Small cases lie — page 9
Analysis
What a missing string of digits leaves behind
Strike from the harmonic series every term whose denominator contains 42 and the rest adds up to 228.45. Strike out 99 instead and it adds up to 253.28. The difference is not about which numbers are lost but about waiting: 99 overlaps itself, so it takes 110 random digits on average to turn up where 42 takes 100, and the sum is almost exactly ln 10 times that wait.
Applied
One, plus a quarter, plus a ninth
Give n workers n jobs with every cost drawn at random, average one, and find the cheapest way to pair them. However large n is, the cheapest total averages less than π²/6, and for every n it is exactly 1 + 1/4 + 1/9 + … + 1/n². Parisi guessed the formula in 1998 from n = 1, 2 and 3; thirteen sizes of random tables, solved exactly, land on it within sampling error.
Computation
Three hundred thousand outputs of nearly nothing
Start the Mersenne twister from a state with a single 1 among its 19,937 bits and it puts out zeros, then sparse ones, and only after about 340,000 outputs anything that looks random. Every linear generator has such a zeroland around the all-zero state; xorshift128 crosses it in 37 outputs, WELL512 in 44, and the twister's three-word twist makes it the slowest of all by four orders of magnitude.
Discrete
How far apart two triangulations can be
Any triangulation of a polygon can be turned into any other by flipping one diagonal at a time. On an octagon seven flips always suffice; on a polygon with m vertices the worst case is 2m − 10 flips once m reaches thirteen, a value Sleator, Tarjan and Thurston proved with hyperbolic geometry in 1988. Breadth-first search over all 58,786 triangulations of the 13-gon finds the sixteen.
Number
Real fields that factorise uniquely
Among the fields Q(√−d) only nine have unique factorisation, and Gauss guessed as much. Among the real fields Q(√p) he guessed the opposite — infinitely many — and the guess is still unproved. Counting cycles of reduced quadratic forms for the 16,900 primes p ≡ 1 mod 4 below 400,000 finds 79 per cent with class number one, drifting slowly down towards the 75.45 per cent that Cohen and Lenstra's heuristic predicts — and finds the reason the real fields behave so differently: their units are enormous.