Streetcar Drama Solution Attempt

Streetcar Drama (from lecture slides) Solution Attempt

I approached this problem by first assuming that person B has 3 children, all older than 1, with each age a positive integer and one eldest, so I need to find 3 integers. The product of all ages is equal to 36 and I set the sum of all ages equal to the variable y. I then set the age of Child 1 to the variable a, the age of Child 2 to the variable b, and the age of Child 3 to the variable c. Initially, my instincts told me to look for clues while attempting to use 3 variable substitution.

I started thinking about possible situations, perhaps person B has twins or even triplets. Person B must have either one eldest and two younger twins, or children all of different ages. I ruled out triplets because the children's ages could not be equal to a cube root of 36 while remaining an integer. I also disregarded the addition of the eldest child playing piano because I felt that it was unnecessary.

I then thought about possible ways that person A might not be able to figure out the ages based on 2 equations, which are:

In the case of all different ages:

a × b × c = 36
a + b + c = y

In the case of one eldest and two younger twins:

In this case a = c, so we can write a in place of c
a × a × b = 36
a + a + b =y


Because of person A's response to the sum of the ages ("That still doesn't tell me how old they are."), it seems clear that this wasn't enough information for person A to use algebra and solve. In the case of twins and one older child, person B should have been able to use substitution to solve (2 unknowns in each of the 2 equations). But she or he did not, which leads me to believe that there must have been 3 unknowns in each of the 2 equations (one can not use algebra to solve this).

This completely stumped me until I began writing factors of 36.

2 × 18
4 × 9
4 × 3 × 3
2 × 2 × 9
2 × 2 × 3 × 3
3 × 12
3 × 2 × 6

I then realized that 3, 2, and 9 are the only set of 3 factors of 36 with all different values. I concluded that these must be the ages of each child.

This being said, I believe my solution is flawed, because if I assume that Person A could potentially do 2 variable substitution using mental math, it would also be reasonable to assume that he or she could figure out 3 different factors that multiply to 36.