Recreational Games

Here are some interesting games to play, and perhaps analyze.
  1. (Nim) This is a game for 2 players. There are three stacks of coins. The players take it in turns to remove some number of coins (greater than zero) from only one of the stacks (the whole stack could be removed). The winner is the player who removes the last coin. Can you generalize to N stacks. With three stacks, the game is called NIM.
  2. This is a two player game. 50 coins are arranged in a line (each of different, arbitrary, denominations). A player may pick a coin from one or other end. Players alternate until there are no more coins left. The loser has the smaller amount of money. Can the first player always guarantee herself not to be the loser? Is there an efficient algorithm to compute the optimal strategy?
  3. Place (say) 20 red points and 20 blue points on a plane piece of paper. No three should be on a line. Two players, the red player and the blue player take turns connecting a pair of points of their color with a straight line so that no lines intersect. The loser is the first player who has to draw a line that intersects an existing line. For any arrangement of the points is it always possible to place the n red lines and blue lines so that no two lines intersect.
  4. Sam Lloyd's matrimony game: Two players alternately pick integers in the range 1-5, and add it to a running total which is initially zero. The constraint is that a player cannot pick the same integer that was picked by her opponent on the previous turn. The loser is the player who picks the integer that takes the running total above 45. Do you wish to go first or second, and what is your strategy?
  5. A game is to be played on a circular table. Quarters are to be placed non overlappingly on the table. You alternate turns with your opponent. The last person to place a quarter on the table wins. Do you want to start or go second?