How to prove prove k-cube graph is distance transitive?

Question

The $k$-cube is the graph with vertex set $\{0,1\}^k$ such that any two vertices $x,y ∈\{0,1\}^k$ are joined by an edge if and only if x and y differ in exactly one coordinate. $$$$ A graph is distance transitive, if $∀,,,∈$ with $(,)=(,)$, then there exists $∈()$, so that $()=$ and $()=$.

Answer 1

Hint:

Induct on $k$, and make note of the case when $d(u,v)=k$.