This question already has an answer here:
Given a puzzle of the following form:
Find a path between the top left corner to the bottom right corner, visiting each spot (.) exactly once. You can only move horizontally or vertically.
x . . . . . . . x
For a 3 x 3 grid, this yields two solutions:
x--.--. | .--.--. | .--.--x x .--. | | | . . . | | | .--. x
There are mul olutions for a 5x5 grid, 7x7, and so on.
But for a 2x2, 4x4, 6x6, and higher n x n (where n is even), this does not seem to yield any solution.
How would you prove that this is the case? That there is no solution for n x n grids where n is even? (Is this even true?)