2022/23 AHSMC Part 1 Problem 7

Problem 7 of 16

The faces of a die show distinct positive integers from one to six and the opposite sides sum to seven. You roll two dice and determine that the product of the two numbers on the top faces is equal to the sum of the two numbers on the bottom faces. What is the sum of the two numbers on the top faces?

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Solution

Suppose the top faces show the integers $x$ and $y$ with $1 \le x, y \le 6$ . Then $xy = (7-x) + (7-y)$ gives $y = (14-x)/(x+1)$ . Substituting $x = 1, 2, 3, 4, 5, 6$ gives $y = 13/2$ , $4$ , $11/4$ , $2$ , $3/2$ , $8/7$ , respectively. Thus $(x,y) = (2,4)$ or $(x,y) = (4,2)$ implying that $x+y = 6$ . The answer is (B).

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