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# Penny Game

## Question

There are three one-dimensional tracks, of length 12, 7, and 5 spaces respectively. You start with pennies in the first space of each track; your opponent starts with pennies in the last space of each track. On your turn, you may move any one of your pennies any number of spaces in either direction along a track (as a chess rook), however you are not permitted to bypass the other player's penny or occupy its space. If a player has no legal move, he loses.
What should your first move be?
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First, notice that moving your penny backwards, towards your own starting position, is always a useless move. If you were losing before moving your penny backwards, your opponent moves their penny to the space immediately in front of yours, and you will still be losing. So for the purposes of forming an optimal strategy, we can ignore the ability to move backwards. This transforms the puzzle into a simple game of nim. There are 10, 5, and 3 empty spaces to be occupied. The binary representations are 1010, 0101, and 0011. With our first move, we want to achieve 0110, 0101, and 0011. This is done by taking away two empty spaces from the largest stack. So we should move our penny in the 12-track two spaces forward.

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