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.