Posted: Dec 16, 07 6:44am

SOLUTIONS TO PUZZLE SET #1

Puzzle #1

The center (circled) digit must be 4. This diagram shows a solution: all rows and columns add up to 13, and the digits 1-9 are each used only once. Other solutions can be obtained by switching the 9 and the 3, the 5 and the 6, and reversing the entire string, for a total of eight different solutions, but all of them have the digit 4 in the middle square.

Puzzle #2

For the ascending sequence, the solution that uses the fewest number of symbols (3) is

123-45-67+89=100

For the descending sequence, the solution that uses the fewest number of symbols (4) is

98-76+54+3+21=100

Puzzle #3

The game described is equivalent to playing tic-tac-toe on a 3x3 magic square where each row, column, and diagonal add up to 15. There are exactly eight ways that three different numbers from one to nine can add up to 15, and these eight ways are represented by the three rows, three diagonals, and two diagonals of a magic square. The best strategy is therefore the same as for tic-tac-toe; for example, if the first player chooses a five / center square, then the best response is to choose an even number / corner square. As with tic-tac-toe, the game is a draw if both players follow their best strategy.

8 1 6

3 5 7

4 9 2

Puzzle #4

Here are the eleven solutions:

532 x 14 = 98 x 76 = 7,448

584 x 12 = 96 x 73 = 7,008

174 x 32 = 96 x 58 = 5,568

158 x 32 = 79 x 64 = 5,056

186 x 27 = 93 x 54 = 5,022

259 x 18 = 74 x 63 = 4,662

146 x 29 = 73 x 58 = 4,234

174 x 23 = 69 x 58 = 4,002

134 x 29 = 67 x 58 = 3,886

138 x 27 = 69 x 54 = 3,726

158 x 23 = 79 x 46 = 3,634

Puzzle #5

The only solution is 3 x 54 = 6 x 27 = 9 x 18