What??? That was possible???

I've recently been browsing Slogs of fellow students and I've stumbled upon a very interesting post by Mandy Xu slog: http://mandyxu.blog.com/

During one of Danny Heap's lectures, a problem presented to us as "Streetcar Drama" was discussed. We did not really go in-depth on this problem and I left with the impression that this problem was unsolvable because there just wasn't enough information.

However, Mandy's slog showed me that this problem did actually have a solution and I should not have been so lazy, taking the easy path of assuming that it was unsolvable so I wouldn't have to think about it much more!

Here is the "Streetcar Drama" problem:

There are two friends having a conversation on a streetcar. We'll label them A and B:

A: Haven't seen you in a long time! How old are your three kids now?

B: The product of their ages (rounded down to nearest) is 36.

A: That doesn't really answer my question.

B: Well, the sum of their ages is - [fire engine goes by]

A: That still doesn't tell me how old they are.

B: Well, the eldest plays piano.

A: Okay, I see: their ages are - [you have to get off the streetcar]



Mandy concisely takes a mathematical approach to solving this problem. The best part of her post is that she does not tell you the answer right away. She simply gives you hints to how you might come close to the answer at first, so you can try to figure it out for yourself first.

Here was my thought process of reading her post:

As I was going along with her post, I prevented myself from scrolling all the way to see the next step or answer and only considering some of her hints to see whether I can solve this myself first.

So the first piece of information that is given to us is "The product of their ages (rounded down to the nearest) is 36." Thus you can represent this mathematically using 3 variables:

x = age of first child
y = age of second child
z = age of third child

thus x*y*z = 36

Before looking at any more of Mandy's solution or hints, I tried to solve this problem myself further first. So I considered the line "Well, the eldest plays piano."

This tells us that of the three children, there is 1 older than the others.
Let x = the eldest child. Then y and z must be < x.
Thus y and z may or may not be the same age, but we know they have to be younger.

since x*y*z = 36

What 3 numbers, one of which must be larger than the other two, gives you a product of 36?

Well, let's consider all the pairs of natural numbers that multiply to give you 36:

1 * 36 = 36
2 * 18 = 36
3 * 12 = 36
4 * 9 = 36
6 * 6 = 36

Now let's try to split any of these numbers so that they're a combination of 3 numbers, with one of the numbers being greater than the other two numbers:

1 * 1 * 36 = 36
1 * 2 * 18 = 36
1 * 3 * 12 = 36
1 * 4 * 9 = 36

2 * 3 * 6 = 36
2 * 2 * 9 = 36

3 * 3 * 4 = 36

Thus one x, y, and z have to be one of the combinations above. What other lines give us more information??
I on my own can not decipher the lines of information much further. The only things I can think of, that may not be entirely concrete are:

• Friend A asked how old are your "kids" now. Usually you wouldn't refer to someone's child as a "kid" if that child was 18 or older. Thus I don't think the eldest can be 18 or 36. This is not a concrete assumption though.
• Friend A mentions how he hasn't seen friend B "in a very long time". Assuming "a very long time" means over a year, this may indicate that if friend B does have any children that is 1 years of age, friend B probably wouldn't have known about it (unless through social networking). Thus I don't think any of the children can be 1 years of age. This is also not a concrete assumption as well.
• Finally, friend B mentions that the eldest plays piano. Children don't go to school until 4, the youngest age children usually start taking piano lessons is age 5 or 6. At that age, they are just starting to learn how to play the piano. Thus to be able to "play" the piano, they must be older than 5 or 6. These assumptions are also not concrete

With the not-for-sure information I gathered through real-life intuition, this would mean any combination with:

• ages >= 18 would not be valid
• age 1 would not be valid
• and the eldest has to be >= 6 (assuming children can start playing the piano after at least a year of lessons)

1 * 1 * 36 = 36

1 * 2 * 18 = 36

1 * 3 * 12 = 36

1 * 4 * 9 = 36

2 * 3 * 6 = 36

2 * 2 * 9 = 36

3 * 3 * 4 = 36

This leaves us with 1 combination: 

2 * 2 * 9

Thus x = 9, y = 2, and z = 2

Since my steps to the solution involves making some very bold assumptions, I am not confident in this solution at all. I am, however, happy with the fact that I even found a solution. This is the point where I looked at Mandy's steps and solutions to compare.

Well surprise surprise. After I looked at Mandy's solutions, I have the same answers!!! HOWEVER, the steps I took were completely wrong and ridiculous! 

Mandy's solution was completely mathematical and the main pieces of information taken in and considered were:

• The product of their ages is 36
• The sum of their ages doesn't give you any exact information
• And there is an eldest

The biggest clue I missed was that the sum of their ages doesn't give you an exact answer.

Thus this means there are two product of 3 numbers, x*y*z, whose sum are the same:

As Mandy demonstrated, just find the possible combinations (as I did earlier), find the two combinations that have an identical sum, and choose the one that has only 1 exact eldest!

I would like to thank Mandy for posting that and for making me realize that I should not be so lazy when it comes to thinking about a complex problem!