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Relation between positions and positions mod M in FTM
Submitted by kociemba on Thu, 07/15/2010 - 05:10.
Tom proposed that I give the relation between the number Nm of positions mod M and the number of positions N in the way N = 48*Nm - constant C. This is indeed possible, because the symmetric positions of the cube are completely analyzed.
If we denote the number of symmetric positions by S and the number of symmetric position mod m by Sm, we have the relation
(Nm-Sm)*48 = N - S, this is because (Nm-Sm)*48 is the number of positions with no symmetry. So we have N = 48*Nm - (Sm*48 - S) = 48*Nm - C. The constant C is tabulated below for all levels from 0 to 20: depth C 0 47 1 78 2 189 3 360 4 1593 5 4788 6 19850 7 72564 8 237656 9 858381 10 3015740 11 9785356 12 35144616 13 122254428 14 436594274 15 1764160807 16 8037257961 17 37547823254 18 95403536079 19 21275288869 20 1140678 |
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