My Dogs

We give a sophisticated approach first and then use a tale to guess and improve.

We have to find three numbers that add up to 15. There are lots of them. We also have to find three numbers whose product is 45. There are not so many of them. The clue here is that 45 is not divisible by too many numbers. For instance, the divisors of 45 are 1, 3, 5, 9, 15, 45. What’s more the three dogs have to each have one of these divisors as their age. And what’s even more, none of their ages is bigger than 15 because they all add up to 15!

So we’re now choosing from 1, 3, 5, 9. But the ages can’t be 1, 3, 9 because 1 x 3 x 9 is not equal to 45. So 5 must be in there somewhere. That leaves two numbers to make 9. They must be 1 and 9. So the three ages are 1, 5, 9. Wait on! Does 1 + 5 + 9 = 15? Yes. That’s all right then.

This table shows how our guesses improve until we have the answer. In this problem we were lucky that one of the dogs was one year old or we may have needed many more guesses.

Dog 1	Dog 2	Dog 3	Sum	Product
1	2	12	15	24
1	3	11	15	33
1	4	10	15	40
1	5	9	15	45