Two MIT math grads bump into each other while shopping at Fry’s. They haven’t seen each other in over 20 years.

**First grad to the second**: “How have you been?”**Second**: “Great! I got married and I have three daughters now.”**First**: “Really? How old are they?”**Second**: “Well, the product of their ages is 72, and the sum of their ages is the same as the number on that building over there…”**First**: “Right, ok… Oh wait… Hmm, I still don’t know.”**Second**: “Oh sorry, the oldest one just started to play the piano.”**First**: “Wonderful! My oldest is the same age!”

How old was the first grad’s daughter?

### Solution

The possible ages ( factors of 72 ) and their sums are shown below:

Ages: Sum of ages:

1 1 72 74

1 2 36 39

1 3 24 28

1 4 18 23

1 6 12 19

1 8 9 18

2 2 18 22

2 3 12 17

2 4 9 15

2 6 6 14

3 3 8 14

3 4 6 13

We can deduce from the man’s confusion over the building number that this wasn’t enough information to solve the problem. The chart shows the sum 14 twice for two different age possibilities, which would explain how knowing the building number alone would not have given him the answer. The clue that the “oldest one” started to play the piano rules out “2 6 6” as an answer, because there is no “oldest”. Since the

first grad was certain with the piano clue, the first grad’s oldest daughter is 8. I’ll leave it up to the reader to figure out why this doesn’t necessarily mean the second grad’s oldest daughter was also 8.