Lothar Collatz1910–1990
Posed the 3n + 1 problem, 1937
German mathematician who worked on numerical analysis and introduced the 3n + 1 problem in 1937, two years after his doctorate. It is still unsolved.
Hailstones
Halve it if it is even, triple it and add one if it is odd, until you reach 1. Grown backwards from 1, thousands of numbers become a coral; drawn as arcs, they rise and fall like hailstones.
About this pattern
Lothar Collatz, 1937
Pick any whole number. If it is even, halve it; if it is odd, multiply it by three and add one. Then do the same to the answer, and keep going. Start at 6 and you get 6, 3, 10, 5, 16, 8, 4, 2, 1, and once you reach 1 you are stuck in a loop of 4, 2, 1. Every number anyone has tried ends at 1, and computers have checked them up past two followed by twenty-one zeros, yet nobody can prove that every number does.
Lothar Collatz introduced the problem in 1937, and it is often called the Collatz conjecture. The numbers are called hailstones, because they rise and fall many times on the way down, like hailstones tossed about in a cloud: 27 takes 111 steps and climbs as high as 9,232 before it falls to 1. Here each step is a half circle along a number line, halving steps above and tripling steps below, on a scale that squeezes the big numbers together so the whole flight fits.
Draw many starting numbers at once and you see that their paths soon join, every one of them funnelling down through 16, 8, 4, 2 and 1; a step that many numbers share is drawn once, and heavier. The coral grows the same paths backwards from 1, turning one way at every halving step and the other way at every tripling step, so that numbers with a shared ending share a stem; it is a favourite sketch among creative coders.
Pick the numbers at random instead, from as far up as a million, and each seed grows its own coral: big numbers take longer roads home, so the stems grow longer and stranger before they join. In motion the steps are drawn in the order they are taken.
Behind the mathematics