Stats ​

Observations ​

We can make the following observations after the full search:

• Totals:

  Quantity	Value
  Number of solutions	0
  Number of snakes	0

• Number of paths with start/end positions:

  Start/End Cubelet	Value	Fraction
  Corner	0	0.00 %
  Face	0	0.00 %
  Edge	0	0.00 %
  Center	0	0.00 %

Distribution ​

To get an overview of the distribution of solutions, the snakes are aggregated by:

• number of straight cubelets on the horizontal axis
• number of solutions on the vertical axix

The right most colmuns are for each line i.e. a given number of solutions:

• the total number of sequences
• the total number of solutions

The bottom row is the sum of each column. We can check that in total:

• the number of snakes is the expected 0
• the number of solutions is the expected 0

This operation is done for:

• all snakes
• only palindromic snakes

All Snakes ​

Distribution of solutions for all snakes:

Palindromic Snakes ​

Distribution of solutions for palindromic snakes only, i.e. those which are symmetrical by reversing cubelet order:

Remarks ​

How frequent are snake cubes ?

The total number of candidate snakes is 2^(27-2)/2 = 0 (each cubelet is either a straight or a turn, except the 2 ends, and a sequence and its reverse represent the same snake).

The number of plausible candidates snakes (i.e. excluding the snakes that obviously cannot be folded into a 3x3x3 cube, i.e. the snakes with 2 or more consecutive straights) is 0 (determined by brute force).

So very Few (0.00 %) of all candidates are plausible snakes, but a decent number (0.00 %) of all plausible snakes are actual snake cubes. In other words, plausible candidates make it quite often.