Next we show the solution to the divertimento The ages of the four daughters.
The divertimento:
Manuela, a mathematician (M), meets an old friend, Antonia (A), whom she has not seen for quite some time, in the street, and the following dialogue is established between them:
– (M): Hello, how are you? Long time no see.
– (A): Yes, it’s true. It’s been many years. In all this time, I’ve got married and I already have 4 daughters.
– (M): Oh yes, and how old are they?
– (A): Well, as you’re a mathematician, you’ll have to calculate them. I’ll give you two clues: the first is that the product of my daughters’ ages is 48, and the second is that, curiously, the sum of those ages is the number of the bus that’s standing right there in front of you.
Manuela thinks for a few seconds, glances at the bus, thinks again for another few seconds and says to Antonia:
– (M): These two clues are not enough. I need you to give me some more information.
And Antonia replies:
– (A): Well, I’ll tell you that the youngest girl has blue eyes.
With those data, would you know how to calculate the age of Antonia’s 4 daughters, which Manuela did immediately?
The solution:
The different possibilities that the product of the ages of the 4 daughters is 48 (not ordered by date of birth of the daughters, but from youngest to oldest), together with the sum of these ages, are shown in the following table:
| Age 1 | Age 2 | Age 3 | Age 4 | Sum |
|---|---|---|---|---|
| 1 | 1 | 1 | 48 | 51 |
| 1 | 1 | 2 | 24 | 28 |
| 1 | 1 | 3 | 16 | 21 |
| 1 | 1 | 4 | 12 | 18 |
| 1 | 1 | 6 | 8 | 16 |
| 1 | 2 | 2 | 12 | 17 |
| 1 | 2 | 3 | 8 | 14 |
| 1 | 2 | 4 | 6 | 13 |
| 1 | 3 | 4 | 4 | 12 |
| 2 | 2 | 2 | 6 | 12 |
| 2 | 2 | 3 | 4 | 11 |
And there are no more possibilities, since the product is commutative. Of all these possibilities, Manuela quickly discards all those that do not add up to 12, since none of the corresponding sums is repeated, and if the number of the bus that was standing in front of her had coincided with one of these sums, she would not have needed any more data. Therefore, when she asked for a new data, the bus was certainly number 12. The possible ages of Antonia’s 4 daughters would then be 1, 3, 4 and 4 or 2, 2, 2 and 6, of which the valid one is the first quadruple, since in the second case there is no one, but three youngest daughters of the same age. Therefore, the ages of Manuela’s daughters are 1, 3, 4 and 4.
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