Question

Bill and Matt are discussing their children. Bill tells Matt that he has 3 kids, and the product of their ages is 72. He also tells him the sum of the ages of his kids. Matt says there is still not enough information to correctly guess the ages of Bill's kids. Finally, Bill says, "My youngest child's name is Joy." Matt can then correctly guess. What are the ages?

Answer 1

To have the answer for this question and know how Matt was able to precisely guess the ages of Bill's children, we can start by writing down all the possible ages of the ages of Bill's children. 

All the possible ages ( of Bill's Children) can be written as the following multiples of 72:
 

possible ages                    Sum of the possible ages

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
From the list of the possible ages of Bill's Children, you can see that we have only two entries that are similar. Since Matt knew the sum of the ages of Bill's children, only these two possibilities could bring him a challenge, that is the sum of 14.

This means that the possible ages of Bill's children is:


2 6 6                                   14 ** 

3 3 8                                   14 ** 

Finally, Bill says, "My youngest child's name is Joy."; This means that there is the youngest child among the three. Of the two possibilities 2, 6, 6 is the only possible ages, because in 3,3,8 there is no single youngest.

Therefore the ages of Bill's children are:

2 years, 6 years and 6 years.