## Friday, July 02, 2010

### Brain Teaser: Find the ages

Question:
I was visiting a friend one evening and remembered that he had three daughters. I asked him how old they were. "The product of their ages is 72," he answered. Quizzically, I asked, "Is there anything else you can tell me?" "Yes," he replied, "the sum of their ages is equal to the number of my house." I stepped outside to see what the house number was. Upon returning inside, I said to my host, "I'm sorry, but I still can't figure out their ages." He responded apologetically, "I'm sorry, I forgot to mention that my oldest daughter likes strawberry shortcake." With this information, I was able to determine all of their ages. How old is each daughter?

The first clue is that the product of their ages is 72. Therefore, the list of all possible sets of numbers whose product is 72 is:

```72 1 1
36 2 1
24 3 1
18 4 1
18 2 2
12 3 2
12 6 1
9 8 1
9 4 2
8 3 3
6 6 2
6 4 3
```
The second clue involves the sums of their ages which are:
```72 + 1 + 1 = 74
36 + 2 + 1 = 39
24 + 3 + 1 = 28
18 + 4 + 1 = 23
18 + 2 + 2 = 22
12 + 3 + 2 = 17
12 + 6 + 1 = 19
9 + 8 + 1 = 18
9 + 4 + 2 = 15
8 + 3 + 3 = 14
6 + 6 + 2 = 14
6 + 4 + 3 = 13
```
Out of all these sums, two of them are equal:

```8 + 3 + 3 = 14
6 + 6 + 2 = 14
```
The final clue is that his "oldest" daughter likes cake, which means that we can eliminate the second one of the above. Therefore, his daughter's ages are:

```8 + 3 + 3 = 14
```
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