The original (3×3×3) Rubik's Cube has eight corners and twelve edges. There are 8! <http://en.wikipedia.org/wiki/Factorial> (40,320) ways to arrange the corner cubes. Seven can be oriented independently, and the orientation of the eighth depends on the preceding seven, giving 37 (2,187) possibilities. There are 12!/2 (239,500,800) ways to arrange the edges, since an odd permutation of the corners implies an odd permutation of the edges as well. Eleven edges can be flipped independently, with the flip of the twelfth depending on the preceding ones, giving 211 (2,048) possibilities.[19]<http://en.wikipedia.org/wiki/Rubik%27s_Cube#cite_note-18> [image: {8! \times 3^7 \times 12! \times 2^{10}} \approx 4.33 \times 10^{19}]
There are exactly 43,252,003,274,489,856,000 permutations<http://en.wikipedia.org/wiki/Permutation>, which is approximately forty-three quintillion<http://en.wikipedia.org/wiki/Quintillion>. The puzzle is often advertised as having only "billions<http://en.wikipedia.org/wiki/1000000000_%28number%29>" of positions, as the larger numbers could be regarded as incomprehensible to many. To put this into perspective, if every permutation of a 57-millimeter<http://en.wikipedia.org/wiki/Millimeter>Rubik's Cube were lined up end to end, it would stretch out approximately 261 light years <http://en.wikipedia.org/wiki/Light_years>. Alternatively, if laid out on the ground, this is enough to cover the earth with 273 layers of cubes, recognizing the fact that the radius of the earth sphere increases by 57 mm with each layer of cubes. The preceding figure is limited to permutations that can be reached solely by turning the sides of the cube. If one considers permutations reached through disassembly of the cube, the number becomes twelve times as large: [image: {8! \times 3^8 \times 12! \times 2^{12}} \approx 5.19 \times 10^{20}.] The full number is 519,024,039,293,878,272,000 or 519 quintillion<http://en.wikipedia.org/wiki/Quintillion>possible arrangements of the pieces that make up the Cube, but only one in twelve of these are actually solvable. This is because there is no sequence of moves that will swap a single pair of pieces or rotate a single corner or edge cube. Thus there are twelve possible sets of reachable configurations, sometimes called "universes" or "orbits<http://en.wikipedia.org/wiki/Orbit_%28group_theory%29>", into which the Cube can be placed by dismantling and reassembling it. *http://en.wikipedia.org/wiki/Rubik%27s_Cube* --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Algorithm Geeks" group. To post to this group, send email to algogeeks@googlegroups.com To unsubscribe from this group, send email to algogeeks+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/algogeeks -~----------~----~----~----~------~----~------~--~---
