Discrete — page 5
A number that is its own square
Square 625 and the answer, 390625, ends in 625. Square 9376 and it ends in 9376. Keep extending them and two endless strings of digits appear, running leftward, each equal to its own square — and their product is nought, though neither is. Base ten does this because ten is two times five: a number in base ten is secretly a pair of numbers, one in base two and one in base five, and these two strings are the pair (one, nought) and (nought, one). In a prime base no such thing can happen.
A search for room that never ends
The textbook way to find the most that a network can carry is to find any route that still has room, push as much along it as it will take, and repeat until no route has room left. With whole-number capacities that must stop. Give one road a capacity of 0.618…, the golden ratio's reciprocal, and choose the routes badly, and it never stops: every push is positive, the flow converges, and it converges to 4.236 when the network can carry 5. The same choice made carelessly in floating point does stop — because of rounding, at the wrong answer, convinced it has finished.