A friend of mine who plays go a lot found this on her go server. Mathematicians post stuff there, she said, and, being mathematicians, they post problems, not answers.
"You have a collection of 11 balls with the property that if you remove any one of the balls, the other 10 can be split into two groups of 5 that have the same weight. If you assume that all the balls have rational weight, there is a cute proof that they all must weigh the same. Can you find a proof? Can you find a way to extend the result to the general case where the balls have real weights?"
Solving the implied system of 11 equations in 11 unknowns is fairly trivial and gives a proof, but it isn’t “cute” by any stretch. I have been unable to come up with either a cute proof, or a proof, cute or otherwise, which depends on the unknowns being rational.
Does anyone see what I missed?