# The 4x4x4 can be solved in 77 single-slice turns

Previously I announced that the 4x4x4 cube could be solved in 79 single-slice turns
by solving it in five stages,
in a manner similar to the Thistlethwaite 4-stage solution for the 3x3x3.
(See
The 4x4x4 can be solved in 79 moves (STM).)
However, I have now realized my solution to the 2nd stage could have allowed
the use of more basic turns than I used.
I have realized that:

< U,u,D,d,L^{2},l^{2},R^{2},r^{2},F^{2},f,B^{2},b >
= < U,u,D,d,L^{2},l,R^{2},r,F^{2},f,B^{2},b >

So with l and r replacing generators l^{2} and r^{2},
you still can not reach any additional positions.
As a result, I should have included the moves { l, l', r, r' }
along with the other 24 allowed slice turns for that stage.

I have therefore recalculated the solution for Stage 2 using the additional basic moves. This new analysis shows that (in the worst case) two less moves are required than what the previous calculation produced. Therefore, the sum of the worst case values for each of the five stages comes to 77 instead of 79 using the new analysis.

For this new analysis, the distribution of positions at various distances from the solved state for Stage 2 are given below.

Stage 2 Slice turns ------------------------ distance positions unique -------- --------- ------ 0 24 14 1 72 17 2 1,368 194 3 18,508 2,455 4 207,996 26,549 5 2,299,348 290,234 6 23,858,972 2,991,456 7 185,459,676 23,195,543 8 895,556,404 111,954,098 9 2,495,445,552 311,942,717 10 3,861,651,288 482,738,888 11 4,430,801,152 553,917,154 12 5,214,924,836 651,928,238 13 3,656,282,908 457,074,674 14 853,364,112 106,680,277 15 2,975,072 372,288 16 112 14 -------------- ------------- 21,622,847,400 2,703,114,810

I note that my results for what I refer to as "twist turns" and "block turns" may also be improved. I will plan to add follow-ups to this post when I complete those analyses.