Problem 6 (Missing Numbers Problem)

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In the following multiplication

every letter and dot stands for one of the numbers 0, 1, 2, ... ,8, and 9 with identical letters standing for identical numbers. Fill in the letters and dots with numbers that make the multiplication correct.

Solution
If you look carefully at the first multiplication of PQQP by P, you will see that since P ² = T with no carry-over number, and so we can conclude that the number P = 0, 1, 2, or 3. But since P ² = T is different from P, we can also conclude P = 2 or 3. Also, there is no carry-over number when P is multiplied by Q and so PQ = Q and hence we have Q = 0. But we can also conclude that P is not 2 since if P = 2, we would have

PQQP x P ² = 2002 x 4 = 8008

(remember P = P ²), which means that T = 8, and contradicts the fact that T is a perfect square. Hence, we conclude that P = 3 and so T = P ² = 9. Hence, we know



from which we can determine the last letters S = 2 and R = 7. We can now fill in the rest of the dots and write

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Last modified on Friday, February 12, 1999