Ages of Three Children puzzle facts for kids
The Ages of Three Children puzzle is a fun logic puzzle. It seems tricky at first because you don't think you have enough clues. But if you think carefully, you can solve it! It's all about using math and logic to figure out the ages of three children.
Contents
The Puzzle
Imagine a census taker, someone who collects information about people. He meets a woman and asks about her children. She tells him, "I have three children. If you multiply their ages together, you get 72." Then she adds, "If you add their ages together, the total is the number written on this gate." The census taker looks at the gate, does some quick math, and says, "Hmm, I still don't have enough information." The woman starts to go inside. Before she closes the door, she says, "Oh, I need to check on my oldest child. They are in bed with measles." The census taker smiles. Now he knows the answer!
This puzzle can be told in different ways. The main clues are always the same:
- You know what their ages multiply to (the product).
- You know what their ages add up to (the sum), but only the census taker knows this number at first.
- There is an "oldest" child.
Solutions
Solving for 72
To solve this puzzle, we first need to find all the possible groups of three numbers that multiply to 72. These are called the factors of 72. The numbers that multiply to 72 are: 2 × 2 × 2 × 3 × 3 = 72.
Here are all the possible age combinations and what they add up to:
| Age 1 | Age 2 | Age 3 | Total (Sum) |
|---|---|---|---|
| 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 |
The census taker knew the sum of the ages (the number on the gate). But he said he didn't have enough information. This means there must have been more than one group of ages that added up to the same number. Looking at the table, only two groups of ages have the same sum:
- A. 2 + 6 + 6 = 14
- B. 3 + 3 + 8 = 14
If the sum on the gate was any other number (like 74, 39, 28, etc.), the census taker would have known the ages right away. Since he was confused, the sum on the gate had to be 14.
Now, let's use the last clue: "I have to see to my eldest child."
- In group A (2, 6, 6), there isn't a single "eldest" child. Two children are the same age (6). Even if one was born a few minutes or months before the other, they are still both 6 years old.
- In group B (3, 3, 8), there is a clear eldest child, who is 8 years old. The other two children are both 3.
Because the woman said she had an "eldest" child, the census taker knew that group B was the correct answer. So, the children's ages are 3, 3, and 8.
Solving for 36
Another version of this puzzle uses 36 as the product of the ages. The numbers that multiply to 36 are: 2 × 2 × 3 × 3 = 36.
Here are the possible age combinations for a product of 36:
| Age 1 | Age 2 | Age 3 | Total (Sum) |
|---|---|---|---|
| 1 | 1 | 36 | 38 |
| 1 | 2 | 18 | 21 |
| 1 | 3 | 12 | 16 |
| 1 | 4 | 9 | 14 |
| 1 | 6 | 6 | 13 |
| 2 | 2 | 9 | 13 |
| 2 | 3 | 6 | 11 |
| 3 | 3 | 4 | 10 |
Just like before, the census taker would only be confused if two sums were the same. In this case, the sum of 13 appears twice:
- (1, 6, 6)
- (2, 2, 9)
When the woman mentions an "eldest" child, it helps solve the puzzle.
- In (1, 6, 6), there are two children aged 6, so no single "eldest."
- In (2, 2, 9), there is a clear eldest child, who is 9 years old.
So, if the product was 36, the children's ages would be 2, 2, and 9.
