P is the product of three natural numbers. Is it possible that decreasing each of these three numbers by 3 can increase the product P by the number 2,025?
Take a, b and c as the three factors in order of magnitude, meaning abc = P. Because the right-hand side of the equation should increase by 2,025, the left-hand side must also increase by this value. This is only possible if subtracting 3 makes two of the factors negative and at least one of them increases in magnitude as a result. There are only two possibilities for this.
In the first possibility, a = 1 and b = 2. This leads to 1 × 2 × c = P and (–2) × (–1) × (c – 3) = P + 2,025. From the first equation, we get c = P / 2, and from the second, 2(c – 3) = P + 2,025. Substituting the first equation into the second, we get c = P / 2, and from the second, 2(P / 2 – 3) = P – 6 = P + 2,025, which is a contradiction.
In the second case, a = b = 1. This leads to 1 × 1 × c = P and (–2) × (–2) × (c – 3) = P + 2025. From the first equation, we get c = P, and from the second, 4(c – 3) = 4(P – 3) = P + 2,025, which simplifies to P = 679. Therefore, the only solution is 1 × 1 × 679 = 679.
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This puzzle originally appeared in Spektrum der Wissenschaft and was reproduced with permission. It was translated from the original German version with the assistance of artificial intelligence and reviewed by our editors.