## Alternative theme for counting equivalence classes

The problem from the previous post https://bbzippo.wordpress.com/2010/02/14/crossing-lines/ has obviously nothing to do with geometry. All we needed to know about lines on the plane is that they intersect if and only if they are not parallel, and that being parallel is an equivalence relation. This is one of those cases when we need to use some very abstract, foundational properties of objects to solve the problem. I already wrote about the most fascinating example of this approach that I know: https://bbzippo.wordpress.com/2009/11/05/i-think-therefore-vector-basis-exists/ .

Anyway, I thought with regards to the problem about intersecting lines, how to reformulate it in terms of “belongs to the same class” rather than “intersects/parallel”. That would remove one layer of abstraction (or rather the layer that hides the abstraction!) and might make this problem easier for kids.

Let’s recall the anime series “Gakkou-no Kawaii Baka” (I don’t know whether an anime with this name actually exists). It is about a bunch of school girls. Two girls consider each other rivals if they both have a crush on the same boy, otherwise they consider each other friends. By the end of Season 1, each girl has exactly 5 friends. How many girls are there? The answer is 10, since the other answer 6 is unrealistic.

Wait, the whole problem is unrealistic. It assumes that a girl cannot have a crush on more than one boy at a time. Let’s reformulate: two girls are rivals if they have the same hair color. Now the answer is 6, because 2 is too small to be the number of different hair colors in an anime series.

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