Self-consistency: let them vote
From Chapter 38 of Large Language Models from the Ground Up. A single reasoning chain is a coin flip. But sample many, extract each final answer, and take a majority vote — and accuracy climbs, with no verifier and no training. It works because the right answer is reached by many roads while each wrong answer arises from its own idiosyncratic slip: wrong answers scatter, the right one repeats.
Sample N chains, then vote
Each sampled chain of thought reaches a final number. A single chain here is correct about 65% of the time. Drag N to draw more chains and watch the vote; the most common answer wins. Wrong slips scatter across many values, so the truth pulls ahead as N grows.
Correct answer: 21 (3×24 = 72; sold 17 + 34 = 51; 72 − 51 = 21)
The sampled chains
Vote over final answers
Accuracy vs. N (each point = 2,000 simulated votes)
The curve rises from the single-chain rate toward the majority-vote ceiling. The dashed line marks single-sample accuracy; the marker is your current N.
This is the book's verifier-free fallback for questions with short, comparable answers. The chapter works the arithmetic with a harder solver — p = 0.4 per attempt, all wrong answers distinct — and shows the vote climbing from 40% to ≈ 71% by n = 5, purely from the model's agreement with itself. Here the per-chain rate is a friendlier 65%, so the curve climbs higher and faster, but the mechanism is identical: voting amplifies whatever the attempts agree on. The book's crucial caveat is the other half — if the model had one systematic wrong answer instead of scattered slips, that shared misconception "votes as one bloc," the same n = 5 vote falls to ≈ 51%, and no amount of sampling dilutes it. More samples help when errors are independent, and barely help when they're systematic.