Test-time compute: the two-line math behind majority vote, best-of-N, and "thinking longer"

A single sample from a language model is a noisy guess. On a hard question, one draw at temperature > 0 is right only part of the time. The whole idea behind test-time compute (inference-time scaling) is that you can trade extra compute at answer time for accuracy — without training a bigger model. Here's how the three main methods work, and the surprisingly simple math that decides where each one plateaus. Model a question by its single-sample accuracy Give each question a probability p that…