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Date: Thu, 23 Jul 1998 18:21:02 -0400
From: michael reid
Message-Id: <199807232221.SAA07643@hilbert.math.brown.edu>
To: cube-lovers@ai.mit.edu
Subject: patterns on 5x5x5 cube
a while ago, rainer asked for patterns on the 5x5x5 cube. here
are some i know (the hardest part seems to be finding the scraps
of paper on which the maneuvers are written).
standard notation uses R and r for 90 degree clockwise
twists of the outer layer and second layer, respectively. i've
found it convenient to have notation for Rr , so i use _R_
(that is, capital R underlined). i guess this notation is more
convenient for handwritten maneuvers, but not so convenient for
e-mail. i'll use _R_ to denote R underlined and
_( F L U B ... )_
to mean that the whole thing inside the parentheses is underlined.
the first pattern is a "double" snake; it meanders onto each face twice.
_R'_ b' _U_ F2 _U'_ b _U_ F2 _(U' R2 F')_ u2 _(F U L')_
u2 _(L U' R' L)_ f _D'_ B2 _D_ f' _D'_ B2 _(D L2 B)_ d2
_(B' D' R)_ d2 _(R' D L)_
if i understand the terminology correctly, this is a continuous
non-chiral isoglyph with the pattern
...*.
**.**
.*...
.*...
.*...
i still remember that when i found this pattern some 10-12 years
ago, i saw the URF faces together. then i turned the cube around,
and was surprised by how it continued on the other three faces.
(i shouldn't have been surprised, but you know how that goes ... )
i came across this pattern accidently. then i went snake hunting,
and found several others:
snake 2
_(R F2 R2 U2)_ r2 _(U2 R2 F2 R' D' F2 B2 D R F2 R2 U2)_ r2
_(U2 R2 F2 R' D')_ r2 _(F2 B2)_ L2 _(R2 U' D F2 B2 U)_
those two have the property that the two snake segments on each face
have the same color. if this condition is relaxed, we also have
snake 3
_(R L' F U2 R F2 R2 U2)_ r2
_(D2 L2 F B' D' F B' U' D F R L D' B2 L' F B' D')_
f2 _(U2 D2)_ f2 _(U' D2)_
this one can be modified slightly; change the U and D faces
.*.*. .*.*.
.*.*. .***.
.*.*. .....
from .*.*. to .***.
.*.*. .*.*.
if only one is changed, then we get two separate snakes.
there's also
snake 4
_(D F2 B2)_ l2 _(F2 B2 R')_ R2 _(F2 R2 U2)_ r2
_(U2 R2 F2 R' D' F2 B2 D R F2 R2 U2)_ r2
_(U' D' F' U2 D2 B U' D L2 B2 L' U2 D F2 B2)_
another interesting pattern is
U R' U' F' _U'_ R' _U_ f _U'_ R _(U F')_ F2 U R U'
_B_ l' _D2_ l _D_ f' _D2_ f _(D' B')_
D' L D B _D_ L _D'_ b' _D_ L' _(D' B)_ B2 D' L' D
_F'_ r _U2_ r' _U'_ b _U2_ b' _(U F)_
which gives a continuous non-chiral isoglyph with the pattern
.*...
.*...
.*...
*****
...*.
the same maneuver produces an analogous pattern on the 4x4x4 cube,
but there's probably an easier maneuver. another isoglyph (also
continuous and non-chiral) with the same pattern is
R f' U2 f U l' U2 l U' R' _D'_
L b2 L' _B'_ U b2 U' _(B D)_
L' b D2 b' D' r D2 r' D L _U_
R' f2 R _F_ D' f2 D _(F' U')_
modifying this pattern is how i found the first double snake.
mike