# Massively Parallel Proof-Number Search for Impartial Games and Beyond

arXiv: https://arxiv.org/abs/2511.10339 · HTML: https://mlsanity.com/paper/2511.10339/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 22 references that are in the map.
- First posted on arXiv 13 Nov 2025.

## Builds on (most cited first, 1 of 1)

- [HyperTree Proof Search for Neural Theorem Proving](https://mlsanity.com/paper/2205.11491/) (2022)

## Abstract (arXiv)

Proof-Number Search is a best-first search algorithm with many successful applications, especially in game solving. As large-scale computing clusters become increasingly accessible, parallelization is a natural way to accelerate computation. However, existing parallel versions of Proof-Number Search are known to scale poorly on many CPU cores. Using two parallelized levels and shared information among workers, we present the first massively parallel version of Proof-Number Search that scales efficiently even on a large number of CPUs. We apply our solver, enhanced with Grundy numbers for reducing game trees of impartial games, to the Sprouts game, a case study motivated by the long-standing Sprouts Conjecture. Our algorithm achieves 332.9× speedup on 1024 cores, significantly improving previous parallelizations and outperforming the state-of-the-art Sprouts solver GLOP by four orders of magnitude in runtime while generating proofs 1,000× more complex. Despite exponential growth in game tree size, our solver verified the Sprouts Conjecture for 42 new positions, nearly doubling the number of known outcomes.