Question 91
Number the vertices and edges of a regular $n$-gon clockwise from 1 to $n$, so that edge $i$ follows vertex $i$ ($1\le i\le n$). Assign nonnegative integers $e_i$ to its vertices and $k_i$ to its edges such that:
(1) $(e_1,e_2,\ldots,e_n)$ is a permutation of $(k_1,k_2,\ldots,k_n)$.
(2) For every $1\le i\le n$, $k_i=|e_{i+1}-e_i|$, where $e_{n+1}=e_1$.
a) Prove that for every natural number $n\ge3$, such tuples exist other than $(0,\ldots,0)$.
b) For each positive natural number $m$, determine the smallest natural number $n$ for which both tuples contain every natural number from $0$ to $m$.
(1) $(e_1,e_2,\ldots,e_n)$ is a permutation of $(k_1,k_2,\ldots,k_n)$.
(2) For every $1\le i\le n$, $k_i=|e_{i+1}-e_i|$, where $e_{n+1}=e_1$.
a) Prove that for every natural number $n\ge3$, such tuples exist other than $(0,\ldots,0)$.
b) For each positive natural number $m$, determine the smallest natural number $n$ for which both tuples contain every natural number from $0$ to $m$.