Question 105
Consider a regular cube with side length $2$. Let $A$ and $B$ be two vertices that are furthest apart. Construct a sequence of points on the surface of the cube $A_1, A_2, \dots, A_k$ so that $A_1 = A$, $A_k = B$ and for any $i = 1, \dots, k-1$, the distance from $A_i$ to $A_{i+1}$ is $3$. Find the minimum value of $k$.