The first shuttle reorders the top labels, the second reorders thus reordered labels, the third changes the label order obtained after the second shuttle, and so on. Since this is also true of the n-th shuttle as well, this is true for any n shuttles. The mathematical term for any reodering operation is a permutation . A simple permutation that only exchanges two elements is known as a transposition . All permutations of the same set of indices form a group of permutations .
Every permutation is a product of transpositions .
So here it is, another example of a good answer to the what question. Easy to understand, entertaining to experiment with, and with a good deal of mathematical contents. It may even serve as an introduction into the theory of permutation groups. I am very satisfied to have put such a nice example up on the Web. To wind up the discussion there is a small question intended for those boys and girls who may stumble onto this page by mere chance.