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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