.edu>
Date: Wed, 19 Nov 1997 17:41:50 -0500
From: Jim Mahoney
Organization: Marlboro College
To: Tenie Remmel
Cc: Cube-Lovers@ai.mit.edu
Subject: Re: Rubiks Revenge moves
References: <19971118.204255.7126.1.tenie1@juno.com>
Tenie Remmel wrote:
> Is there an easy way to cycle three adjacent top edges on the
> Rubiks Revenge? I can't find one shorter than 62 moves, but if
> there was a short one I could simplify my solution greatly.
>
> . b c . . a b .
> a . . . => c . . .
> . . . . . . . .
> . . . . . . . .
You can cycle these three edges on the 4x4x4 in 14 quarter turns
without disturbing the corners.
With the "up" and "front" faces like this (in a kind of projection view;
the corners are given by "*"; the "right" face is not shown),
* . . *
. . . .
. . . C
* A B *
. . . .
. . . .
* . . *
a procedure to cyle A,B,C is as follows:
(1) 2 preparation moves which put C on "down" slice and B on "up/back"
(2) 3 moves to get A off top slice and replace with C.
(3) 2 moves (1/2 rotate) the top slice to put B where C (orginally A) was.
(4) undo (2), restoring bottom layers and bring A back to top, in new spot.
(5) undo (3)
(6) undo (1), the prep moves.
(In each case "undo" means to do the inverse of the same moves in the
opposite order; that is, "undo" ABC means C'B'A' where C' is the inverse
of move C.)
The hardest part I have of describing the specifics of each of
these is the notation; each of the 6 steps is only a couple of moves.
Let me define U,R,F as the Up, Right, and Front faces, and
number the slices by integers 1 to 4, so for example (F1,F2,F3,F4)
are clockwise quarter turns on the 4 slices (front to back)
parallel to the front face. Counterclockwise turns are indicated
with either a ' (indicating "inverse") or lower case, so
F1' = a counterclockwise quarter turn of the front face.
In pictures,
* - - * * - - *
. 1 2 . . 3 1 .
. 3 4 . . 4 2 .
* - - * ==> U1 ==> * - - *
| a b | | a b |
| c d | | c d |
* - - * * - - *
* - - * * - - *
. 1 2 . . 1 2 .
. 3 4 . . 3 4 .
* - - * ==> F1' ==> * - - *
| a b | | b d |
| c d | | a c |
* - - * * - - *
where the letters are on the "front" face
and the numbers are on "up" face.
Then with this notation, steps (1) through (6) of this procedure are
(1) F2 R2
(2) F3' U4' F3
(3) U1 U1
(4) F3' U4 F3'
(5) U1' U1'
(6) R2' F2'
which is 14 moves.
[Moderator's note: Certainly (4) should be F3' U4 F3, but that still
cycles the wrong edges. With (2)=R2 U4' R2', (4)=R2 U4 U2' we cycle
the correct triple of edges, but in inverse order. ]
I confess that I don't have a 4x4x4 anymore and so can't try this -
I may have visualized one of the details wrong. Hope not.
> I'd also like to see some short moves for the following 3-cycles:
>
> . * * . . . * . . . . . . . * .
> . . . . * . . * * . . * . . . *
> * . . . . . . . . . . * * . . .
> . . . . . . . . . . . . . . . .
You can do any of these as variations of the method I give above
with different "preperation" moves to get the edges into the
proper positions, namely two on the same slice which can be turned
into one another (the top slice in these cases),
and the third on another slice (usually the bottom slice) which
can replace one of the edges from the top slice.
I have a discussion of the NxNxN cube which includes in section (VI)
this recipe for 3-cycles of any kind of edge, corner, or face
piece; you can read it at
http://www.marlboro.edu/~mahoney/cube/NxN.txt if you're interested.
Regards,
Jim Mahoney (mahoney@marlboro.edu)
Marlboro College
From cube-lovers-errors@mc.lcs.mit.edu Thu Nov 20 21:52:54 1997
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Mail-from: From cube-lovers-request@life.ai.mit.edu Wed Nov 19 11:58:12 1997
Date: Wed, 19 Nov 1997 11:53:14 -0500
Message-Id: <000E9C40.001706@scudder.com>
From: jdavenport@scudder.com (Jacob Davenport)
Subject: Re: Rubiks Revenge moves
To: Cube-Lovers@ai.mit.edu
Forget three adjacent top edges, I just want to cycle two of them. I've
been solving a 5x5x5, and finally figured out how to make it look like a
3x3x3 so that I could solve nearly all of it. However, the one place where
I cannot do that is solving the second and fourth edges from any side, and
have been using a short move that cycles three of them:
.axx. .bxx.
y.... z....
y.... => z....
b.... a....
.czz. .cyy.
The move is 2L F' L F 2L' (where 2L means the second layer from the
left)
This works great for getting nearly all the edges in place, but I have
two edges that are switched, and every time I use this move to put
them in place, I either leave two other edges out of place or leave
four edges out of place. That is, I have the following:
.baa. .aaa.
b.... c....
b.... which I can only make into c....
a.... b....
.ccc. .cbb.
which does not help.
I believe that my move works on 4x4x4 edges, and any move that helps a 4x4x4
cube will probably help me.
From cube-lovers-errors@mc.lcs.mit.edu Sat Nov 22 22:56:58 1997
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