Problem 5
Two squirrels, Bushy and Jumpy, have collected walnuts for the winter. Jumpy numbers the walnuts from through and digs little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts by performing a sequence of moves. In the -th move, Jumpy swaps the positions of the two walnuts adjacent to walnut . Prove that there exists a value of such that, on the -th move, Jumpy swaps some walnuts and with .
Step 1 of 5: Assume no move ever straddles its own number
Detailed analysis
Suppose for contradiction that for every , the two walnuts swapped on move satisfy or , i.e. never .