A disease one person in two hundred carries, and a test with no false negatives at all. The reflex answer to a positive result is above ninety percent, and it is wrong by more than a factor of ten. A crowd of a thousand people shows why before the algebra does, and Bayes puts the exact figure at 100/1493.
A stock with zero volatility, a call struck at the money, and the reflex answer — zero — that fails real trading interviews. One arbitrage argument prices it three different ways, and Black–Scholes agrees on the way out.
One condition, a two-line recurrence, and the fifth power falls out as a clean 123 with no radicals left. Climb the same ladder far enough and the golden ratio and the Lucas numbers are hiding underneath.
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.
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.
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.
A line with rational coefficients maps ℚ onto ℚ. Nothing curved ever does. Three filters — interpolation, shape, denominators — leave the full classification.
A counterfeit coin that might be heavy or light, a balance that only reports which side falls, and a hundred dollars a weighing. Counting rules out four; only a construction gets you five.