Affine Mess

Problem solving usually involves getting from a set of premises to a set of conclusions. Sometimes the road can be long – meaning the problem can be quite hard – even though there is another point in “idea space” such that the following peculiar circumstances are true: getting from to is really easy, and getting from to is really easy, too. This in spite of the fact that getting from to is quite hard: the challenge is to find the right .

In a very vague sense this can be regarded as saying that mathematical reasoning doesn’t obey the triangle inequality: the “distance” from to is not necessarily smaller than the sum of the distances from to and from to . Hence, mathematical reasoning is non-metric

I feel that the following almost-funny problem embodies this principle:

Given 51 distinct integers between 1 and 100, prove that two of them are relatively prime.

If you’ve never seen this before and you’re able to solve it within seconds, you’ll have to take my word that this problem can stump good problem solvers for a good number of hours. However, the right makes this problem embarrassingly easy; can you find it?

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