simple games with cute winning strategies?

Question

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

Answer 1

Person $A$ and person $B$ take turns placing pennies flat on a circular table (which can accommodate at least $1$ penny). A person loses if there is no valid placement possible.

Winning strategy: Person $A$ places a penny in the middle of the table, and then mirrors all of person $B$'s moves.

Answer 2

Here's one from an old Putnam.

Player $A$ and player $B$ take turns filling the entries of a $2n\times 2n$ matrix $X$. Player $A$ goes first. Player $A$ wins if $X$ is invertible and player $B$ wins if $X$ is not invertible.

Winning strategy: Player $B$ writes the negative of player $A$'s last number somewhere in the same row (even dimension is necessary to make this possible). Then the columns of $X$ sum to $\bf 0$ and $B$ wins.