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. Mar

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 called Justice." Mark can then correctly determine the ages of Bob's children. What are the ages?

Answer 1

1. Let X, Y and Z be the ages of the the three children.
2. Then X*Y*Z = 72
3. Three unknowns, one equation.  So, not enough info to resolve the problem
4. Now we also know X+Y+Z = SUM.   We weren't explicity told the value of SUM, but apparently Mark has been given the information.
5. So we now have three equations, and two unknowns.
6. Let's see what we have so far, by creating a table listing out all the unique possibilities of three numbers whose product is 72.  It might help to factor 72.  72 = 3 * 3 * 2 * 2 * 2.
1. X = 1, Y = 1, Z = 72  {this would be an unlikely scenario, but possible nevertheless}
2. X = 1, Y = 2, Z = 36
3. X = 1, Y = 3, Z = 24
4. continue doing this.  Note that you do not need to worry about the possibilities that are duplicates of the values but just in a different order.  In other words, no need to list X = 72, Y = 1, Z =1. 
7. When you have completed this table, add another column that shows the Sum:
1. For the entry where X = 1, Y = 1, and Z =72, the SUM is 74
2. For the entry where X = 1, Y = 2, and Z = 36, the SUM is 39
3. For the entry where X = 1, Y = 3, and Z = 24, the SUM is 28
4. Complete the table by figuring out the sums of the other entries
8. Now, at this point, if all the entries in the table had a unique SUM, then Mark would be able to identify the ages of the three children.  However, since Mark says this is not enough information, it must be because there are at least two entries (may be more) in the table where the SUM is the same number.
9. So now Mark needs a way to distinguish between two table entries that have the same sum.
10. What do you think that third clue is giving you.  Obviously, the name of the child is not relevant to find the ages.  However, Bob uses the term "youngest".  Doesn't this help eliminate some of the possibilities?  For example, in the very first entry of your table, the possible ages are 1, 1 and 72.   Is there a youngest child?  (There actually would be, since even twins cannot be born at the same instant of time.  But the convention for these type of problems is that, with twins, there is no "youngest" if the lowest number age is the one that is duplicated )
11. You can let me know what answer you got, and I'll tell you if it's correct.   Good luck