George Duke wrote on Tue, May 20, 2014 03:44 PM UTC:
Of course it's better to have correspondence to normal Chess playing
space. There is readable 6x6 board on the surface. Instead, how to
generate visualizable 8x8 64 squares equitably from surface having 54
squares? Or map 54 -> 64 squares the best way?
Fortunately, the Rubik's surface it's easy, given logic of the geometry.
Think perimetres.
(1) Open and flatten the surface of 6 faces each given Rubik's possibility
into natural "Cross," lengthwise 9x3 with adjacent 3x3 above and 3x6
below. Topological equivalence keeps with pre-existing Rubik's state
complete exterior. There is only one natural flattening once top-side
accepted.
(2) The wanted commonplace 8x8 has 4 decreasing perimetres inwardly: 28,
20, 12 and 4 (the ballied central four).
(3) The Rubik's variant chessboard 2-d generated in '1' above has two
perimetres: 38 outside and 16 inside by inspection.
(4) Map by algorithm, pre-determined as to starting points, first, the 38
of given Rubik's to 28 exterior of 8x8. That leaves perfect '10' for
the 8x8 second perimetre (of 20). Use it again, the 10, that is use the
sub-leg of 10 from Rubik's exterior twice to fill in the 8x8 second
perimetre. Finally, the 16 of "Rubik's inside" and 16 of "64-square
inside" happen numerically to match exactly, to complete the map.
(5) So any state of (3x3x3) Rubik's 5 x 10^26 possibilities, or so by
differing reckonings, gives one and only one potential Chessboard
piece-arranged. For piece set-ups, no colours having been used yet,
Frolov's colouring represents the 6 Chess pieces for problems. Only small
fraction will be useful and fulfill Chess problem with Solution. By
Frolov's method, just looking at any given classical Rubik's permutation, six-coloured as each one is,
should immediately bring to anyone's mind a unique Orthodox Chess board position too -- to solve or not.
/// Converse could be challenging. There are far, far more Chess positions than Rubik's possibilities. So most cannot uniquely transform to a Rubik's state. Yet prove or show that some even very conveniently-chosen Chess position(s) thus is not representational accurately by "a Rubik's."
