Symmetry

Problem4. Two players take turns putting pennies on a round table, without piling one penny on the top of another. The player who cannot place a penny looses.

Solution. In this game the first player can always win, no matter how big the table may be! To do so, he must play the first penny so that it's center coincides with the center of the table. After this, he replies to each move of the second player by placing a penny in a position symmetric to the penny placed by the second player with respect to the center of the table. Notice, that in such a strategy the positions of the two players are symmetric after each move of the first player. It follows, that if there is a possible turn for the second player, then there is a possible response for the first player, who will therefore win.

Problem5. Two players take turns placing bishops on the squares of a chessboard, so that they cannot capture each other. (The bishops may be placed on squares of any color.) The player who cannot move loses.

Solution. Since a chessboard is symmetric with respect to it's center, it is natural to try a symmetric strategy. But this time, since one cannot place a bishop at the center of the chess boar, the symmetry will help the second player. It might seem, from analogy with the previous problem, that such a strategy would alow the second player to win. However, if he follows it, he cannot even make a second move! The bishop placed by the first player can take a bishop placed in the symmetric square.

This example shows that in employing a symmetric strategy one must take into account that a symmetric move can be blocked or prevented by a move the opponent has just made. Because of the symmetry, moves made earlier cannot affect a players move. To solve a game using a symmetric strategy, one must find a symmetry such that the previous move does not destroy the chosen strategy.

Therefore, to solve last problem we must look not to the point symmetry of the chessboard, but ti its line symmetry. We can choose, for example, the line between fourth and fifth rows as the line of symmetry. Squares which are symmetric with respect to this line will be of different colors, and therefore a bishop on one square cannot take a bishop on the symmetric square. Therefore, the second player can win this game.

Problem6. There are two piles of 7 stones each. At each turn, a player may take as many stones as he chooses but only from one of the piles. The loser is the player who cannot move.

Problem7. There are 20 points on a circle. Players take turns connecting two out of these 20 points by segments so that the new segments do not intersect segments that are already drawn. The player who cannot draw a segment loses. Define a winning strategy. Create similar problems with different starting numbers of points.