# EndsWith Values, Corners Group, Quarter Turn Metric

My calculations of God's algorithm have generally included an analysis of the distribution of EndsWith values. My best God's algorithm results for the quarter turn metric are out to 13q. Tom Rokicki has since calculated out to 15q. Out to 13q, it is the case that for the vast majority of positions x we have |EndsWith(x)| = 1. Tom may be able to speak to the 14q and 15q cases, but I cannot.

Given the predominance of |EndsWith(x)| = 1 out to 13q, I have assumed that for the vast majority all of cube space it's probably the case that |EndsWith(x)| = 1. I no longer believe that assumption is correct. Which is to say, I now have an EndsWith distribution for the entirety of the corners group in the quarter turn metric. Beyond a certain distance from Start in the corners group, there are no instances of |EndsWith(x)| = 1 whatsoever, and overall the |EndsWith(x)| = 1 cases are very much in the minority. It now seems to me that the same is probably true for the complete cube group including corners and edges. I suspect the reason we have not seen this effect for the complete cube group is simply that we have not yet been able to calculate out far enough from Start.

I use EndsWith as an attribute of a position, not of a maneuver. We might look at the maneuver FB' and say that EndsWith(FB')={B'} and we might look at the maneuver B'F and say that EndsWith(B'F)={F}. But the maneuvers FB' and B'F both effect the same position, so we have EndsWith(FB')={F,B').

EndsWith is important for a couple of reasons. For one thing, it is closely related to branching factors. For example, if |EndsWith(x)| = 3, then we know that from position x in the quarter turn metric there are 3 moves that go closer to Start and 9 moves that go further from Start. Essentially, the branching factor for position x is 9. Some of those 9 positions might be duplicated with positions effected by a different maneuver, but the maximum number of new positions further from Start that might be reached from position x is 9.

EndsWith is also closely related to syllables and non-trivial identities. In particular, no position for which |EndsWith(x)| = 1 can be the midpoint of a non-trivial identity. For example, we have |EndsWith(FB'U)| = {U} and |EndsWith(B'FU)| = 1. If we take the walk FB'U through cube space, there are two minimal ways back to Start, namely U'BF' and U'F'B. But either way back to Start yields a trivial identity, namely (FB'U)(U'BF') or (FB'U)(U'F'B), because the UU' in the middle immediately cancels out.

Given my assumed predominance of |EndsWith(x)| = 1 positions in cube space, I was also assuming that most positions in cube space could not serve as the midpoint of a non-trivial identity. But I now believe that far enough from Start, the vast majority of positions can serve as the mid-point of a non-trivial identity. And because most positions are far from Start, I now believe that the vast majority of positions overall can serve as the mid-point of a non-trivial identity.

Here follow the results for the corners group that have caused me to change my mind for the complete cube group about the predominance of positions for which |EndsWith(x)| = 1.

|EndsWith(x)| Corners of 3x3x3, Quarter Turn Metric 0 1 2 3 4 5 6 7 8 9 10 11 12 Total |x| 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 12 0 0 0 0 0 0 0 0 0 0 0 12 2 0 96 18 0 0 0 0 0 0 0 0 0 0 114 3 0 672 192 60 0 0 0 0 0 0 0 0 0 924 4 0 4032 1920 480 51 0 56 0 0 0 0 0 0 6539 5 0 19104 14904 3792 984 216 384 144 0 0 0 0 0 39528 6 0 71184 90984 16656 13212 1872 3936 528 1344 0 96 0 114 199926 7 0 123360 478008 42768 117576 7824 16656 1680 9096 1536 6552 480 600 806136 8 0 23328 1911312 9024 643536 2736 121872 384 23232 96 7584 720 17916 2761740 9 0 0 5573376 0 2327616 0 558336 0 167616 0 19008 0 10200 8656152 10 0 0 11167488 0 7057440 0 2818176 0 1020384 0 235584 0 35040 22334112 11 0 0 4661568 0 8314272 0 8893248 0 6746688 0 2986560 0 818112 32420448 12 0 0 19008 0 123840 0 591744 0 2202912 0 6189120 0 9654240 18780864 13 0 0 0 0 0 0 0 0 288 0 39168 0 2127264 2166720 14 0 0 0 0 0 0 0 0 0 0 0 0 6624 6624 Total 1 241788 23918778 72780 18598527 12648 13004408 2736 10171560 1632 9483672 1200 12670110 88179840 <0.01% 0.27% 27.12% 0.08% 21.09% 0.01% 14.75% <0.01% 11.54% <0.01% 10.75% <0.01% 14.37%