Kinesys touches down gently in a little square on Veritas, in front of a glass house with a big number on the door. “At least here you can read the addresses,” Liz remarks. “On Veritas nothing is hidden, not even house numbers.”
On a bench, two elderly Veritasians meet again after many years and hug, deeply moved. They are old college classmates, mathematicians and perfect logicians like everyone around here: neither would ever claim to know something he doesn’t. M00N, enchanted by the scene, floats closer to listen. Dord follows reluctantly, muttering something about other people’s privacy.
The two chat about their children, and soon the conversation turns into a logic problem. Dord stops muttering and leans in: finally, an interesting conversation.
The Riddle
The first mathematician asks: “So you have three children? How old are they?” The other replies: “Taking their ages as whole numbers, their product is 36, and their sum is the number of that house right there.”
The first one thinks for a while and blurts out: “Well, you certainly haven’t given me enough information!” The second replies: “You’re right: the oldest has blue eyes.”
How old are the three children?
Hint
List every triple of ages whose product is 36 and work out each sum. Why can’t the first mathematician, who can see the house number, answer anyway?
Solution
The three children are 9, 2, and 2 years old.
Here are all the triples of whole numbers whose product is 36, with their sums:
| Ages | Sum |
|---|---|
| 36-1-1 | 38 |
| 18-2-1 | 21 |
| 12-3-1 | 16 |
| 9-4-1 | 14 |
| 9-2-2 | 13 |
| 6-6-1 | 13 |
| 6-3-2 | 11 |
| 4-3-3 | 10 |
The first mathematician knows the sum, because he can see the house number, and yet he can’t answer. That means the sum is 13, the only value that appears twice in the table: the ages could be 9-2-2 or 6-6-1.
The second mathematician’s remark, “the oldest has blue eyes,” says there is a single oldest child. In the 6-6-1 case there would instead be two six-year-old twins, both the oldest. So the answer is 9-2-2.
Dord nods, satisfied: on Veritas, even eye color can be decisive information.