At Penn Alexander’s math club yesterday, the students worked on a fun puzzle that I’d never seen before. It goes like this:
You have five bales of hay.
For some reason, instead of being weighed individually, they were weighed in all possible combinations of two. The weights of each of these combinations were written down and arranged in numerical order, without keeping track of which weight matched which pair of bales. The weights, in kilograms, were 80, 82, 83, 84, 85, 86, 87, 88, 90, and 91.
How much does each bale weigh? Is there a solution? Are there multiple possible solutions?
Unfortunately, the problem seemed a little beyond them (or at least, they thought it was beyond them, so they quickly lost interest) but this seems like a great problem to use in middle school or high school math classes. In middle school, keep them talking and focus on the methods they employ to try to solve it. In high school, perhaps once they solve it you could get them to try generalizing the problem (to other sets of weights, more than five bales, etc.).