Word problem help

Question

Bob and Mark talk about their families. Bob says he has 3 kids, the product of their ages is 72. He gives another clue: the sum of the ages of his children. Mark points out that there is still not enough information to accurately guess. Finally, Bob says "my youngest child is called justice." Mark can then correctly determine the ages of Bob's children. What are the ages?

Answer 1

Here are absolutely all of the sets of 3 positive integers (in ascending order) that may be multiplied to produce a product of 72.

Age1        Age2      Age3

1                1              72

1                2              36

1                3              24

1                4              18

1                6              12

1                8              9

2                2              18

2                3              12

2                4              9

2                6              6

3                3              8

3                4              6

 

That first clue is not sufficient to produce a single solution, so Bob reveals the sum of their ages to Mark (sums sorted in ascending order):

 

Age1        Age2      Age3              Sum

3                4              6                      13

2                6              6                      14

3                3              8                      14

2                4              9                      15

2                3              12                   17

1                8              9                      18

1                6              12                   19

2                2              18                   22

1                4              18                   23

1                3              24                   28

1                2              36                   39

1                1              72                   74

 

Since Mark says that there is still insufficient information, there must be duplicate sets of ages that produce a single product.  Here are the two:

    

Age1        Age2      Age3              Sum

2                6              6                      14

3                3              8                      14

 

Now, the words, “My youngest” restricts the solution to {2,6,6} because “My youngest” cannot be twins.