Im thinking of games of two players ($A$ goes first and $B$ second) like the following:
There are 35 chips in a table, during each turn a player can remove 1,2,3 or 4 chips. Prove player $B$ can always win (here the trick is to for $B$ to always leave $A$ with a multiple of 5 number of chips.
There are two piles in a table, one with $2013$ chips and the other with $4017$ chips. During each turn a player must select a pile and remove a positive integer number of chips, the player that removes all the chips wins. Prove player $A$ can always win. (here the trick is for player $A$ to always leave both piles with the same number of chips.
The nim game.
In each turn a player places a knight in a position not threatened by another knight. Prove player $B$ can always win (player $B$ always choses the spot that is mirrored by $A$ over the diagonal, so if $A$ picks $(x,y)$ $B$ picks $(8-x,8-y)$.
and other examples