91Ó°ÊÓ

Show that in an unbalanced game of Nim in which the largest unbalanced bit is the \(j\) th bit, player I can always balance the game by removing coins from any heap the base 2 numeral of whose number has a 1 in the \(j\) th bit.

A game is played between two players, alternating turns as follows: The game starts with an empty pile. When it is his turn, a player may add either 1,2, 3. or 4 coins to the pile. The person who adds the 100 th coin to the pile is the winner. Determine whether it is the first or second player who can guarantee a win in this game. What is the winning strategy?

A Latin square of order \(n\) is idempotent provided the integers \(\\{1,2, \ldots, n\\}\) occur in the diagonal positions \((1,1),(2,2), \ldots,(n, n)\) in the order \(1,2, \ldots, n\), and is symmetric provided the integer in position \((i, j)\) equals the integer in position \((j, i)\) whenever \(i \neq j .\) There is no symmetric, idempotent Latin square of order 2. Construct a symmetric, idempotent Latin square of order 3. Show that there is no symmetric, idempotent Latin square of order \(4 .\) What about order \(n\) in general, where \(n\) is even?

Take any set of \(2 n\) points in the plane with no three collinear, and then arbitrarily color each point red or blue. Prove that it is always possible to pair up the red points with the blue points by drawing line segments connecting them so that no two of the line segments intersect.

Consider an \(n\) -by- \(n\) board and \(L\) -tetrominoes ( 4 squares joined in the shape of an L). Show that if there is a perfect cover of the \(n\) -by- \(n\) board with \(L\) -tetrominoes, then \(n\) is divisible by 4 . What about \(m\) -by- \(n\) -boards?