## Tuesday, October 21, 2014

### Challenging problem?

Challenging problem?  I ran across this post by Terence Tao, a math professor at UCLA, that poses a puzzle.  It sounds very difficult (or even impossible), but it turns out to have an easy answer:
Three farmers were selling chickens at the local market.  One farmer had 10 chickens to sell, another had 16 chickens to sell, and the last had 26 chickens to sell.  In order not to compete with each other, they agreed to all sell their chickens at the same price.  But by lunchtime, they decided that sales were not going so well, and they all decided to lower their prices to the same lower price point.  By the end of the day, they had sold all their chickens.  It turned out that they all collected the same amount of money, \$35, from the day's chicken sales.  What was the price of the chickens before lunchtime and after lunchtime?﻿
As is often the case with such problems, the description (probably deliberately) makes the problem sound a bit more complex than it actually is.  You can simplify all that verbiage down to this equation:
a(52-x) + bx - 105 = 0
The problem is to find a and b.  Though the problem didn't actually say this, it turns out the answers are non-zero integers, which I just assumed was the case (for puzzles, this is the sort of answer one would expect).  Once I had this equation, I reasoned that the average price per chicken was 105/52, or slightly over \$2.  So I tried 3 and 2 for a and b, and I had the answer (plug them into that equation, and you'll get x = 51).  Total time: under 5 minutes.

Now check out the comments on that post.  Many of the people tackling this problem made it far more complicated than it actually is.  You can almost see their knowledge of mathematical techniques getting in the way of their reasoning about the problem.  You've heard the old expression “If you have a hammer, then everything looks like a nail.”  I think there's a lot of that going on here :)