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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

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Try using the math.se search box and looking for "winning strategy" or something like that. –  David May 25 '14 at 4:36
Good idea, but I've seen many of the questions about winning strategies are either about complicated games, or don't have a clear winning strategy –  dREaM May 25 '14 at 4:40
How about this one? There is a small amount of information which needs to be memorised in order to play a perfect game, but not all that much. –  David May 25 '14 at 4:52

2 Answers 2

up vote 10 down vote accepted

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.

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oh, this is nice, I like how you can't approach it by looking at specific moves. –  dREaM May 25 '14 at 4:44
This is actually a strategy used occasionally in Go. It is not a winning strategy (in particular due to the use of komi), but it can take some skill to defeat. The first player will usually search for an opportune moment to break the symmetry. Similarly in chess mirroring your opponent's moves is known to be disadvantageous, but many games have been won by initially mimicking and then choosing a moment to break symmetry. Mimicking in either game can produce a psychological effect on the opponent. –  PrimeRibeyeDeal May 25 '14 at 15:43

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.

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