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ASU 262 All Soviet Union MO 1978 checkers on a chessboard

Source:

July 11, 2019
game strategycombinatoricsChessboard

Problem Statement

The checker is standing on the corner field of a n×nn\times n chess-board. Each of two players moves it in turn to the neighbour (i.e. that has the common side) field. It is forbidden to move to the field, the checker has already visited. That who cannot make a move losts.
a) Prove that for even nn the first can always win, and if nn is odd, than the second can always win.
b) Who wins if the checker stands initially on the neighbour to the corner field?
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