s1: Simple test-time scaling

When OpenAI's o1 came out, its main novelty was a curve: accuracy on hard math rose as the model was allowed to think for longer. Replications used reinforcement learning or large distilled datasets. The s1 paper asks for the simplest recipe that produces both a strong reasoning model and that curve.

The recipe has two parts. First, data. The authors collected 59,029 questions from math, science and quantitative puzzles, generated a reasoning trace and answer for each with Google's Gemini 2.0 Flash Thinking, and filtered down to 1,000 (s1K). They removed traces with formatting problems, removed questions that either Qwen2.5-7B or Qwen2.5-32B could already solve, then sampled across 50 subject domains, favoring longer traces. Qwen2.5-32B-Instruct fine-tuned on these 1,000 examples became s1-32B, in 26 minutes on 16 H100 GPUs.

Second, budget forcing, a decoding trick. To cap thinking, insert the end-of-thinking marker when the budget is reached, so the model has to answer. To extend thinking, block the marker when the model tries to stop and append "Wait", which often makes it re-check and fix an answer. With budget forcing, s1-32B rose from 50.0% to 56.7% on AIME24, above o1-preview's 44.6%, and scored 93.0% on MATH500. AIME24 has only 30 problems, so 50.0% to 56.7% is 15 to 17 correct answers.

s1 is a practical recipe more than a research breakthrough, and each of its parts maps to a decision on a customer project.

A thousand good examples can beat fifty thousand ordinary ones per GPU-hour. Training on all 59K examples scored 53.3% against s1K's 50.0% on AIME24, within the paper's own bootstrap interval, while costing 57 times the compute. Random, longest-trace or most-diverse selections of 1,000 did worse. When a customer wants a fine-tune, spend the effort on choosing examples: hard for the base model, spread across the task's types, and well formatted.

Thinking length is a dial you can set. Budget forcing works on any model with an explicit end-of-thinking marker. It gives exact control of cost and latency per request, which prompt instructions such as "think for 500 tokens" did not: the model could not count its own tokens.

Read the denominators. Many headline numbers here rest on AIME24's 30 problems, where one answer is 3.3 points. The authors also found that greedy evaluation with vLLM changed between runs with different batch sizes, and used full precision to reduce it. Ask for the question count and the run-to-run spread before accepting a reasoning benchmark claim.