Why Random Triangles Catch the Circle's Center ¼ of the Time
Drop three random points on a circle and the triangle contains the center exactly a quarter of the time. Two proofs: an honest average, then two coins wearing a geometry costume.
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Support Vector Machines: the Widest Street Wins
Infinitely many lines separate the same data; only the widest street generalizes. And once it's found, almost none of your points were holding it up.
Every Tournament Contains a Winning Chain (Rédei's Theorem)
Round-robin results can be pure chaos, cycles everywhere, no champion. A one-move induction still lines everyone up so each player beat the next.
Which Polynomials Map ℚ Onto ℚ? The Complete Answer
A line with rational coefficients maps ℚ onto ℚ. Nothing curved ever does. Three filters — interpolation, shape, denominators — leave the full classification.