# Salesman Problem

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When I first read this problem, my gut instinct was to claim that it was impossible and that there was no way to solve this problem. Instead, I reassured myself that I love solving problems like this and began to work through it using my fail-safe method of breaking it down into smaller pieces, writing down what I know, and writing down what I want to know. Once I had translated the problem into a format that I was more comfortable looking at, the solution came to me pretty quickly. I had a little fun with this problem and pictured myself in the shoes of the salesman.

Here I am standing at this woman’s door and she tells me that the product of her children’s ages is 36. So this tells me that step one is to write down all possible combinations of 3 numbers that multiply to 36. Awesome. Once that’s done we go into the crazy information that includes no numbers. Time to pull out the logic!

So I’m standing at the door and the woman tells me that the sum of her children’s ages add up to the house number next door. Here is where I got stuck for a while. Obviously I then took all of my combinations and wrote down their sums. But I’m not at the house, so I have no clue what number I’m looking for and suddenly I have 7 different possibilities. Is is supposed to be an odd number? If we’re on the same side of the street as this woman’s house, and her house number is an odd number then my sum should be? Or should it be an even number? I pondered these for a minute, but they lead me nowhere. So instead I stopped and listened to the words that were being exchanged around the room as others were working on the problem. This actually got me nowhere. I found no inspiration, only frustration! I stared at the paper for a little while longer, reared the problem a few times and then it hit me. He needed more information! There was a third clue! Therefore, if I were to put myself back into the salesman’s shoes, I know that I had to have more than one answer that gave me the house number next door. Otherwise I would have known right away and the problem would have been solved! So it seemed as though the key to this problem was to stay in the salesman’s shoes and experience the problem firsthand. So I had 4 sets of ages that produced repeated results. Two gave me the sum of 13 and two gave me the sum of 16.

The woman’s next clue was that her oldest plays piano. A brief moment of frustration again as I wonder what in the world this could possibly do with getting the solution to this! So I looked at my groups of ages: I had a 1, 3, 12 and a 1, 6, 6, and a 2, 2, 9 and a 3, 4, 9. My husband is a piano player so my mind instead starting thinking about what ages kids usually are when they’re playing the piano and yet all the oldest children in these situations could very easily be playing piano and so that thought lead me nowhere. I thought about this for a long time until I realized my one key point that lead me to my solution: this clue leads to an obvious solution, assuming that I new that the house number was 16 or 13. So let’s think about this in that way. Suppose the house number was 16. I then have the two possibilities of children with ages 1, 3, 12 or 3, 4, 9. In either of those cases, being told that the eldest plays piano leads to no logical conclusion. If the house number was 13 then I would have the two possibilities of 1, 6, 6 or 2, 2, 9. If I was told that the OLDEST child is playing piano, than that means that there is a singular oldest child. No oldest twins! So the 1, 6, 6, has been ruled out and my solution is 2, 2, 9! Woohoo! Joy!

I felt pretty accomplished when I solved this one!!