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Date: Wed, 27 Sep 1995 17:50:00 -0400
From: michael reid
To: cube-lovers@ai.mit.edu
Subject: number of positions of square1
mark says
> I know of no Square 1 calculations whatsoever, but would be
> interested in seeing anyone's calculations.
i also haven't seen any figures for square1 (although i guess that
ideal toy corp.'s generic "more than three billion" might apply).
but it's not hard to derive some figures. i guess there might be
some question about what constitutes a "position". i think it's
reasonable to consider only configurations where all three axes
are free to turn.
note that up to rotation, there are 29 different shapes for the
top face. they occur as 19 symmetric shapes and 5 mirror image
pairs. these are grouped into five different classes according to
the number of 60 degree pieces ("corners"?), which i'll call doubles.
for each shape, we also count the number of rotationally distinct
orientations, as well as the number of orientations where the
locations of the doubles do not block the half-turn axis.
description # rotational # orientations
name of shape symmetries # orientations which allow half-turn
type 6-0
A 222222 6 2 1
type 5-2
B 2222211 1 12 6
C 2222121 1 12 4
D 2221221 1 12 2
type 4-4
E 22221111 1 12 6
F 22122111 1 12 4
G 22112211 2 6 4
H 22212111 1 12 4
H' 22211121 1 12 4
I 22211211 1 12 6
J 22121211 1 12 6
J' 22112121 1 12 6
K 22121121 1 12 4
L 21212121 4 3 2
type 3-6
M 222111111 1 12 6
N 221211111 1 12 6
N' 221111121 1 12 6
O 221121111 1 12 8
O' 221111211 1 12 8
P 221112111 1 12 6
Q 212121111 1 12 8
R 212112111 1 12 6
R' 212111211 1 12 6
S 211211211 3 4 2
type 2-8
T 2211111111 1 12 8
U 2121111111 1 12 8
V 2112111111 1 12 8
W 2111211111 1 12 8
X 2111121111 2 6 5
the top and bottom faces have complementary type, (i.e. are 6-0 and
2-8, 5-2 and 3-6, 4-4 and 4-4, 3-6 and 5-2, or 2-8 and 6-0).
type total # valid orientations
6-0 1
5-2 12
4-4 46
3-6 62
2-8 37
thus we have 1 * 37 + 12 * 62 + 46 * 46 + 62 * 12 + 37 * 1 = 3678
valid possibilities of the doubles. each permutation of the doubles
and singles is possible, and the middle layer has two orientations.
any combination of these is possible. therefore we get a final count
of 3678 * 2 * 8! * 8! = 11958666854400 positions.
mike