# Essays/InsanityCube

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The perlmonks node Challenge: "Insanity" Cube Puzzle presents an interesting puzzle game.

The puzzle consists of four cubes. Each cube has each six sides colored in one of four colors: green, purple, red, yellow. The patterns are the the following, listed for each cube in the order *left, front, right, back, top, bottom*.

pgpygr rprrgy ppryyg rrygpy

The goal is to build a tower of four cubes rotating the four cubes in such a way that on any side of the tower you see four different colors.

The puzzle is easy to solve with a brute force solution in J, so here's what I (B Jonas) wrote (it is also posted in that thread).

NB. Insanity cube puzzle: http://www.perlmonks.com/?node_id=559228 NB. Solution by ambrus. NB. Very inefficent wrt memory and somewhat inefficent wrt speed, NB. but no need to optimize on a modern machine. NB. a cube is a list of 6 sides (each atoms): NB. left front right back top bottom. NB. the first four of these sides are visible on the sides in a tower. NB. a towers is a list of 4 cubes. NB. colorings. input =: >;: 'pgpygr rprrgy ppryyg rrygpy' NB. all 24 rotations of a cube. gens =: (i.6),'NSKTFL'i.'FSLTKN',:'SKTNFL' rots =: ([:~.[:/:~[:,/{"1/)^:_~ gens NB. all 24 rotations of each 4 cube. posn =: rots {"_ 1 input NB. stack a list of 4 lists of 24 cubes to a tower of 4 cubes in each 24^4 way, NB. order of 4 cubes doesn't matter. stack =: [:>[:>[:,[:{[:<"1<"1 towers =: stack posn NB. check if a cube is good or bad -- rank 2, gives bool atom. good =: [: (*./"1) 4 ({."1) 4 = [: (#@:~."1) |:"2 solns =: (good # ]) towers NB. each solution gives 8 trivially equivalent solutions because you can NB. rotate all cubes simultanously around the vertical axis or twice around NB. the horizontal axis. (this may be less than 8 for degenerate cubes.) NB. rota y gives all 8 rotations of a tower y rott =: (2{gens)&({"_ 1) rotu =: ({~1{gens)&({"_ 1) rota =: ([: ,/ [: rott^:(<4) rotu^:(<2))"2 NB. verify that each rotation of each solution is indeed among the solutions. assert =: 3 :'assert. y' assert *./ solns e.~ ,/ rota solns NB. now normalize the solutions by changing a tower to the NB. lexicographically first out of these eight variants, NB. and weed out repetitions. norml =: ([:{.[:/:~rota)"2 solns1 =: ~./:~ norml solns NB. output the solutions. there's just one if you use the sample NB. cube colorings in the post. echo solns1 echo #solns1 NB. end

The output follows (there's just one solution for this pattern).

ggrypp ypgrrr ryppyg prygyr 1

See also Doc/Articles/Play173

-- B Jonas <<DateTime(2008-08-06T19:57:49Z)>>