Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

This is a Tough Nut. To crack it you have to prove that, for all frameworks of the type described in this question, no two paths from different starting points ever end up at the foot of the same upright. In other words we have to show that the system described always produces a permutation (a re-ordering) of the numbers $1$ to $n$ which occur at the top of the uprights. In this case the permutation is

$ 12345 \rightarrow 53124 $