Permutations
There are 30 edges, each of which can be flipped into two positions, giving a theoretical maximum of 30!×230 permutations. This value is not reached for the following reasons:
- Only even permutations of edges are possible, reducing the possible edge arrangements to 30!/2.
- The orientation of the last edge is determined by the orientation of the other edges, reducing the number of edge orientations to 229.
- Since opposite sides of the solved puzzle are the same color, each edge piece has a duplicate. It would be impossible to swap all 15 pairs (an odd permutation), so a reducing factor of 214 is applied.
- The orientation of the puzzle does not matter (since there are no fixed face centers to serve as reference points), dividing the final total by 60. There are 60 possible positions and orientations of the first edge, but all of them are equivalent because of the lack of face centers.
This gives a total of possible combinations.
The precise figure is 72 431 714 252 715 638 411 621 302 272 000 000 (roughly 72.4 decillion on the short scale or 72.4 quintilliard on the long scale).
Read more about this topic: Alexander's Star
Famous quotes containing the word permutations:
“Motherhood in all its guises and permutations is more art than science.”
—Melinda M. Marshall (20th century)
“The new shopping malls make possible the synthesis of all consumer activities, not least of which are shopping, flirting with objects, idle wandering, and all the permutations of these.”
—Jean Baudrillard (b. 1929)