Three sons

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On sunny afternoon in September, Richard and Katie were chatting over coffee, after not seeing each other in many years.

"Do you have any children?" asked Katie?

"Yes. I have three boys," replied Richard.

"Oh my! What are their ages?"

"Well, if you multiply all their ages together, you get 36," Richard said cryptically. "Oh, and if you add up all their ages, you happen to get the current day of the month."

Katie checked her watch to see what the date was. "Hmm. I still don't know what their ages are."

"Oh. Well did I mention that the oldest loves math?"

"Ah. Well of course! Now I know." Katie exclaims.

What ages are Richard's three kids?


The puzzle also works if the three ages multiply to 72.


For which product of ages will this puzzle work? Infinitely many?