Mathematical Game – Put Down
Put Down is a game where two players alternate taking turns. The last player to make a legal move is the winner. The goal is to place pennies on a bare rectangular surface until no more pennies can be placed. They cannot overlap, but can almost touch each other. The winner is the person who places the last penny.
Using notions of symmetry, player A (the first player) should place a penny in the center of the rectangle. Then after player B (the second player) places a penny anywhere in the rectangle, player A should place the third penny using 180 degree rotational symmetry from the second penny. This guarantees that player A will win. As long as there is room for player B to place a penny, player A will also have a spot for the rotated penny. When player B runs out of spots for pennies, player A will win because he placed the last penny.
Things are not so assured if you are not the first player. (Always good to be the first player!) If the other player doesn’t pick the center, just rotate what he does and you will win. If he picks the center, all you can hope for is that he doesn’t rotate your next move. And even then, you may lose.
Knowing that symmetry is good, it makes sense to pick the center first. The rectangle has 180 degree rotational symmetry as does the penny. Picking the center keeps the rotational symmetry. Then as you mirror each move the other person makes you also maintain rotational symmetry.