The Collatz conjecture

The statement. Start with any positive integer. If it is even, halve it; if odd, multiply by 3 and add 1. Repeat. The conjecture says you always reach 1. From 27 it takes 111 steps and climbs to 9,232 on the way.

In plain words. The rule is simple enough for a child, the behaviour looks random, and no one can prove the process never escapes to infinity or loops forever somewhere. Checked for every start below 2⁶⁸.

What you need first. Stage 1 (parity, division), Stage 10 (modular arithmetic, binary representation), Stage 8 (why “on average it shrinks” is a probabilistic statement, not a proof).

The parts. (1) The heuristic: an odd step multiplies by about 3/2 (3n + 1 is even, so it is immediately halved), an even step by 1/2; on average the sequence shrinks by a factor of √3/2 per step — so it “should” fall to 1. (2) Cycles: any loop other than 4 → 2 → 1 would have to be enormous (more than 10¹¹ elements). (3) Density results: almost all starting values eventually dip below any fixed bound (Tao, 2019) — almost all, not all. (4) Why it resists: the problem mixes the additive structure (+1) with the multiplicative (×3, ÷2), and mathematics has few tools for that mix.

What is known. Verified to 2⁶⁸; Tao's almost-all result; no proof, and Erdős's verdict that “mathematics is not yet ripe for such problems”.