KGS math club/solutio prince

Only for the 1x1 board.

Consider a board with at least two rows and columns and label the squares in the upper right resp. lower left as follows.

Now, assume a Hamiltonian circuit, as in the problem description. Given the available moves, the prince may enter the square B only with a move to the east and may leave it only with a move to the south. That is, the short path A -> B -> C is a forced part of the circuit.

Similarly, there is a forced path X -> Y -> Z on the circuit.

Now, on the circuit, there has to be some path P from C to X. Similarly, there has to be a path Q from Z to A. 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. This is easily seen as follows. Draw the circuit as a piecewise linear curve from square center to square center. It is clear, that two such line segments may only intersect when they are diagonal and perpendicular to each other. But the prince has only one diagonal move at his disposal.

Note: This argument applies not only to rectangular boards but to any board having an upper right resp. lower left corner in the sense of this argument.