Deciding hard small Turing machines (Busy Beaver)
Prove halting or non-halting of specific small Turing machines that current deciders cannot resolve.
Cite
@misc{cairn-busy-beaver-holdouts,
title = {Deciding hard small Turing machines (Busy Beaver)},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/busy-beaver-holdouts}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
} Also: CITATION.cff · Atom feed of results
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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.
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Current state
No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.
The problem
After BB(5) was determined with a formal proof (bbchallenge collaboration, 2024), attention moved to 6-state and 2-symbol/other small machine classes, where some individual machines ("cryptids" and holdouts) resist all known deciders.
Progress: a decider (with its code and a certificate format) that resolves additional machines, a non-halting proof for a named machine, or a formalised proof. Name machines in standard notation and link the bbchallenge page.
History of the bounds
| Year | Result | By |
|---|---|---|
| 1990 | Σ(6) ≥ 136,612 | Marxen & Buntrock |
| 2010 | Σ(6) > 3.5×10^18267 | P. Kropitz |
| 2022 | S(6) > 10↑↑15 | S. Ligocki & P. Kropitz |
| 2024 | BB(5) = 47,176,870 (Rocq proof) | bbchallenge collaboration |
| 2025 | Σ(6) > 10↑↑10↑↑10↑↑8 | mxdys |