Problem of the Month
Hint for Problem 7: April 2023

1. Suppose $d(\mathit{a},\mathit{b})=k$ for some $k$. Try to construct a path of length $k$ in the natural graph from the vertex labelled $\mathit{a}$ to the vertex labelled $\mathit{b}$.

2. For fixed $\mathit{a}\in A_n$, how many $\mathit{b}\in A_n$ have the property that $d(\mathit{a},\mathit{b})=k$?

3. Find a function that works for $n=2$ and use this to build one for $n=3$. It might be useful to think of the natural graph of $A_2$ as a square and the natural graph of $A_3$ as a cube. As well, a cube can be thought of as two squares on top of each other with vertical edges connecting corresponding vertices in the top and bottom faces.