Lars Vandenbergh's CubeZone

Speedcubing taken one step further

Extended cross study

This short article describes the results of a computer analysis I did for solving the extended cross (a.k.a. x-cross) in the least number of moves. In this study we are trying to determine the required number of moves to solve the cross and one F2L pair simultaneously 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" extended cross solver would need to solve the extended cross from a random state.

Fixed cross color, fixed pair extended cross solving

When solving the cross on a fixed cross color and always solving the same specific pair with it, there is one single goal state.

There are 5 edge pieces and 1 corner piece that we need to take into account. The edge pieces be oriented in 2 ways and can placed in 12 locations. The corner piece can be oriented in 3 ways and can be placed in 8 locations. The amount of cases we need to explore in this scenario is 2⁵ x (12 x 11 x 10 x 9 x 8) x 3 x 8 = 72,990,720. In the following table you can see how many cases can optimally be solved in a certain number of moves in face turn metric:

Face turn metric

Depth	# cases	Distribution	Cummulative
0	1	< 0.01%	< 0.01%
1	15	< 0.01%	< 0.01%
2	172	< 0.01%	< 0.01%
3	1,950	< 0.01%	< 0.01%
4	21,535	0.03%	0.03%
5	220,368	0.30%	0.33%
6	1,989,591	2.73%	3.06%
7	13,431,990	18.40%	21.46%
8	40,963,892	56.12%	77.58%
9	16,325,184	22.37%	99.95%
10	36,022	0.05%	100.00%

Average: 7.98 moves

Fixed cross color, any pair extended cross solving

When solving the extended cross on a fixed cross color and solving any of the four F2L pairs with it that are available, there are four possible goal states:

There are 8 edge pieces and 4 corner pieces that we need to take into account. The edge pieces can each be oriented in 2 ways and can placed in 12 locations. The corner pieces can each be oriented in 3 ways and can placed in 8 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) x 3⁴ x (8 x 7 x 6 x 5) = 695,280,402,432,000. In the following table you can see how many cases can optimally be solved in a certain number of moves in face turn metric:

Face turn metric

Depth	# cases	Distribution	Cummulative
0	37,908,599	<0.01%	<0.01%
1	540,245,986	<0.01%	<0.01%
2	7,051,475,522	<0.01%	<0.01%
3	78,051,801,545	0.01%	0.01%
4	818,309,752,050	0.12%	0.13%
5	8,056,952,195,999	1.16%	1.29%
6	65,868,170,742,174	9.47%	10.76%
7	305,428,202,215,150	43.93%	54.69%
8	305,686,723,917,403	43.97%	98.66%
9	9,336,359,900,311	1.34%	>99.99%
10	2,277,261	<0.01%	100.00%

Average: 7.34 moves