Concept

Collatz

The rule that halves an even number and sends an odd one to three times itself plus one. Whether every starting value reaches one is unproved, though almost every value is known to fall below its own start.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

The Collatz orbit of 27. Halve an even number, triple an odd one and add one; the sequence plotted on a logarithmic scale.

The question nobody can answer

Halve it if it is even, triple it and add one if it is odd. Every number anyone has tried comes down to one. Nobody can prove they all do, and the reason is not that the problem is hard to state.

dynamics · Collatz

Almost every number comes down

The Collatz conjecture is open and a great deal about it is not. Whether a number falls below its own start in the first few steps is decided entirely by its remainder on division by a power of two, and the share of numbers for which it happens can be counted exactly.

dynamics · Collatz

Downhill on average, and never on purpose. The logarithms of 4 Collatz orbits plotted against step number, each wandering upward and downward and each ending at one. A separate sample of four thousand starts gives an average fall of -0.15 per step.

The heuristic that cannot be a proof

There is a two-line argument that the Collatz conjecture is true, it is convincing, and everybody who works on the problem believes it. It also cannot be turned into a proof, and understanding exactly where it fails is more instructive than the argument itself.

dynamics · Collatz

Named alongside it

The objects these essays reach for when they reach for this one.

Iteration Heuristic Logarithm Parity Binary Counting-two ways Density Drift Expectation Modular arithmetic Open problem Orbit

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