An absent-minded bank teller switched the dollars and cents when he cashed a check for Mr. Brown, giving him dollars instead of cents and cents instead of dollars. After buying a five-cent newspaper, Mr. Brown discovered that he had exactly twice the amount of his original check left. What was the amount of the check?
To determine the value of Mr. Brown’s check, let x stand for the dollars and y for the cents. The problem can now be expressed by the following equation: 100y + x – 5 = 2(100x + y). This reduces to 98y – 199x = 5, a Diophantine equation with an infinite number of integral solutions. Only one solution, however, meets the problem’s condition that the value of y be less than 100. This solution is x = 31 and y = 63, making Mr. Brown’s check $31.63.
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A version of this puzzle originally appeared in the May 1959 issue of Scientific American.