Three daughters whose ages multiply to 72

Question

An old friend tells you: "I have three daughters. The product of their ages, in whole years, is 72. And the sum of their ages equals the number on that door across the street."

You glance at the door, think for a moment, and say, "That's not enough, I still can't tell their ages."

Your friend replies, "Fair enough. Here's one more fact: my eldest daughter plays the violin."

Now you know the ages. What are the three ages?

Accepted Answer

First list every way three whole-number ages can multiply to 72, and note each sum: - 1, 1, 72 → sum 74 - 1, 2, 36 → sum 39 - 1, 3, 24 → sum 28 - 1, 4, 18 → sum 23 - 1, 6, 12 → sum 19 - 1, 8, 9 → sum 18 - 2, 2, 18 → sum 22 - 2, 3, 12 → sum 17 - 2, 4, 9 → sum 15 - 2, 6, 6 → sum 14 - 3, 3, 8 → sum 14 - 3, 4, 6 → sum 13 You could see the door number, so you knew the exact sum, yet you still couldn't decide. That only happens if the sum leaves more than one option open. Scanning the list, every sum is unique except 14, which appears twice: (2, 6, 6) and (3, 3, 8). So the door number must be 14, a…

Three daughters whose ages multiply to 72

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