KGS math club/solutio prince

Only for the 1x1 board.

Consider a board with at least two rows and columns and assume a Hamiltonian circuit as described in the problem. Let B be the square in the upper right corner of the board, A the square to the left of it and C the square below it. The square B may only be entered with a move to the east and may only be left with a move to the south. That is, the short path A -> B -> C is a forced part of the given circuit. Similarly, there is a forced path X -> Y -> Z, where Y is the square in the lower left corner, X the square above it and Z the square to the right of it.

Now, on the circuit, there has to be some path P from Z to A. Similarly, there has to be a path Q from C to X. For topological reasons, P and Q have to intersect somewhere on the board. But given the available moves, the prince can never cross its own path.