Question 56
Consider a cube of edge $1~\mathrm{cm}$ and two vertices $A$ and $B$ that are diagonally opposite on a face of the cube. A path of length $n$ is a sequence of $n+1$ vertices such that two consecutive ones are at distance $1~\mathrm{cm}$. Then: which of the following numbers is larger: the number of paths of length $1000~\mathrm{cm}$ that start and end at $A$, or the number of paths of length $1000~\mathrm{cm}$ that start at $A$ and end at $B$? Justify the answer.