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Date: Tue, 3 Aug 1999 02:11:45 -0700 (PDT)
From: Tim Browne
To: Cube-Lovers@ai.mit.edu
Subject: Notes on the Bandaged Cube... combinations and other musings.
In-Reply-To:
I just picked up a bandaged cube the other day, and I get the feeling that
this might be one of the few puzzles which would beat even the Square-1
for solving difficulty. For those of you who haven't seen this nightmare
of a cube, Hendrik Haak has a picture of it in the museum section of his
Puzzle Shop (he calls it a "Bicube"). After playing around with it for a
bit, I figured I'd try and work out the number of possible combinations.
The cube is constructed from a standard Rubik's Cube mechanism, so all the
standard Rubik's Cube restrictions apply here.
The cube is made up of 13 pieces. One of them is created by fusing an edge
piece between 2 centres, effectively turning it into a 2x2 piece. Because
of this piece, 2 axes of motion are effectively cut off permanently,
making a maximum of 4 axes of rotation. There are 4 pieces which are made
up by fusing an edge to a centre, 7 pieces are a corner/edge fusion,
leaving us with one standard corner piece. These fusions make the puzzle
much more difficult than it first appears, as the contortions of all of
these 1x2 pieces effectively block axes of rotation which were easily
accessible only a quarter turn ago, sometimes getting so bad as to make
the only available axis the one you just turned, in some extreme cases
even forcing you to back out following exactly the same path you used to
enter the current state. Needless to say, this makes solving the puzzle
very frustrating indeed. Anyhoo, back to the combinations...
Two of the centres are effectively locked into place for all time, leaving
us with 4 edge/centre fusions which can be rotated one of 4 ways, giving
us a factor of 4^4 combinations. Piece rotations in place are impossible.
The first potential restriction would be creating more than one area where
only the corner cube could fit, but the 2x2 piece makes this an
impossibility. The second would be potential collisions between pieces.
There are 5 pairs of adjacent sides where you have a 1:16 chance of a
conflict, and 2 ways of creating a double conflict, so we reduce this
amount by 5*256/16-2=82, leaving us with 174 possible arrangements of the
edge/centre fusions.
The edge/corner fusions and the lone corner piece will be handled
together. The corner piece can be placed into any one of 8 corner slots,
while the edge/corner fusions fit tightly into the remaining slots
surrounding it, giving us a factor of 8!=40,320 combinations. Now for the
restrictions... let's start with a simple swap first. Let's take an
example side. Like numbers in the table are bandaged together.
112
345
345
Would it be possible to swap piece 1 and piece 3? Given the restrictions
carried over from the Rubik's Cube, the only way you can swap a pair of
edges or a pair of corners is if you also swap either a second pair of the
same type, or a pair of its opposite. To see this latter case, rotate a
slice of a solved cube 90 degrees either way and then "reconstruct" it
using your favourite patterns. God's Algorithm in this case is expressly
prohibited. Since the corners and edges are fused, a swap of one expressly
implies a swap of the other, so this is OK.
How about swapping pieces 3 and 5? This one's a bit more difficult. Not
only are you swapping the pieces, but you're rotating them as well. When
you swap these 2, both edge pieces are inverted, maintaining the even
parity, so that's OK... the corner half of piece 3 is given a positive
rotation, while the corner half of piece 5 is given a negative rotation.
Modulo 3 parity is maintained, so this is also OK, meaning the whole swap
is OK. Swapping any 2 edge/corner fusions on the cube can be broken down
into a compound movement of either of these, so any of these pieces can be
swapped with any other similar piece without restriction.
The second potential restriction happens with edge/centre fusions. In
certain special cases, the centres can rotate in such a way they they box
in a 1x1x1 area, limiting it to 1 possible place. This can happen in one
of two possible ways, bringing our combinations from 174x40,320 down to
172x40,320 + 2x5,040.
The final possible restriction happens when we swap an edge/centre fusion
with an edge/corner fusion. What happpens when we swap pieces 4 and 5? So
you don't have to scroll back, here it is again...
122 122
345 --> 344 ?
345 355
There are 3 problems with this: 1) you're swapping a single pair of edge
pieces, 2) you're flipping a single edge piece, and 3) You're rotating a
single corner in place, *all* of which are definite no-no's on the Rubik's
Cube, either solo or in any combination, so you definitely can't do it
here. The corner rotation can be accounted for easily enough by rotating
the 1x1x1 piece in place to compensate, which you've probably noticed that
I conveniently left out of all calculations to this point, to save
multiplying and dividing by 3 unnecessarily. ALL of these problems can be
compensated for by either simply not doing it, or by doing the same thing
with another set, giving us 2 pairs of swapped edges and 2 edges flipped,
compensating for the corner rotation again with the 1x1x1. This
effectively slashes the potential combinations in half, bringing us down
to 86x40,320 + 1x5,040 = (86x8+1)x5,040 = 689 x 5,040 =
5040
689
----
45360
403200
3024000
--------
3472560 possible combinations.
Most puzzles of this type are difficult because of the sheer number of
combinations. This is perhaps the only one of this type which is so
difficult because of its limits. If you want to take it apart and put it
back together randomly, you've got a 1 in 6 1/1920th chance of doing so
correctly, perhaps this puzzle's one advantage over the cube. However, if
you want to tough it out and devise a pattern for it, then you'll need to
work out 172x8+2 = 1,378 patterns to get it back to its default shape,
followed by at least 6 more patterns to restore the edge/corner fusions to
their proper positions, making at least 1,384 patterns to work out for a
general solution. I've graciously decided to leave this as an exercise for
the reader. ;-) L8r.