Proofs are Fun - by Grigore Rosu

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Some people think of a mathematical proof as a burden: a burden to go through in order to complete a homework assignment, or a burden to go through in order to convince your colleagues or even yourself that the result that you claim is correct. I think that in fact proofs are fun. On this page I intend to collect nice proofs, like the ones below. If you know of similar interesting or nice proofs and do not mind if I put them in the same format as the ones below please do not hesitate to contact me. Have fun:

  • Coin problem: If you use a coin to draw three circles that intersect in one point, then you can use the same coin to draw a fourth circle that goes through the other three intersection points. What I like about this problem is that all you need to explain it is a coin. There are many possible proofs. Try to find your own; I bet your proof will be different from mine.
  • Sums: The sums of the numbers from 1 to n and of their squares are, respectively,
1 + 2 + \cdots + n = \frac{n(n+1)}{2}
1^2 + 2^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}
There is, of course, nothing deep about these sums. However, they admit interesting geometrical proofs. The second one is more interesting, the first is just a warmup. The proof of the first is sort of folklore. The first proof of the second sum was done by myself, but the second I heard from a former colleague from University of California, Kai Lin.
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