Consider a path of length 2n with San = 0.

We order the sides in circular order by identifying 0 and 2n with the result that the first and the last side become adjacent. Applying a cyclical permutation amounts to viewing the same closed path with (k, S,) as origin. Show that this preserves maxima, but moves them k steps ahead. Conclude that when all 2n cyclical permuta- tions are applied the number of times that a maximum occurs at r is independent of r. Consider now a randomly chosen path with Sn = 0 and pick the place of the maximum if the latter is unique; if there are several maxima, pick one at random. This procedure leads to a number between 0 and 2n - 1.

Show that all possibilities are equally probable.