a constructible regular polygon
/ Gauss -> GOWSS /
We can build a regular triangle, square, pentagon, and hexagon with compass and straightedge, but try the regular 7-gon and you will fail no matter how clever you are. Which regular polygons can be constructed, and which simply cannot? For two thousand years this was a puzzle; the teenage Gauss settled it completely, and it is one of the most beautiful results in geometry.
Constructing a regular n-gon amounts to dividing a circle into n equal arcs, which in turn requires constructing a particular length tied to cos(360/n degrees). Gauss proved, and Pierre Wantzel later finished, that a regular n-gon is constructible exactly when n is a power of 2 times a product of distinct Fermat primes — primes of the form 2^(2^k) + 1. The known Fermat primes are 3, 5, 17, 257, and 65537. So 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, ... are constructible, while 7, 9, 11, 13, 14, 18 are not. The famous trophy is the regular 17-gon (the heptadecagon), which Gauss constructed at age 19 and was so proud of that he asked for it on his tombstone.
The reason 7 and 9 fail is the same machinery behind the three impossible problems: building the side amounts to solving an equation whose solution does not lie in any tower of square-root extensions, so it cannot be reached by compass and straightedge. The result is exact and proven, not a matter of 'no one has found a way yet' — for the 7-gon and 9-gon, no way exists. Note: with extra tools (a marked ruler, paper folding) these become possible; the impossibility is specific to the classical instruments.
The regular 17-gon is constructible because 17 = 2^(2^2) + 1 is a Fermat prime; Gauss showed cos(360/17 degrees) can be written using only integers, the four operations, and nested square roots. The regular 7-gon is not, because 7 is not of the allowed form.
Gauss's criterion: a regular n-gon is constructible exactly when n is a power of 2 times distinct Fermat primes.
Only five Fermat primes are known (3, 5, 17, 257, 65537), and whether any others exist is an open problem. The impossibility of, say, the regular 7-gon is proven for compass and straightedge specifically — other tools can build it.