Lars Vandenbergh's CubeZone

Speedcubing taken one step further

Cross study

This short article describes the results of a computer analysis I did for solving the cross of the first layer in the least number of moves. In this study we are trying to determine the required number of moves to the solve the cross for all possible cases if we would always be able to see an optimal solution (God's algorithm). From that information we can then calculate the average and maximum number of moves that a "perfect" cross solver would need to solve the cross from a random state.

Most cross-first speedcubers always start with the same color when building the cross. A lot of people can still comfortably build the cross on the opposite face of their prefered color, and a few people are completely color-neutral and start building the cross on whichever face it seems the easiest. In this computer study we have investigated all three scenarios.

Fixed color cross solving

When solving the cross on the same face, there are 4 edge pieces that we need to take into account. They can each be oriented in 2 ways and can placed in 12 locations. The amount of cases we need to explore in this scenario is 2⁴ x (12 x 11 x 10 x 9) = 190,080. In the following table you can see how many cases can optimally be solved in a certain number of moves, both in face turn metric and quarter turn metric:

Face turn metric

Quarter turn metric

Depth	# cases	Distribution	Cummulative
0	1	0.00%	< 0.01%
1	15	0.01%	0.01%
2	158	0.08%	0.09%
3	1,394	0.73%	0.82%
4	9,809	5.16%	5.99%
5	46,381	24.40%	30.39%
6	97,254	51.16%	81.55%
7	34,966	18.40%	99.95%
8	102	0.05%	100.00%

Depth	# cases	Distribution	Cummulative
0	1	0.00%	< 0.01%
1	10	0.01%	0.01%
2	73	0.04%	0.04%
3	500	0.26%	0.31%
4	3,078	1.62%	1.93%
5	15,528	8.17%	10.10%
6	57,180	30.08%	40.18%
7	91,654	48.22%	88.40%
8	21,849	11.49%	99.89%
9	207	0.11%	100.00%

Average: 5.81 moves

Average: 6.59 moves

Opposite color cross solving

When solving the cross on either of two opposite faces, there are 8 edge pieces that we need to take into account. They can each be oriented in 2 ways and can placed in 12 locations. The amount of cases we need to explore in this scenario is 2⁸ x (12 x 11 x 10 x 9 x 8 x 7 x 6 x 5) = 5,109,350,400. In the following table you can see how many cases can optimally be solved in a certain number of moves, both in face turn metric and quarter turn metric:

Face turn metric

Quarter turn metric

Depth	# cases	Distribution	Cummulative
0	53,759	0.00%	< 0.01%
1	806,253	0.02%	0.02%
2	8,484,602	0.17%	0.18%
3	74,437,062	1.46%	1.64%
4	506,855,983	9.92%	11.56%
5	2,031,420,585	39.76%	51.32%
6	2,311,536,662	45.24%	96.56%
7	175,751,822	3.44%	> 99.99%
8	3,672	0.00%	100.00%

Depth	# cases	Distribution	Cummulative
0	53,759	0.00%	< 0.01%
1	537,496	0.01%	0.01%
2	3,920,873	0.08%	0.09%
3	26,775,612	0.52%	0.61%
4	162,620,494	3.18%	3.80%
5	773,798,728	15.14%	18.94%
6	2,260,615,130	44.24%	63.18%
7	1,794,284,224	35.12%	98.30%
8	86,731,327	1.70%	> 99.99%
9	12,757	0.00%	100.00%

Average: 5.39 moves

Average: 6.15 moves

Color neutral cross solving

When solving the cross on any of the six faces, there are 12 edge pieces that we need to take into account. They can each be oriented in 2 ways and can placed in 12 locations. However since we are using all the edge pieces and edge pieces can only be flipped in pairs, the orientation of the 12th edge can be derived from the orientation all the others. The amount of cases we need to explore in this scenario is 2¹¹ x 12! = 980,995,276,800. In the following table you can see how many cases can optimally be solved in a certain number of moves, both in face turn metric and quarter turn metric:

Face turn metric

Quarter turn metric

Depth	# cases	Distribution	Cummulative
0	30,942,374	0.00%	< 0.01%
1	462,820,266	0.05%	0.05%
2	4,839,379,314	0.49%	0.54%
3	41,131,207,644	4.19%	4.74%
4	239,671,237,081	24.43%	29.17%
5	543,580,917,185	55.41%	84.58%
6	151,019,930,400	15.39%	99.97%
7	258,842,496	0.03%	> 99.99%
8	40	0.00%	100.00%

Depth	# cases	Distribution	Cummulative
0	30,942,374	0.00%	< 0.01%
1	308,828,676	0.03%	0.03%
2	2,244,689,022	0.23%	0.26%
3	15,116,501,844	1.54%	1.80%
4	86,723,043,456	8.84%	10.64%
5	333,077,773,019	33.95%	44.60%
6	475,482,906,734	48.47%	93.07%
7	67,953,971,216	6.93%	99.99%
8	56,619,244	0.01%	> 99.99%
9	1,215	0.00%	100.00%

Average: 4.81 moves

Average: 5.50 moves