Ten Billion Barrel Puzzle

This puzzle was designed by Gunpei Yokoi and originally appeared in 1980. The physical puzzle consists of five rotating rings of beads in a clear plastic cylinder with a plunger mechanism that shifts three of the columns up and down. Counting the number of positions that can be achieved requires some clarification: as mentioned above, all 23!/2 (approximately ) even permutations of the beads can be achieved, but in the physical puzzle, beads of a given color are indistinguishable, so a more realistic count would reduce that by a factor of , leaving approximately visually distinct states. Since an odd permutation of the beads can be changed to an even permutation by transposing a pair of indistinguishable beads, any desired arrangement of colors can be achieved in the puzzle when labels are turned off. One further twist: the physical puzzle is symmetric top-to-bottom, and can be turned upside down to effect an odd permutation of the beads (hence making all permutations of the beads possible).

More information, including several algorithms for solving the puzzle, can be found at Jaap Scherphuis's ten billion barrel page, which is also the source for several of the pretty patterns provided here.