Well now, seekers, do you remember the pretty combinatorial graphs I was playing with a few weeks ago? I only showed you one of them before, which I call the elephant because it looks like an elephant seen from the underneath.
The other one is here (the edges represent transposition of adjacent elements). I'm not quite sure what to call it - it reminds me a little of a trampoline for some reason.
Print out one of these graphs, and choose two vertices which are an odd number of edges apart. Now take a highlighter pen, and try and draw a continuous line which
It's good fun, the results are pretty, it can be suprisingly difficult, and there's even a serious point to it.
[[ The graphs have a lot of symmetry, so there are only five essentially different ways to choose an odd pair. So you only have to solve ten of these problems (five for each graph), and you can be sure that it's always possible. Furthermore, any way of breaking all four-element permutations into pairwise element swaps can be reduced to one of these two graphs. Then you can use a fairly simple inductive argument to extend the result to any such graph of the permutations of four or more elements, and that's the central result of Maurice Tchuente's paper. ]]