I think it starts like this:<p>1) at most nb of stone increase by one each time one player plays<p>2) nb of stone cannot decrease unless the board is full of stones<p>3) no one can lose unless the board has been full of stone at least once;
this is because until the board has been full of stones it is possible to increase the nb of stone by adding one and this is a new position because of 2)<p>4) as soon as there are n - 1 stones, the player who puts the last stone wins; because there are n - 1 different positions left with n - 1 stones and n - 1 is even