A robot is placed on the top-left vertex of a 3x3 grid. At each move the robot can take one step (up, down, left or right) along the edge of the grid to the adjacent vertex, but he cannot go outside the grid. The robot can revisit vertices and edges. What is the least number of moves required for it to visit every edge of the grid? Good luck!
The minimum number of steps is
How to do it
Proof that this is minimal
The graph has $24$ edges in all.
Eight of the vertices have order $3$. This means that these vertices must be visited at least twice and there is an overlapping edge whenever the second incidence occurs. This means there will be at least four edges visited twice and $24 + 4 = 28$ and this minimum is achieved when we ensure the ends of the overlapping edges are on vertices of order $3$.