1. Construct a magic square of order 6.
2. (1) Is 4-pile Nim with heaps of sizes 22. 19. 14, and 11 balanced or unbalanced?
(2) Player ¥°'s first move is to remove 6 coins from the heap of size 19. What should player ¥±'s first move be?
3. Consider an m-by-n chessboard with m and n both odd. To fix the notation, suppose that the square in the upper left-hand corner os colored white. Show that if a white square is cut out anywhere on the board, the resulting pruned board has a perfect cover by dominos.
4. Show how to cut a cube, 3 feet on an edge, into 27 cubes. 1 foot on an edge, using exactly 6 cuts, but making a non-trivial rearrangement of the pieces between two of the cuts.
5. A bag contains 100 apples, 100 bananas, 100 oranges, and 100 pears. If I pick one piece of fruit out of the bag every minute, how long will it be before I am assured of having picked at least a dozen pieces of fruit of the same kind ?
6. Let n be a positive integer divisible by 4. say n=4m. Consider the following construction of an n-by-n array.
(i) Proceeding from left to right and from first row to nth row. fill in the place of the array with the integer 1,2,¡¦¡¦,n2 in order.
(ii) Partition the resulting square array into m2 4-by-4 smaller arrays. Replace each number a on the two diagonals of each of the 4-by-4 arrays with its "complement" n2+1-a. Verify that this construction produces a magic square of order n when n=4 and n=8.
7. Suppose we change the object of Nim so that the player who takes the last coin loses (the misere version). Show that the following is a winning strategy. Play as in ordinary Nim until all but exactly one heap contains a single coin. Then remove either all or all but one of the coins of the exceptional heap so as to leave an odd number of heaps of size 1.