This is a question from the German Math Olympiad from 2020. We were given that problem in math club, but weren't provided with a solution, so I would appreciate any help!
Bob moves a knight on an $n * n$ chessboard $(n ≥ 3)$ with as few moves as possible, from the field in the lower left corner to the field in the lower right corner. Then Anne takes this knight and moves from the square in the lower left corner with as few moves as possible to the square in the upper right corner. For which $n$ do both need the same number of moves?
My Progress so far: We call $B(n)$ the lowest number of possible moves for Bob and $A(n)$ the lowest amount of possible moves for Anne.
We designate the field in the lower left corner with $l_u$, the one in the lower right corner with $r_u$ and that in the upper right corner with $r_h$. On a $7*7$ and a $3*3$ chessboard we label the field at the bottom left with 0, then all fields that we can reach from here with a knight move with "1", then all fields that can be reached by such a field with another move can be reached and are not yet labeled, with "2" etc. until we have reached $r_u$ and $r_h$. With this we quickly see that $B (7) = A (7) = 4$ and $2 = B (3) ≠ A (3) = 4$
I was thinking that for $n>3$ it must always be true that $n=7$, but I don't know how to prove that.