The Collatz conjecture, also known as the 3x+1 problem, needs only two rules: halve an even number, or triple an odd number and add one. First proposed by Lothar Collatz in 1937, it remains unsolved despite computers checking every starting value up to roughly 2.36 sextillion. This episode plays the game with the numbers 12 and 27, explains why the resulting sequences are called hailstone numbers, and walks through the probabilistic 3/4 heuristic that suggests the house always wins without actually proving it.

From there the conversation moves to Terence Tao's 2019 logarithmic density breakthrough, the repeating loops that appear when the rules are applied to negative numbers, the fractals and Julia sets that emerge in the complex plane, and John Conway's 1972 proof that generalized Collatz problems are algorithmically undecidable. The epis ...