BusyBeaverWiki - User contributions [en]

Main Page

2026-08-04T14:35:35Z

BB fan weird: Ho messo il TMBR d'agosto

The [[Busy Beaver function]] BB (called ''S'' originally) was introduced by [[Tibor Radó]] in 1962 for 2-symbol [[Turing machines]] and later generalised to ''m''-symbol Turing machines:<ref>Radó, T. (1962), On Non-Computable Functions. Bell System Technical Journal, 41: 877-884. https://doi.org/10.1002/j.1538-7305.1962.tb00480.x</ref><ref>Brady, Allen H, and the Meaning of Life, 'The Busy Beaver Game and the Meaning of Life', in Rolf Herken (ed.), The Universal Turing Machine: A Half-Century Survey (Oxford, 1990; online edn, Oxford Academic, 31 Oct. 2023), https://doi.org/10.1093/oso/9780198537748.003.0009, accessed 8 June 2024.</ref>

{| class="wikitable"
|-
| BB(''n'', ''m'') = Maximum number of steps taken by a halting ''n''-state, ''m''-symbol Turing machine starting from a blank (all 0) tape
|}

The 2-symbol case BB(''n'', 2) is abbreviated as BB(''n''). The busy beaver function is not computable, but a few of its values are known:

{| class="wikitable"
|+ Small busy beaver values<ref>P. Michel, "[https://bbchallenge.org/~pascal.michel/ha.html Historical survey of Busy Beavers]".</ref>
! !!2-state!!3-state !!4-state!!5-state!!6-state
!7-state
|-
! 2-symbol
| [[BB(2)]] = 6
| [[BB(3)]] = 21
| [[BB(4)]] = 107
| [[BB(5)]] = 47,176,870
| style="background: orange;" | [[BB(6)]] > <math>2 \uparrow \uparrow \uparrow 5</math>
| style="background: orange;" | [[BB(7)]] > <math>2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|-
! 3-symbol
| [[BB(2,3)]] = 38
| style="background: orange;" | [[BB(3,3)]] > <math>10^{17}</math>
| style="background: #ffe4b2;" | [[BB(4,3)]] > <math>10 \uparrow^{4} 4</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 4-symbol
| [[BB(2,4)]] = 3,932,964
| style="background: #ffe4b2;" | [[BB(3,4)]] > <math>2 \uparrow^{15} 5</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 5-symbol
| style="background: orange;" | [[BB(2,5)]] > <math>10\uparrow\uparrow 4</math>
| style="background: #ffe4b2;" | [[BB(3,5)]] > <math> f_\omega(2 \uparrow^{15} 5)</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 6-symbol
| style="background: #ffe4b2;" | [[BB(2,6)]] > <math>10 \uparrow\uparrow\uparrow 3</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|}

In the above table, cells are highlighted in orange when there are known [[Cryptids]] (mathematically-hard machines) in that class, and cells are highlighted in light orange when the existence of a Cryptid is given by using a known one with fewer states or symbols.

1-state domains and 1-symbol domains are omitted in the table as [[BB(1,m)]] = 1 and [[BB(n,1)]] = n.

== About bbchallenge ==
[[bbchallenge]] is a massively collaborative research project whose general goal is to obtain more knowledge on the [[Busy Beaver function]]. In practice, it mainly consists in collaboratively building [[Deciders]], programs that automatically prove that some Turing machines do not halt. Other efforts also include:

* Formalising results using theorem provers (such as [https://en.wikipedia.org/wiki/Rocq Rocq])
* Maintaining [[Holdouts lists]] for small busy beaver values
* [[Analysis Tools and Techniques|Proving]] the behavior of [[:Category:Individual Machines|Individual machines]]
* Finding [[Cryptids]] (mathematically-hard machines)
* Searching for new [[Champions]]
* Building [[Accelerated Simulator]]s to simulate halting machines faster
* Writing papers and giving talks about busy beaver, see [[Papers & Talks]]

In June 2024, bbchallenge achieved a significant milestone by proving in [https://en.wikipedia.org/wiki/Rocq Rocq] (previously known as Coq) that the 5th busy beaver value, [[BB(5)]], is equal to the lower bound found in 1989: 47,176,870.<ref>H. Marxen and J. Buntrock. Attacking the Busy Beaver 5.
Bulletin of the EATCS, 40, pages 247-251, February 1990. https://turbotm.de/~heiner/BB/mabu90.html</ref>

== This Month in Beaver Research (TMBR) ==
[[:Category:This Month in Beaver Research|This Month in Beaver Research]] (TMBR, pronounced "timber") is a monthly summary of Busy Beaver research progress. Here are the three most recent released entries:

* [[TMBR: April 2026]]
* [[TMBR: March 2026]]
* [[TMBR: February 2026]]

[[TMBR: May 2026]], [[TMBR: June 2026]], [[TMBR: July 2026]] and TMBR: August 2026 are currently work in progress, This Year in Beaver Research (TYBR) is a yearly summary of Busy Beaver research progress. Its first edition, [[TYBR: 2025]] is currently work in progress.

==Notes==
<references />

Main Page

2026-07-10T12:09:01Z

BB fan weird: Ho messo il TMBR di luglio

The [[Busy Beaver function]] BB (called ''S'' originally) was introduced by [[Tibor Radó]] in 1962 for 2-symbol [[Turing machines]] and later generalised to ''m''-symbol Turing machines:<ref>Radó, T. (1962), On Non-Computable Functions. Bell System Technical Journal, 41: 877-884. https://doi.org/10.1002/j.1538-7305.1962.tb00480.x</ref><ref>Brady, Allen H, and the Meaning of Life, 'The Busy Beaver Game and the Meaning of Life', in Rolf Herken (ed.), The Universal Turing Machine: A Half-Century Survey (Oxford, 1990; online edn, Oxford Academic, 31 Oct. 2023), https://doi.org/10.1093/oso/9780198537748.003.0009, accessed 8 June 2024.</ref>

{| class="wikitable"
|-
| BB(''n'', ''m'') = Maximum number of steps taken by a halting ''n''-state, ''m''-symbol Turing machine starting from a blank (all 0) tape
|}

The 2-symbol case BB(''n'', 2) is abbreviated as BB(''n''). The busy beaver function is not computable, but a few of its values are known:

{| class="wikitable"
|+ Small busy beaver values<ref>P. Michel, "[https://bbchallenge.org/~pascal.michel/ha.html Historical survey of Busy Beavers]".</ref>
! !!2-state!!3-state !!4-state!!5-state!!6-state
!7-state
|-
! 2-symbol
| [[BB(2)]] = 6
| [[BB(3)]] = 21
| [[BB(4)]] = 107
| [[BB(5)]] = 47,176,870
| style="background: orange;" | [[BB(6)]] > <math>2 \uparrow \uparrow \uparrow 5</math>
| style="background: orange;" | [[BB(7)]] > <math>2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|-
! 3-symbol
| [[BB(2,3)]] = 38
| style="background: orange;" | [[BB(3,3)]] > <math>10^{17}</math>
| style="background: #ffe4b2;" | [[BB(4,3)]] > <math>10 \uparrow^{4} 4</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 4-symbol
| [[BB(2,4)]] = 3,932,964
| style="background: #ffe4b2;" | [[BB(3,4)]] > <math>2 \uparrow^{15} 5</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 5-symbol
| style="background: orange;" | [[BB(2,5)]] > <math>10\uparrow\uparrow 4</math>
| style="background: #ffe4b2;" | [[BB(3,5)]] > <math> f_\omega(2 \uparrow^{15} 5)</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 6-symbol
| style="background: #ffe4b2;" | [[BB(2,6)]] > <math>10 \uparrow\uparrow\uparrow 3</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|}

In the above table, cells are highlighted in orange when there are known [[Cryptids]] (mathematically-hard machines) in that class, and cells are highlighted in light orange when the existence of a Cryptid is given by using a known one with fewer states or symbols.

1-state domains and 1-symbol domains are omitted in the table as [[BB(1,m)]] = 1 and [[BB(n,1)]] = n.

== About bbchallenge ==
[[bbchallenge]] is a massively collaborative research project whose general goal is to obtain more knowledge on the [[Busy Beaver function]]. In practice, it mainly consists in collaboratively building [[Deciders]], programs that automatically prove that some Turing machines do not halt. Other efforts also include:

* Formalising results using theorem provers (such as [https://en.wikipedia.org/wiki/Rocq Rocq])
* Maintaining [[Holdouts lists]] for small busy beaver values
* [[Analysis Tools and Techniques|Proving]] the behavior of [[:Category:Individual Machines|Individual machines]]
* Finding [[Cryptids]] (mathematically-hard machines)
* Searching for new [[Champions]]
* Building [[Accelerated Simulator]]s to simulate halting machines faster
* Writing papers and giving talks about busy beaver, see [[Papers & Talks]]

In June 2024, bbchallenge achieved a significant milestone by proving in [https://en.wikipedia.org/wiki/Rocq Rocq] (previously known as Coq) that the 5th busy beaver value, [[BB(5)]], is equal to the lower bound found in 1989: 47,176,870.<ref>H. Marxen and J. Buntrock. Attacking the Busy Beaver 5.
Bulletin of the EATCS, 40, pages 247-251, February 1990. https://turbotm.de/~heiner/BB/mabu90.html</ref>

== This Month in Beaver Research (TMBR) ==
[[:Category:This Month in Beaver Research|This Month in Beaver Research]] (TMBR, pronounced "timber") is a monthly summary of Busy Beaver research progress. Here are the three most recent released entries:

* [[TMBR: April 2026]]
* [[TMBR: March 2026]]
* [[TMBR: February 2026]]

[[TMBR: May 2026]], [[TMBR: June 2026]] and TMBR: July 2026 are currently work in progress, This Year in Beaver Research (TYBR) is a yearly summary of Busy Beaver research progress. Its first edition, [[TYBR: 2025]] is currently work in progress.

==Notes==
<references />

Main Page

2026-07-08T09:35:40Z

BB fan weird: Ho commesso un errore fatale ma ho sistemato un pochino, correggete quel punto che ho messo involontariamente (e stato bbchallenge!)

The [[Busy Beaver function]] BB (called ''S'' originally) was introduced by [[Tibor Radó]] in 1962 for 2-symbol [[Turing machines]] and later generalised to ''m''-symbol Turing machines:<ref>Radó, T. (1962), On Non-Computable Functions. Bell System Technical Journal, 41: 877-884. https://doi.org/10.1002/j.1538-7305.1962.tb00480.x</ref><ref>Brady, Allen H, and the Meaning of Life, 'The Busy Beaver Game and the Meaning of Life', in Rolf Herken (ed.), The Universal Turing Machine: A Half-Century Survey (Oxford, 1990; online edn, Oxford Academic, 31 Oct. 2023), https://doi.org/10.1093/oso/9780198537748.003.0009, accessed 8 June 2024.</ref>

{| class="wikitable"
|-
| BB(''n'', ''m'') = Maximum number of steps taken by a halting ''n''-state, ''m''-symbol Turing machine starting from a blank (all 0) tape
|}

The 2-symbol case BB(''n'', 2) is abbreviated as BB(''n''). The busy beaver function is not computable, but a few of its values are known:

{| class="wikitable"
|+ Small busy beaver values<ref>P. Michel, "[https://bbchallenge.org/~pascal.michel/ha.html Historical survey of Busy Beavers]".</ref>
! !!2-state!!3-state !!4-state!!5-state!!6-state
!7-state
|-
! 2-symbol
| [[BB(2)]] = 6
| [[BB(3)]] = 21
| [[BB(4)]] = 107
| [[BB(5)]] = 47,176,870
| style="background: orange;" | [[BB(6)]] > <math>2 \uparrow \uparrow \uparrow 5</math>
| style="background: orange;" | [[BB(7)]] > <math>2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|-
! 3-symbol
| [[BB(2,3)]] = 38
| style="background: orange;" | [[BB(3,3)]] > <math>10^{17}</math>
| style="background: #ffe4b2;" | [[BB(4,3)]] > <math>10 \uparrow^{4} 4</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 4-symbol
| [[BB(2,4)]] = 3,932,964
| style="background: #ffe4b2;" | [[BB(3,4)]] > <math>2 \uparrow^{15} 5</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 5-symbol
| style="background: orange;" | [[BB(2,5)]] > <math>10\uparrow\uparrow 4</math>
| style="background: #ffe4b2;" | [[BB(3,5)]] > <math> f_\omega(2 \uparrow^{15} 5)</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 6-symbol
| style="background: #ffe4b2;" | [[BB(2,6)]] > <math>10 \uparrow\uparrow\uparrow 3</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|}

In the above table, cells are highlighted in orange when there are known [[Cryptids]] (mathematically-hard machines) in that class, and cells are highlighted in light orange when the existence of a Cryptid is given by using a known one with fewer states or symbols.

1-state domains and 1-symbol domains are omitted in the table as [[BB(1,m)]] = 1 and [[BB(n,1)]] = n.

== About bbchallenge ==
[[bbchallenge]] is a massively collaborative research project whose general goal is to obtain more knowledge on the [[Busy Beaver function]]. In practice, it mainly consists in collaboratively building [[Deciders]], programs that automatically prove that some Turing machines do not halt. Other efforts also include:

* Formalising results using theorem provers (such as [https://en.wikipedia.org/wiki/Rocq Rocq])
* Maintaining [[Holdouts lists]] for small busy beaver values
* [[Analysis Tools and Techniques|Proving]] the behavior of [[:Category:Individual Machines|Individual machines]]
* Finding [[Cryptids]] (mathematically-hard machines)
* Searching for new [[Champions]]
* Building [[Accelerated Simulator]]s to simulate halting machines faster
* Writing papers and giving talks about busy beaver, see [[Papers & Talks]]

In June 2024, bbchallenge achieved a significant milestone by proving in [https://en.wikipedia.org/wiki/Rocq Rocq] (previously known as Coq) that the 5th busy beaver value, [[BB(5)]], is equal to the lower bound found in 1989: 47,176,870.<ref>H. Marxen and J. Buntrock. Attacking the Busy Beaver 5.
Bulletin of the EATCS, 40, pages 247-251, February 1990. https://turbotm.de/~heiner/BB/mabu90.html</ref>

== This Month in Beaver Research (TMBR) ==
[[:Category:This Month in Beaver Research|This Month in Beaver Research]] (TMBR, pronounced "timber") is a monthly summary of Busy Beaver research progress. Here are the three most recent released entries:

* [[TMBR: April 2026|TMBR: May 2026]]
* [[TMBR: March 2026|TMBR: April 2026]]
* [[TMBR: February 2026|TMBR: March 2026]]
* [[TMBR: June 2026]] and TMBR: July 2026 are currently work in progress, This Year in Beaver Research (TYBR) is a yearly summary of Busy Beaver research progress. Its first edition, [[TYBR: 2025]] is currently work in progress.

==Notes==
<references />

Holdouts lists

2026-06-16T18:42:02Z

BB fan weird: Ho cambiato il numero di holdouts su BB(2,5)

A '''holdout''' (or undecided machine) is a [[Turing machine]] for which it is not known whether the machine halts or not from an all-0 input tape. Holdouts are the machines which [[Decider|deciders]] are yet unable to decide. Contributors have shared lists of holdouts. Some lists exclude known equivalent machines, i.e., machines whose halting status is implied by the halting status of another machine. If a holdout machine with more than one undefined transitions is found to reach a undefined transition, the number of holdouts may increase as a consequence of [[Tree Normal Form|TNF enumeration]].

{| class="wikitable"
|+Number of holdouts
!
!2-state
!3-state
!4-state
!5-state
!6-state
!7-state
|-
!2-symbol
|0
|0
|0
|0
|1104
|15,743,264
|-
!3-symbol
|0
|4
|5,127,263
|
|
|
|-
!4-symbol
|0
|11,362,197
|
|
|
|
|-
!5-symbol
|65
|
|
|
|
|
|-
!6-symbol
|412,086
|
|
|
|
|
|}

== Downloadable Holdouts Lists ==
This is a table where people have added lists with no restriction or independent verification. For some of the entries there is a reference to a spreadsheet that documents what was run to achieve the result. For others, there is additional documentation on the specific BB pages.
{| class="wikitable sortable"
|+
!BB space
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[[BB(7)]]
|[https://drive.google.com/drive/folders/1_lIqfvj4_J7WWl5LOBUp_pntoI99QYui May 25th, 2026]
|
|17,823,260
|[https://drive.google.com/file/d/1GFumm3tzm58Yxza_sD_BdJ3RMY_DKmRv/view 7x2_holdouts_17823260.txt.zip]
|
|-
|[[BB(3,4)]]
|[https://drive.google.com/drive/folders/1uGWJMYYuVGJj581GX_vlm3Wv1adYbcRs May 11th, 2026]
|@XnoobSpeakable
|11,362,197
|[https://drive.google.com/file/d/1-Fnexjfq0mWyZDPa2MNFmU64EpSu-3Ig/view 3x4_holdouts_11362197.zip]
|Work done by XnoobSpeakable
|-
|[[BB(4,3)]]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW- May 9th, 2026]
|@Andrew Ducharme
|5,127,263
|[https://drive.google.com/file/d/1xHukBgJnm1zctZkOMR0UzdQyzr4Fx5pC/view 4x3_holdouts_5127263.txt.zip]
|Work done by Andrew Ducharme
|-
|[[BB(3,4)]]
|[https://drive.google.com/drive/folders/1lNKc_GF2sa5ZfosleAlMNBQ3uJV4qClo April 30th, 2026]
|@Andrew Ducharme
|12,049,358
|[https://drive.google.com/file/d/1xCtI58doiCoI3lU_oQjK_21zsLFLF9g3/view 3x4_holdouts_12049358.txt.zip]
|Work done by Andrew Ducharme
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1499000732236382358 April 29th, 2026]
|@mxdys
|1104
|[https://wiki.bbchallenge.org/w/images/b/bf/BB6_holdouts_1104.txt BB6_holdouts_1104.txt]
|More reduction by equivalences
|-
|[[BB(4,3)]]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW- April 26th, 2026]
|@Andrew Ducharme
|5,641,006
|[https://drive.google.com/file/d/1z8oBXUECL5dpUNAIqik2z_yI5JGmW79t/view 4x3_holdouts_5641006.txt.zip]
|Work done by Andrew Ducharme
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1497651809773289552 April 25th, 2026]
|@mxdys
|1119
|[https://wiki.bbchallenge.org/w/images/a/ab/BB6_holdouts_1119.txt BB6_holdouts_1119.txt]
|Reduction by equivalences, one halting TM
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1486207538843222116 March 25th, 2026]
|@mxdys
|1161
|[https://wiki.bbchallenge.org/w/images/2/2a/BB6_holdouts_1161.txt BB6_holdouts_1161.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1477224991136419983 February 28th, 2026]
|@mxdys
|1214
|[https://wiki.bbchallenge.org/w/images/0/0e/BB6_holdouts_1214.txt BB6_holdouts_1214.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1473945366821998684 February 19th, 2026]
|@mxdys
|1226
|[https://wiki.bbchallenge.org/w/images/8/80/BB6_holdouts_1226.txt BB6_holdouts_1226.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1466438332677619956 January 29th, 2026]
|@mxdys
|1314
|[https://discord.com/channels/960643023006490684/1239205785913790465/1466438332677619956 BB6_holdouts_1314.txt]
|
|-
|[[BB(3,4)]]
|[https://drive.google.com/drive/folders/1pj0MJBCA1hk6NxSj0mYvu5iXbTtVRt64 December 31st, 2025]
|
|12,435,284
|[https://drive.google.com/file/d/15e-Lwl_F6Jn_HoYFPA2a94jLTjX1mmHd/view 3x4st10c_holdouts_12435284.txt]
|Work done by [[User:XnoobSpeakable|XnoobSpeakable]] and [[User:WarpedWartWars|Lúkos]].
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1455369448264568904 December 30th, 2025]
|@mxdys
|1326
|[https://wiki.bbchallenge.org/w/images/f/f0/BB6_holdouts_1326.txt BB6_holdouts_1326.txt]
|
|-
|[[BB(6)]]
|November 22nd, 2025
|Robin Rovenszky
|Variable
|[https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=1330361301#gid=1330361301 Google Sheets]
|Annotated list, including links to Discord discussions, always up-to-date.
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1450455364179857410 December 16th, 2025]
|@mxdys
|1343
|[https://wiki.bbchallenge.org/w/images/9/9a/BB6_holdouts_classified_1343.txt BB6_holdouts_1343.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1443989662019354647 November 28th, 2025]
|@mxdys
|1416
|[https://wiki.bbchallenge.org/w/images/9/98/BB6_holdouts_1416.txt BB6_holdouts_1416.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1438983076662477004 November 14th, 2025]
|@mxdys
|1534
|[https://wiki.bbchallenge.org/w/images/0/02/BB6_holdouts_1534.txt BB6_holdouts_1534.txt]
|
|-
|[[BB(4,3)]]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW- October 26th, 2025]
|
|9,401,447
|[https://drive.google.com/file/d/1r99v3dcdNTgGkmaydLKiZCp5KD3jGWeE/view 4x3_holdouts_9401447.txt.gz]
|
|-
|[[BB(7)]]
|[https://drive.google.com/drive/folders/1_lIqfvj4_J7WWl5LOBUp_pntoI99QYui October 25th, 2025]
|
|20,405,295
|[https://drive.google.com/file/d/12riXHC3Vz4tk_5mMV7__gZsF7hqAsZ_S/view bb7_holdouts_20405295.txt.gz]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1429892916763033601 October 20th, 2025]
|@mxdys
|1618
|[https://wiki.bbchallenge.org/w/images/e/e3/BB6_holdouts_1618.txt BB6 holdouts 1618.txt]
|
|-
|[[BB(4,3)]]
|[https://drive.google.com/drive/folders/1qNssnvK3W2jJ68VBq9FJZMy9TvwbQk4_ October 20th, 2025]
|@tjligocki
|18,138,027
|[https://drive.google.com/file/d/1vHufk09pI7JcTSAOj1XOzcZXJoX1Wa2a/view 4x3_holdouts_18138027.txt.gz]
|
|-
|[[BB(2,6)]]
|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT October 6th, 2025]
|@tjligocki
|870,085
|[https://drive.google.com/file/d/1xZqEZFPke02pwBHX0EYsyDVaNU5o_OuO/view 2x6_holdouts_870085.txt.zip]
|Work done by Andrew Ducharme.
|-
|[[BB(3,4)]]
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i October 4th, 2025]
|@tjligocki
|64,777,377
|[https://drive.google.com/file/d/1I3s3w-T4NPLn-eCdaZLPgC2Mya51pyvH/view?usp=drive_link 3x4_holdouts_64777377.txt.gz]
|Work done by [[User:XnoobSpeakable|XnoobSpeakable]] and [[User:WarpedWartWars|Lúkos]].
|-
|[[BB(7)]]
|[https://discord.com/channels/960643023006490684/1369339127652159509/1423806362072256676 October 4th, 2025]
|Andrew Ducharme
|22,721,168
|[https://drive.google.com/file/d/1xAFSPh6qAR8VxsF4QVipmdc0UpdPC1hn/view bb7_holdouts_22721168.txt.zip]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1419290767767240734 September 21, 2025]
|@mxdys
|1691
|[https://discord.com/channels/960643023006490684/1239205785913790465/1419290767767240734 BB6_holdouts_1691.txt]
|
|-
|[[BB(2,6)]]
|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT September 16, 2025]
|
|970,101
|[https://drive.google.com/file/d/1Vjm3Qf-4a-Zr-E3EocjPWI5bjqH2YGil/view 2x6_holdouts_970101.txt.gz]
|Work done by Terry Ligocki
|-
|[[BB(2,6)]]
|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT September 12, 2025]
|
|20,358,011
|[https://drive.google.com/file/d/1YOoh4Zh2Jn5zaq9-C638S6VR_4jzd6pa/view 2x6_holdouts_20358011.zip]
|Work done by Andrew Ducharme
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1411308408220549140 August 30 2025]
|@mxdys
|2592
|[[:File:BB6 holdouts 2592.txt|BB6 holdouts 2592.txt]]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1399783936019664896 July 29 2025]
|@mxdys
|2728
|[[:File:BB6 holdouts 2728.txt|BB6 holdouts 2728.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1393808719615361205 July 13, 2025]
|@mxdys
|2891
|[https://discord.com/channels/960643023006490684/1239205785913790465/1393808719615361205 BB6_holdouts_2891.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1390953522379489401 July 5, 2025]
|@mxdys
|3094
|[https://discord.com/channels/960643023006490684/1239205785913790465/1390953522379489401 BB6_holdouts_3094.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1384957116376158228 June 18, 2025]
|@mxdys
|3335
|[https://discord.com/channels/960643023006490684/1239205785913790465/1384957116376158228 BB6_holdouts_3335.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1259770421046411285/1355593937531961365 March 29 2025]
|@mxdys
|83
|[[:File:BB2x5 Coq holdouts 83.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1345246656769163284 March 1, 2025]
|@mxdys
|3571
|[https://discord.com/channels/960643023006490684/1239205785913790465/1345246656769163284 BB6_holdouts_3571.txt]
|
|-
|[[BB(4,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1320172124509311004 December 21, 2024]
|@tjligocki
|460,916,384
|[https://drive.google.com/file/d/1hKy0TuPuI62rN95j6ZLjXgE-Pue8tRsK/view?usp=drive_link 4x3_holdouts_460916384.txt.gz]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW-?usp=drive_link Google Drive directory for 4x3 TMs]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1310648046576730124 November 25, 2024 (@icy)]
|@tjligocki
|4319
|[https://docs.google.com/spreadsheets/d/1grhW_0neb2I8TfceN5-v70_3W42Z2U159r9L6FhPGf8/edit?usp=sharing Spreadsheet of holdouts]
|Keeping track of BB(6) progress - informal.
|-
|[[BB(3,3)]]
|November 10, 2024
|
|6
|[[:File:3x3 holdout 6.txt|3x3 holdout 6.txt]]
|
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1304303803213942846 November 8, 2024]
|@mxdys
|4408
|[[:File:BB6 holdouts 4408.txt|BB6_holdouts_4408.txt]]
|
|-
|[[BB(2,6)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1303219184221683733 November 4, 2024]
|@tjligocki
|22,302,296
|[https://drive.google.com/file/d/1xRLIjdiqImFP2SL38gvhxVlAaX0L1cYO/view?usp=drive_link 2x6_holdouts_22302296.txt.gz]
|[https://drive.google.com/drive/folders/1p9b5g-Id3WEMUYIwEnaKWRBGIW66ADjM?usp=drive_link Google Drive directory for 2x6 TMs]
|-
|[[BB(3,4)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1302767449476694188 November 3, 2024]
|@tjligocki
|434,787,751
|[https://drive.google.com/file/d/1PLzN3wLw-MRgk1OFmYh4RTwNc30nflR7/view?usp=drive_link 3x4_holdouts_434787751.txt.gz]
|[https://drive.google.com/drive/folders/1bZxl7jg5q9IVvHQNZwItx1kPusAWznZk?usp=drive_link Google Drive directory for 3x4 TMs]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1297163115179278336 October 19, 2024]
|@mxdys
|4521
|[https://discord.com/channels/960643023006490684/1239205785913790465/1297163115179278336 BB6_holdouts_4521.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1287580515938865225 September 23, 2024]
|@mxdys
|4741
|[https://discord.com/channels/960643023006490684/1239205785913790465/1287580515938865225 BB6_holdouts_4741.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1282002734912110655 September 7, 2024]
|@mxdys
|4986
|[https://discord.com/channels/960643023006490684/1239205785913790465/1282002734912110655 BB6_holdouts_4986.txt]
|
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1280185195877634098 September 2, 2024]
|@mxdys
|5394
|[[:File:BB6 holdouts 5394.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1269612923127599164 August 4, 2024]
|@mxdys
|5877
|[[:File:BB6 holdouts 5877.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1267013907910758451 July 28, 2024]
|@mxdys
|6417
|[https://discord.com/channels/960643023006490684/1239205785913790465/1267013907910758451 BB6_holdouts_6417.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262276904887521300 July 15, 2024]
|@mxdys
|7013
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262276904887521300 BB6_holdouts_7013.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262126923321180250 July 14, 2024]
|@Justin Blanchard
|7255
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262126923321180250 BB6_holdouts_7255.txt]
|Work done by Justin Blanchard
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259131753176498216 July 6, 2024]
|@mxdys
|7296
|[[:File:BB6 holdouts 7296.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259011186192552068 July 6, 2024]
|@Justin Blanchard
|7430
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259011186192552068 BB6_holdouts_7430.txt]
|Work done by Justin Blanchard
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1258810921879605298 July 5, 2024]
|@mxdys
|7449
|[https://discord.com/channels/960643023006490684/1239205785913790465/1258810921879605298 BB6_holdouts_7449.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1256254912573079572 June 28, 2024]
|@mxdys
|9366
|[https://discord.com/channels/960643023006490684/1239205785913790465/1256254912573079572 BB6_holdouts_9366.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1254684723955896362 June 24, 2024]
|@Justin Blanchard
|9853
|[https://discord.com/channels/960643023006490684/1239205785913790465/1254684723955896362 BB6_holdouts_9853.txt]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253478694798884934 June 21, 2024]
|@mxdys
|10,020
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253478694798884934 BB6_holdouts_10020.txt]
|Work done by Justin Blanchard
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253466035189846056 June 20, 2024]
|@mxdys
|10,944
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253466035189846056 BB6_holdouts_10944.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1252989316175499284 June 19, 2024]
|@mxdys
|217
|[[:File:2x5_holdouts_217.txt]]
|273 holdouts minus machines solved by CTL
|-
|[[BB(2,5)]]
|June 15th 2024
|@dyuan01
| 273
|[[:File:2x5_holdouts_273.txt]]
|@Justin Blanchard's 499 holdouts minus machines solved by @mxdys
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1250895665719148595 June 13, 2024]
|@tjligocki
| 12,091
|[[:File:BB6 holdouts 12091.txt]]
|Work done with @Shawn Ligocki
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1249142547217907772 June 9, 2024]
|@Justin Blanchard
|22
|[[:File:3x3.todo.txt]], [[:File:Mugshots small.pdf]]
|
|-
|BB(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1248708916381220954 June 7, 2024]
|@mxdys
|12,325
|[[:File:BB6 holdouts 12325.txt]]
|Some equivalent machines are removed.
|-
|[[BB(5)]]
|June 2024
|BBChallenge
|0
|
|'''BB(5) is SOLVED!'''
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1245116022873264129 May 28, 2024]
|@mxdys
|18,560
|[https://discord.com/channels/960643023006490684/1239205785913790465/1245116022873264129 BB6_holdouts_18560.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1242679236142170203 May 22, 2024]
|@Justin Blanchard
|499
|[[:File:2x5.todo.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1239628496184279080 May 13, 2024]
|@mxdys
|35,788
|[https://discord.com/channels/960643023006490684/1239205785913790465/1239628496184279080 BB6_holdouts_35788.txt]
|C++ version of the deciders from [[Coq-BB5]] ran on Shawns 181,851 holdouts<ref>https://discord.com/channels/960643023006490684/1239205785913790465/1239275863699361813</ref>
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1233154495635259543 April 25, 2024]
|@Justin Blanchard
|920
|[https://discord.com/channels/960643023006490684/1084047886494470185/1233154495635259543 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1232667326126489710 April 24, 2024]
|@Justin Blanchard
|1045
|[https://discord.com/channels/960643023006490684/1084047886494470185/1232667326126489710 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1231941302975987844 April 22, 2024]
|@Justin Blanchard
|1588
|[https://discord.com/channels/960643023006490684/1084047886494470185/1231941302975987844 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230737657076650066 April 19, 2024]
|@Justin Blanchard
|1835
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230737657076650066 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230491507287523409 April 18, 2024]
|@Justin Blanchard
|1921
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230491507287523409 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230132897583857756 April 17, 2024]
|@Justin Blanchard
|2354
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230132897583857756 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229770471898087506 April 16, 2024]
|@Justin Blanchard
|2438
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229770471898087506 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229406059274305546 April 15, 2024]
|@Justin Blanchard
|2564
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229406059274305546 2x5.todo.txt]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1228351493745152083 April 12, 2024]
|@Justin Blanchard
|3364
|[https://discord.com/channels/960643023006490684/1084047886494470185/1228351493745152083 2x5.todo.txt]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|925
|[[:File:2023 06 08.3x3.holdouts intersect sligocki iijil 925.txt]]
|Intersection of @sligocki and @Iijil from below
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|2,480
|[[:File:2023 06 08.3x3.holdouts iijil 2380.txt]]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116178334620070000 June 7, 2023]
|@sligocki
|2,417
|[[:File:2023 06 07.3x3.holdouts 2417.txt]]
|
|-
|[[BB(6)]]
|May 27, 2023
|@sligocki
|181,851
|[https://drive.google.com/file/d/1YNwFCN6XJeDNKxxK5KbGHOAFOdIBvDb6/view?usp=drive_link 6x2.holdouts_181851.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(6)]]
|May 10, 2023
|@sligocki
|1,458,704
|[https://drive.google.com/file/d/14bDnBt0OwuHATFBiubc_5jub0220EyXf/view?usp=drive_link 6x2.holdouts_1458704.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(5)]]
|[https://skelet.ludost.net/bb/nreg.html circa May 13th, 2003]
|Georgi Georgiev (Skelet)
|43
|[https://bbchallenge.org/skelet List of 43 holdouts]
|
|}

== Busy Beaver adjacent holdouts lists ==

=== Beeping Busy Beaver ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBB(4)
|[https://discord.com/channels/960643023006490684/1385498968011575366/1387585253928992831 June 26, 2025]
|@mxdys
|3713
|[https://wiki.bbchallenge.org/w/images/6/62/BBB4_holdouts_3713.txt BBB4_holdouts_3713.txt]
|
|}

=== Reversible Turing machines ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBrev(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1400313138611490897 July 31, 2025]
|@Sligocki
|388
|[https://wiki.bbchallenge.org/w/images/8/86/BBrev6_holdouts_388.txt BBrev6_holdouts_388.txt]
|
|-
|BBrev(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1400253700986900692 July 31, 2025]
|@Sligocki
|409
|[https://discord.com/channels/960643023006490684/1239205785913790465/1400253700986900692 BBrev6_holdouts_409.txt]
|
|}

=== Instruction-limited Busy Beaver ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBi(8)
|[https://discord.com/channels/960643023006490684/1084047886494470185/1425234398919266405 October 7, 2025]
|@Shawn Ligocki
|348
|[https://discord.com/channels/960643023006490684/1084047886494470185/1425234398919266405 bbi8_holdouts_348.txt]
|
|-
|BBi(8)
|[https://discord.com/channels/960643023006490684/1084047886494470185/1424786698537271347 October 6, 2025]
|@Peacemaker II
|441
|[https://discord.com/channels/960643023006490684/1084047886494470185/1424786698537271347 BBi8_holdouts_441.txt]
|
|}

=== Fractran ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout June 3, 2026]
|@-d
|21295
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_21295.txt sz23_21295.txt]
|
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout June 2, 2026]
|@-d
|21320
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_21320.txt sz23_21320.txt]
|
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout June 2, 2026]
|@-d
|28100
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_28100.txt sz23_28100.txt]
|
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout June 2, 2026]
|@-d
|29188
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_29188.txt sz23_29188.txt]
|
|-
|BBf(23)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1510749997392724190 May 31, 2026]
|@-d
|29250
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_29250.txt sz23_29250.txt]
|
|-
|BBf(23)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1510749997392724190 May 31, 2026]
|@-d
|94367
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_94367.txt sz23_94367.txt]
|
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout May 30, 2026]
|@-d
|790,335
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_790335.txt sz23_790335.txt]
|
|-
|BBf(22)
|[https://github.com/int-y1/BBFractran/tree/main/holdout April 4, 2026]
|@-d
|3
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz22_3.txt sz22_3.txt]
|Holdout reduction by Claude Opus 4.6
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 January 25, 2026]
|@-d
|2003
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz22_2003_unofficial.txt sz22_2003_unofficial.txt]
|Result of the beeping permutation decider being run on sz22_5682.txt
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 January 25, 2026]
|@-d
|140
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz21_140_unofficial.txt sz21_140_unofficial.txt]
|Result of the beeping permutation decider being run on sz21_345.txt
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1463862950203887811 January 22, 2026]
|@-d
|143
|[https://discord.com/channels/960643023006490684/1438019511155691521/1463862950203887811 sz21_143_workinprogress.txt]
|
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1456498841334317210 January 2, 2026]
|@-d
|5682
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz22_5682.txt sz22_5682.txt]
|
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1456498841334317210 January 2, 2026]
|@-d
|345
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz21_345.txt sz21_345.txt]
|
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 December 15, 2025]
|@Sligocki
|394
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 sz21_394.txt]
|
|-
|BBf(20)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 December 15, 2025]
|@Sligocki
|6
|[https://discord.com/channels/960643023006490684/1438019511155691521/1450186358474936533 sz20_6.txt]
|BBf(20) has since been solved
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1449536000098308236 December 13, 2025]
|@Sligocki
|10441
|[https://discord.com/channels/960643023006490684/1438019511155691521/1449536000098308236 sz22_10441.txt]
|17 known halters are removed
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448806255261913199 December 11, 2025]
|@Sligocki
|10458
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448806255261913199 sz22_10457.txt]
|Known halters are removed
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448729669263163596 December 11, 2025]
|@dyuan01
|11130
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448729669263163596 message.txt]
|
|-
|BBf(22)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1448214077028569098 December 10, 2025]
|@-d
|91123
|[https://raw.githubusercontent.com/int-y1/BBFractran/refs/heads/main/holdout/sz22_91123.txt sz22_91123.txt]
|Initial enumeration
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439799813758648421 November 17, 2025]
|@Sligocki
|760
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439799813758648421 sz21_760.txt]
|23 halting machines are removed
|-
|BBf(21)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439725735211303003 November 16, 2025]
|@dyuan01
|783
|[https://discord.com/channels/960643023006490684/1438019511155691521/1439725735211303003 message.txt]
|
|-
|BBf(20)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438560626329784321 November 13, 2025]
|@dyuan01
|279
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438560626329784321 message.txt]
|
|-
|BBf(20)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438483146864590900 November 13, 2025]
|@-d
|902
|[https://discord.com/channels/960643023006490684/1438019511155691521/1438483146864590900 sz20_902.txt]
|
|}

=== Terminating Turmites ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|TT(2,4)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1400609116103118870 August 1, 2025]
|@creeperman7002
|2298
|[https://discord.com/channels/960643023006490684/1243312334907375676/1400609116103118870 holdout_2x4_2298.txt]
|
|-
|TT(4)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1393363926871310508 July 12, 2025]
|@creeperman7002
|99
|[https://discord.com/channels/960643023006490684/1243312334907375676/1393363926871310508 holdout4x2.txt]
|
|}

=== Busy Beaver for Lambda Calculus ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBλ(40)
|[https://discord.com/channels/960643023006490684/1355653587824283678/1514278632695005224 June 10, 2026]
|@C++DSucker
|Variable (3)
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=741510300#gid=741510300 Spreadsheet]
|-
|BBλ(39)
|[https://discord.com/channels/960643023006490684/1355653587824283678/1492187654429868062 April 10, 2026]
|@Sligocki
|0
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=1717120706#gid=1717120706 Spreadsheet]
|'''BBλ(39) was solved on June 10th, 2026'''
|-
|BBλ(38)
|[https://discord.com/channels/960643023006490684/1355653587824283678/1362975027154129056 April 19, 2025]
|@Sligocki
|0
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=563816005#gid=563816005 Spreadsheet]
|'''BBλ(38) was solved on April 13th, 2026'''
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354938497848443032 March 27, 2025]
|@Racheline
|92
|[https://docs.google.com/spreadsheets/d/1yoiez7lDjZ-vRzj8NFWKWOxGlkK9IE_KWMUjeSWYWkw/edit?gid=0#gid=0 BBλ(37) 92 holdouts progress]
|BBλ(37) has since been solved
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354421317792694292 March 26, 2025]
|@Racheline
|216
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354421317792694292 BBL(37)_216.txt]
|
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354398010527711345 March 26, 2025]
|@Racheline
|278
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354398010527711345 BBL2(37)_278.txt]
|
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354355688763031616 March 26, 2025]
|@Sam
|317
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354355688763031616 BBL2(37)_317.txt]
|
|-
|BBλ(37)
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354352262562189344 March 26, 2025]
|@Sam
|333
|[https://discord.com/channels/960643023006490684/1243312334907375676/1354352262562189344 BBL(37).txt]
|
|}

== References ==

User talk:BB fan weird

2026-05-30T09:39:24Z

BB fan weird:

La busy beaver challenge è hard, ma bb(n,m) e per creare numeri ancora piu grandi, bb(3,5)~f omega (2^[15 volte ^] 5, bb(6) e un atomo confrontato con bb(3,5), bb(7) e un pochino pochino piu grande ma bb(3,5) e ĺ intero multiverso!

the busy beaver challenge (or BBC) is a collaborative project to find the busy beaver function values for the maximum possibile, the june 2026 of TMBR is important, becase the bb(5) discovered in the luil (or the 7° month pf year) 1 year, i predict that in the december 2026 the bb(7) value that >2[15]5[12]4, goodbuy of busy beaver(6) !

Quindi, il febbraio 2026 non sta andando come speravo, e ĺ 11° giorno di febbraio, per ora c'è solo la riduzione di macchine di Turing per bb(7), prevedo che entro il luglio 2026, il valore di bb(7)≥2[15]5[13]3 (piu di quello che ho detto sulla mia pagina utente), grazie per aver guardato questo

Hi, i am very happy, i decide one thing: i am a novice of turing machine, not a new open to bbchallenge.org, and the 30° may is my birthday, i have now 9 years.

User:BB fan weird

2026-05-30T09:36:22Z

BB fan weird:

Welcome to my user page!Hi everyone! I am bb weird fan. I am 9 years old and I am super excited about the Busy Beaver challenge! Even though I am still a beginner in theoretical computer science and I'm learning how Turing Machines work, I love making predictions about giant numbers. My current predictions: BB(6) > \(2↑↑↑6 BB(7) >2[11]5[11]3 (I predict this involves about 13 to 15 arrows!) I have already created 17 pages on this wiki. I might be young, but I'm committed to helping the experts to solve BB(7)!

I have much happy, thanks for correction of my errors, thanks!

Main Page

2026-05-08T13:32:03Z

BB fan weird: Ho fatto una modifica per gli aggiornamenti, ma ho cercato di sistemare perche avevo fatto un po male.

The [[Busy Beaver function]] BB (called ''S'' originally) was introduced by [[Tibor Radó]] in 1962 for 2-symbol [[Turing machines]] and later generalised to ''m''-symbol Turing machines:<ref>Radó, T. (1962), On Non-Computable Functions. Bell System Technical Journal, 41: 877-884. https://doi.org/10.1002/j.1538-7305.1962.tb00480.x</ref><ref>Brady, Allen H, and the Meaning of Life, 'The Busy Beaver Game and the Meaning of Life', in Rolf Herken (ed.), The Universal Turing Machine: A Half-Century Survey (Oxford, 1990; online edn, Oxford Academic, 31 Oct. 2023), https://doi.org/10.1093/oso/9780198537748.003.0009, accessed 8 June 2024.</ref>

{| class="wikitable"
|-
| BB(''n'', ''m'') = Maximum number of steps taken by a halting ''n''-state, ''m''-symbol Turing machine starting from a blank (all 0) tape
|}

The 2-symbol case BB(''n'', 2) is abbreviated as BB(''n''). The busy beaver function is not computable, but a few of its values are known:

{| class="wikitable"
|+ Small busy beaver values<ref>P. Michel, "[https://bbchallenge.org/~pascal.michel/ha.html Historical survey of Busy Beavers]".</ref>
! !!2-state!!3-state !!4-state!!5-state!!6-state
!7-state
|-
! 2-symbol
| [[BB(2)]] = 6
| [[BB(3)]] = 21
| [[BB(4)]] = 107
| [[BB(5)]] = 47,176,870
| style="background: orange;" | [[BB(6)]] ≥ <math>2 \uparrow \uparrow \uparrow 5</math>
| style="background: orange;" | [[BB(7)]] ≥ <math>2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|-
! 3-symbol
| [[BB(2,3)]] = 38
| style="background: orange;" | [[BB(3,3)]] ≥ <math>10^{17}</math>
| style="background: #ffe4b2;" | [[BB(4,3)]] ≥ <math>10 \uparrow^{4} 4</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 4-symbol
| [[BB(2,4)]] = 3,932,964
| style="background: #ffe4b2;" | [[BB(3,4)]] ≥ <math>2 \uparrow^{15} 5</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 5-symbol
| style="background: orange;" | [[BB(2,5)]] ≥ <math>10\uparrow\uparrow 4</math>
| style="background: #ffe4b2;" | [[BB(3,5)]] ≥ <math> f_\omega(2 \uparrow^{15} 5)</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 6-symbol
| style="background: #ffe4b2;" | [[BB(2,6)]] ≥ <math>10 \uparrow\uparrow\uparrow 3</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|}

In the above table, cells are highlighted in orange when there are known [[Cryptids]] (mathematically-hard machines) in that class, and cells are highlighted in light orange when the existence of a Cryptid is given by using a known one with fewer states or symbols.

1-state domains and 1-symbol domains are omitted in the table as [[BB(1,m)]] = 1 and [[BB(n,1)]] = n.

== About bbchallenge ==
[[bbchallenge]] is a massively collaborative research project whose general goal is to obtain more knowledge on the [[Busy Beaver function]]. In practice, it mainly consists in collaboratively building [[Deciders]], programs that automatically prove that some Turing machines do not halt. Other efforts also include:

* Formalising results using theorem provers (such as [https://en.wikipedia.org/wiki/Rocq Rocq])
* Maintaining [[Holdouts lists]] for small busy beaver values
* [[Analysis Tools and Techniques|Proving]] the behavior of [[:Category:Individual Machines|Individual machines]]
* Finding [[Cryptids]] (mathematically-hard machines)
* Searching for new [[Champions]]
* Building [[Accelerated Simulator]]s to simulate halting machines faster
* Writing papers and giving talks about busy beaver, see [[Papers & Talks]]

In June 2024, bbchallenge achieved a significant milestone by proving in [https://en.wikipedia.org/wiki/Rocq Rocq] (previously known as Coq) that the 5th busy beaver value, [[BB(5)]], is equal to the lower bound found in 1989: 47,176,870.<ref>H. Marxen and J. Buntrock. Attacking the Busy Beaver 5.
Bulletin of the EATCS, 40, pages 247-251, February 1990. https://turbotm.de/~heiner/BB/mabu90.html</ref>

== This Month in Beaver Research (TMBR) ==
[[:Category:This Month in Beaver Research|This Month in Beaver Research]] (TMBR, pronounced "timber") is a monthly summary of Busy Beaver research progress. Here are the three most recent released entries:

* [[TMBR: March 2026]]
* [[TMBR: February 2026]]
* [[TMBR: January 2026]]

TMBR: May 2026 is currently work in progress, This Year in Beaver Research (TYBR) is a yearly summary of Busy Beaver research progress. Its first edition, [[TYBR: 2025]] is currently work in progress.

==Notes==
<references />

Main Page

2026-05-03T08:31:48Z

BB fan weird: Ho messo il TMBR di maggio

The [[Busy Beaver function]] BB (called ''S'' originally) was introduced by [[Tibor Radó]] in 1962 for 2-symbol [[Turing machines]] and later generalised to ''m''-symbol Turing machines:<ref>Radó, T. (1962), On Non-Computable Functions. Bell System Technical Journal, 41: 877-884. https://doi.org/10.1002/j.1538-7305.1962.tb00480.x</ref><ref>Brady, Allen H, and the Meaning of Life, 'The Busy Beaver Game and the Meaning of Life', in Rolf Herken (ed.), The Universal Turing Machine: A Half-Century Survey (Oxford, 1990; online edn, Oxford Academic, 31 Oct. 2023), https://doi.org/10.1093/oso/9780198537748.003.0009, accessed 8 June 2024.</ref>

{| class="wikitable"
|-
| BB(''n'', ''m'') = Maximum number of steps taken by a halting ''n''-state, ''m''-symbol Turing machine starting from a blank (all 0) tape
|}

The 2-symbol case BB(''n'', 2) is abbreviated as BB(''n''). The busy beaver function is not computable, but a few of its values are known:

{| class="wikitable"
|+ Small busy beaver values<ref>P. Michel, "[https://bbchallenge.org/~pascal.michel/ha.html Historical survey of Busy Beavers]".</ref>
! !!2-state!!3-state !!4-state!!5-state!!6-state
!7-state
|-
! 2-symbol
| [[BB(2)]] = 6
| [[BB(3)]] = 21
| [[BB(4)]] = 107
| [[BB(5)]] = 47,176,870
| style="background: orange;" | [[BB(6)]] ≥ <math>2 \uparrow \uparrow \uparrow 5</math>
| style="background: orange;" | [[BB(7)]] ≥ <math>2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|-
! 3-symbol
| [[BB(2,3)]] = 38
| style="background: orange;" | [[BB(3,3)]] ≥ <math>10^{17}</math>
| style="background: #ffe4b2;" | [[BB(4,3)]] ≥ <math>10 \uparrow^{4} 4</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 4-symbol
| [[BB(2,4)]] = 3,932,964
| style="background: #ffe4b2;" | [[BB(3,4)]] ≥ <math>2 \uparrow^{15} 5</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 5-symbol
| style="background: orange;" | [[BB(2,5)]] ≥ <math>10\uparrow\uparrow 4</math>
| style="background: #ffe4b2;" | [[BB(3,5)]] ≥ <math> f_\omega(2 \uparrow^{15} 5)</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 6-symbol
| style="background: #ffe4b2;" | [[BB(2,6)]] ≥ <math>10 \uparrow\uparrow\uparrow 3</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|}

In the above table, cells are highlighted in orange when there are known [[Cryptids]] (mathematically-hard machines) in that class, and cells are highlighted in light orange when the existence of a Cryptid is given by using a known one with fewer states or symbols.

1-state domains and 1-symbol domains are omitted in the table as [[BB(1,m)]] = 1 and [[BB(n,1)]] = n.

== About bbchallenge ==
[[bbchallenge]] is a massively collaborative research project whose general goal is to obtain more knowledge on the [[Busy Beaver function]]. In practice, it mainly consists in collaboratively building [[Deciders]], programs that automatically prove that some Turing machines do not halt. Other efforts also include:

* Formalising results using theorem provers (such as [https://en.wikipedia.org/wiki/Rocq Rocq])
* Maintaining [[Holdouts lists]] for small busy beaver values
* [[Analysis Tools and Techniques|Proving]] the behavior of [[:Category:Individual Machines|Individual machines]]
* Finding [[Cryptids]] (mathematically-hard machines)
* Searching for new [[Champions]]
* Building [[Accelerated Simulator]]s to simulate halting machines faster
* Writing papers and giving talks about busy beaver, see [[Papers & Talks]]

In June 2024, bbchallenge achieved a significant milestone by proving in [https://en.wikipedia.org/wiki/Rocq Rocq] (previously known as Coq) that the 5th busy beaver value, [[BB(5)]], is equal to the lower bound found in 1989: 47,176,870.<ref>H. Marxen and J. Buntrock. Attacking the Busy Beaver 5.
Bulletin of the EATCS, 40, pages 247-251, February 1990. https://turbotm.de/~heiner/BB/mabu90.html</ref>

== This Month in Beaver Research (TMBR) ==
[[:Category:This Month in Beaver Research|This Month in Beaver Research]] (TMBR, pronounced "timber") is a monthly summary of Busy Beaver research progress. Here are the three most recent released entries:

* [[TMBR: March 2026]]
* [[TMBR: February 2026]]
* [[TMBR: January 2026]]

[[TMBR: April 2026]] and TMBR: May 2026 are currently a work in progress.

This Year in Beaver Research (TYBR) is a yearly summary of Busy Beaver research progress. Its first edition, [[TYBR: 2025]] is currently work in progress.
==Notes==
<references />

Main Page

2026-04-01T12:29:06Z

BB fan weird: Ho cercato di "ripulire" un po'

The [[Busy Beaver function]] BB (called ''S'' originally) was introduced by [[Tibor Radó]] in 1962 for 2-symbol [[Turing machines]] and later generalised to ''m''-symbol Turing machines:<ref>Radó, T. (1962), On Non-Computable Functions. Bell System Technical Journal, 41: 877-884. https://doi.org/10.1002/j.1538-7305.1962.tb00480.x</ref><ref>Brady, Allen H, and the Meaning of Life, 'The Busy Beaver Game and the Meaning of Life', in Rolf Herken (ed.), The Universal Turing Machine: A Half-Century Survey (Oxford, 1990; online edn, Oxford Academic, 31 Oct. 2023), https://doi.org/10.1093/oso/9780198537748.003.0009, accessed 8 June 2024.</ref>

{| class="wikitable"
|-
| BB(''n'', ''m'') = Maximum number of steps taken by a halting ''n''-state, ''m''-symbol Turing machine starting from a blank (all 0) tape
|}

The 2-symbol case BB(''n'', 2) is abbreviated as BB(''n''). The busy beaver function is not computable, but a few of its values are known:

{| class="wikitable"
|+ Small busy beaver values<ref>P. Michel, "[https://bbchallenge.org/~pascal.michel/ha.html Historical survey of Busy Beavers]".</ref>
! !!2-state!!3-state !!4-state!!5-state!!6-state
!7-state
|-
! 2-symbol
| [[BB(2)]] = 6
| [[BB(3)]] = 21
| [[BB(4)]] = 107
| [[BB(5)]] = 47,176,870
| style="background: orange;" | [[BB(6)]] ≥ <math>2 \uparrow \uparrow \uparrow 5</math>
| style="background: orange;" | [[BB(7)]] ≥ <math>2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|-
! 3-symbol
| [[BB(2,3)]] = 38
| style="background: orange;" | [[BB(3,3)]] ≥ <math>10^{17}</math>
| style="background: #ffe4b2;" | [[BB(4,3)]] ≥ <math>10 \uparrow^{4} 4</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 4-symbol
| [[BB(2,4)]] = 3,932,964
| style="background: #ffe4b2;" | [[BB(3,4)]] ≥ <math>2 \uparrow^{15} 5</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 5-symbol
| style="background: orange;" | [[BB(2,5)]] ≥ <math>10\uparrow\uparrow 4</math>
| style="background: #ffe4b2;" | [[BB(3,5)]] ≥ <math> f_\omega(2 \uparrow^{15} 5)</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 6-symbol
| style="background: #ffe4b2;" | [[BB(2,6)]] ≥ <math>10 \uparrow\uparrow\uparrow 3</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|}

In the above table, cells are highlighted in orange when there are known [[Cryptids]] (mathematically-hard machines) in that class, and cells are highlighted in light orange when the existence of a Cryptid is given by using a known one with less states or symbols.

1-state domains and 1-symbol domains are omitted in the table as [[BB(1,m)]] = 1 and [[BB(n,1)]] = n.

== About bbchallenge ==
[[bbchallenge]] is a massively collaborative research project whose general goal is to obtain more knowledge on the [[Busy Beaver function]]. In practice, it mainly consists in collaboratively building [[Deciders]], programs that automatically prove that some Turing machines do not halt. Other efforts also include:

* Formalising results using theorem provers (such as [https://en.wikipedia.org/wiki/Rocq Rocq])
* Maintaining [[Holdouts lists]] for small busy beaver values
* [[Analysis Tools and Techniques|Proving]] the behavior of [[:Category:Individual Machines|Individual machines]]
* Finding [[Cryptids]] (mathematically-hard machines)
* Searching for new [[Champions]]
* Building [[Accelerated Simulator]]s to simulate halting machines faster
* Writing papers and giving talks about busy beaver, see [[Papers & Talks]]

In June 2024, bbchallenge achieved a significant milestone by proving in [https://en.wikipedia.org/wiki/Rocq Rocq] (previously known as Coq) that the 5th busy beaver value, [[BB(5)]], is equal to the lower bound found in 1989: 47,176,870.<ref>H. Marxen and J. Buntrock. Attacking the Busy Beaver 5.
Bulletin of the EATCS, 40, pages 247-251, February 1990. https://turbotm.de/~heiner/BB/mabu90.html</ref>

== This Month in Beaver Research (TMBR) ==
[[:Category:This Month in Beaver Research|This Month in Beaver Research]] (TMBR, pronounced "timber") is a monthly summary of Busy Beaver research progress. Here are the three most recent released entries:

* [[TMBR: January 2026|TMBR: February 2026]]
* [[TMBR: December 2025|TMBR: January 2026]]
* [[TMBR: November 2025|TMBR: December 2025]]

[[TMBR: March 2026]] and [[TMBR: April 2026]] are currently work in progress.

This Year in Beaver Research (TYBR) is a yearly summary of Busy Beaver research progress. Its first edition, [[TYBR: 2025]] is currently work in progress.
==Notes==
<references />

Main Page

2026-03-06T18:11:53Z

BB fan weird: Ho cercato di fare precisione, potrei aver sbagliato, eh!

The [[Busy Beaver function]] BB (called ''S'' originally) was introduced by [[Tibor Radó]] in 1962 for 2-symbol [[Turing machines]] and later generalised to ''m''-symbol Turing machines:<ref>Radó, T. (1962), On Non-Computable Functions. Bell System Technical Journal, 41: 877-884. https://doi.org/10.1002/j.1538-7305.1962.tb00480.x</ref><ref>Brady, Allen H, and the Meaning of Life, 'The Busy Beaver Game and the Meaning of Life', in Rolf Herken (ed.), The Universal Turing Machine: A Half-Century Survey (Oxford, 1990; online edn, Oxford Academic, 31 Oct. 2023), https://doi.org/10.1093/oso/9780198537748.003.0009, accessed 8 June 2024.</ref>

{| class="wikitable"
|-
| BB(''n'', ''m'') = Maximum number of steps taken by a halting ''n''-state, ''m''-symbol Turing machine starting from a blank (all 0) tape
|}

The 2-symbol case BB(''n'', 2) is abbreviated as BB(''n''). The busy beaver function is not computable, but a few of its values are known:

{| class="wikitable"
|+ Small busy beaver values<ref>P. Michel, "[https://bbchallenge.org/~pascal.michel/ha.html Historical survey of Busy Beavers]".</ref>
! !!2-state!!3-state !!4-state!!5-state!!6-state
!7-state
|-
! 2-symbol
| [[BB(2)]] = 6
| [[BB(3)]] = 21
| [[BB(4)]] = 107
| [[BB(5)]] = 47,176,870
| style="background: orange;" | [[BB(6)]] > <math>2 \uparrow \uparrow \uparrow 5</math>
| style="background: orange;" | [[BB(7)]] > <math>2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|-
! 3-symbol
| [[BB(2,3)]] = 38
| style="background: orange;" | [[BB(3,3)]] > <math>10^{17}</math>
| style="background: #ffe4b2;" | [[BB(4,3)]] > <math>10 \uparrow^{4} 4</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 4-symbol
| [[BB(2,4)]] = 3,932,964
| style="background: #ffe4b2;" | [[BB(3,4)]] > <math>2 \uparrow^{15} 5</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 5-symbol
| style="background: orange;" | [[BB(2,5)]] > <math>10\uparrow\uparrow 4</math>
| style="background: #ffe4b2;" | [[BB(3,5)]] > <math> f_\omega(2 \uparrow^{15} 5)</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 6-symbol
| style="background: #ffe4b2;" | [[BB(2,6)]] > <math>10 \uparrow\uparrow\uparrow 3</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|}

In the above table, cells are highlighted in orange when there are known [[Cryptids]] (mathematically-hard machines) in that class, and cells are highlighted in light orange when the existence of a Cryptid is given by using a known one with less states or symbols.

1-state domains and 1-symbol domains are omitted in the table as [[BB(1,m)]] = 1 and [[BB(n,1)]] = n.

== About bbchallenge ==
[[bbchallenge]] is a massively collaborative research project whose general goal is to obtain more knowledge on the [[Busy Beaver function]]. In practice, it mainly consists in collaboratively building [[Deciders]], programs that automatically prove that some Turing machines do not halt. Other efforts also include:

* Formalising results using theorem provers (such as [https://en.wikipedia.org/wiki/Rocq Rocq])
* Maintaining [[Holdouts lists]] for small busy beaver values
* [[Analysis Tools and Techniques|Proving]] the behavior of [[:Category:Individual Machines|Individual machines]]
* Finding [[Cryptids]] (mathematically-hard machines)
* Searching for new [[Champions]]
* Building [[Accelerated Simulator]]s to simulate halting machines faster
* Writing papers and giving talks about busy beaver, see [[Papers & Talks]]

In June 2024, bbchallenge achieved a significant milestone by proving in [https://en.wikipedia.org/wiki/Rocq Rocq] (previously known as Coq) that the 5th busy beaver value, [[BB(5)]], is equal to the lower bound found in 1989: 47,176,870.<ref>H. Marxen and J. Buntrock. Attacking the Busy Beaver 5.
Bulletin of the EATCS, 40, pages 247-251, February 1990. https://turbotm.de/~heiner/BB/mabu90.html</ref>

== This Month in Beaver Research (TMBR) ==
[[:Category:This Month in Beaver Research|This Month in Beaver Research]] (TMBR, pronounced "timber") is a monthly summary of Busy Beaver research progress. Here are the three most recent released entries:

* [[TMBR: December 2025|TMBR Genuary 2026]]
* [[TMBR: November 2025|TMBR December 2025]]
* [[TMBR: October 2025|TMBR November 2025]]

[[TMBR: February 2026|TMBR February 2026]] and [[TMBR: March 2026]] are currently work in progress.

This Year in Beaver Research (TYBR) is a yearly summary of Busy Beaver research progress. Its first edition, [[TYBR: 2025]] is currently work in progress.

==Notes==
<references />

TMBR: January 2026

2026-03-06T18:09:36Z

BB fan weird: E solo che e gia stato fatto il TMBR di gennaio, e stato completato

{{TMBRnav|December 2025|February 2026}}

This is the first edition of TMBR in 2026.

TODO: mxdys released C++ implementation of FAR decider [https://discord.com/channels/960643023006490684/1239205785913790465/1462674863398326362 on 1/18], also add [[Code repositories]] page (is the FAR decider present there?)

== Champions ==

* nickdrozd discovered a new [[BLB|BLB(3,3)]] [[champion]] ({{TM|1RB2RB1LA_2LC0LB2LB_2RC2RA0LC}}) which blanks the tape after running for more than 1042,745 steps.
* creeperman7002 discovered <code>1TB1PA_1PC0PA_1TA0PD_---1TA</code>, a [[TT|TT(4,2)]] TM which runs for 48,186 steps and <code>1TA2PB3TB---_3TA1PB1TA1PA</code>, a [[TT|TT(2,4)]] TM which runs for more than 3.467*1015 steps.
* A new lower bound of <math>f_{\varepsilon_0 \omega^{\omega^{3}}}\left(4\right)</math> was computed for the [[Busy Beaver for lambda calculus#Champions|BBλ(91)]] champion.
* New champions were discovered for BBλ(61), BBλ(86), BBλ(90), BBλ(94), BBλ(96) and BBλ(100).

== Blog Posts ==

* 16 Jan 2026. Nick Drozd. [https://nickdrozd.github.io/2026/01/16/ai-lin-busy-beaver.html AI Reproduction of Lin's Busy Beaver Proof].
* 28 Jan 2026. John Tromp. [https://tromp.github.io/blog/2026/01/28/largest-number-revised The largest number representable in 64 bits - Revised].

== BB Adjacent ==

* [https://discord.com/channels/960643023006490684/1458010522967609425/1458010522967609425 Uniform-Action Busy Beavers] were introduced, and lower bounds have been given up to BBu(6) (with BBu(2)=6, BBu(3)=17 and BBu(4)=29)

== Theory ==
@ConePine [https://discord.com/channels/960643023006490684/960643023530762341/1458876275749032232 shared an idea] on how the value of BB(5) can be proved without enumerating Turing machines.

== Holdouts ==

{| class="wikitable"
|+BB Holdout Reduction by Domain
!Domain
!New Holdout Count
!Previous Holdout Count
!Holdout Reduction
!% Reduction
|-
|[[BB(2,5)]]
|74
|75
|1
|1.3%
|-
|[[BB(6)]]
|1314
|1326
|12
|0.9%
|-
|[[BB(7)]]
|19,303,801
|20,387,509
|1,083,708
|5.32%
|-
|[[BB(2,6)]]
|558,039
|870,085
|312,046
|35.86%
|}
* [[BB(6)]]: '''15''' solved machines. 1311 holdouts. 2 informally solved this month.
** Progress has been made in reducing the list of machines not simulated up to 1e13, by Alistaire and aparker: see [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 spreadsheet]. Current count: '''205'''. (Total reduction: '''73''' (+1 later). 39 machines[https://discord.com/channels/960643023006490684/1456755170527543420/1456755170527543420 <nowiki>[1]</nowiki>][https://discord.com/channels/960643023006490684/1456757189787127971/1456757189787127971 <nowiki>[2]</nowiki>][https://discord.com/channels/960643023006490684/1456757980203585678/1456757980203585678 <nowiki>[3]</nowiki>][https://discord.com/channels/960643023006490684/1456758622762696815/1456758622762696815 <nowiki>[4]</nowiki>][https://discord.com/channels/960643023006490684/1456759118818971709/1456759118818971709 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1456761765688901724/1456761765688901724 <nowiki>[6]</nowiki>][https://discord.com/channels/960643023006490684/1456762231491657830/1456762231491657830 <nowiki>[7]</nowiki>][https://discord.com/channels/960643023006490684/1456310285462802587/1456760768472289321 <nowiki>[8]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1456932659740545217 <nowiki>[9]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1456933097009320089 <nowiki>[10]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1456933714629099593 <nowiki>[11]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1456934113943621653 <nowiki>[12]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1457147385309565021 <nowiki>[13]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1457409447084429594 <nowiki>[11 more]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1457497643478683770 <nowiki>[4 more]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1457644806984437760 <nowiki>[10 more]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1457992404597739672 <nowiki>[11 more]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1458413343088840734 <nowiki>[17 more]</nowiki>][https://discord.com/channels/960643023006490684/1458063260959113279/1468676578455064742 <nowiki>[aparker's machine]</nowiki>] simulated out, plus 6 solved machines).
** Alistaire found a halting machine in the list mentioned above, see [https://discord.com/channels/960643023006490684/1239205785913790465/1456317703156531211 Discord]. Approximate score: 4e12. Later, he found another halting machine in the same list, see [https://discord.com/channels/960643023006490684/1239205785913790465/1456381492019069192 Discord] - approximate score: 1.5e18.
** @mxdys shared a list of machines that seem to be provable. See [https://discord.com/channels/960643023006490684/1460495597386731643/1460495597386731643 Discord].
** Dyuan [https://discord.com/channels/960643023006490684/1460495597386731643/1463044134263853066 shared an informal proof] for two machines nonhalting, using [[Longitudinal Analysis]].
** @Peacemaker II [https://discord.com/channels/960643023006490684/1239205785913790465/1466337165305974806 found] two machines to halt with a bespoke [[accelerated simulator]].
** @mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1466438332677619956 released] a [[holdouts list]] of 1314 machines up to equivalence, which is a 1% reduction (12 machines) from last month. 4 of these were not previously simulated up to 1e13 steps, thus that count has been lowered to 206.
** @mxdys [https://discord.com/channels/960643023006490684/1460495597386731643/1466440885700006123 announced] the start of translating three proofs for halting machines into [[wikipedia:Rocq|Rocq]] (two machines found by @Peacemaker II this month, one proof for [[1RB0RC 0LC0LB 0LD1LC 0LE1LA 0LF--- 1RF1RA|a machine]] found to halt by Racheline in July 2024, which has previously been missed when accounting for the informal holdout count, therefore the informal holdout count is now one less).
** All machines of the "Unknown" class have been simulated up to 1e13 steps, with aparker simulating the last machine to 1.15e13: [https://discord.com/channels/960643023006490684/1458063260959113279/1468676578455064742 Discord].
*[[BB(7)]]:
**Andrew Ducharme further continued reducing the number of holdouts, from 20,387,509 to '''20,197,978''' TMs, a 0.93% reduction. This puts the total compute time on the current BB(7) pipeline just over '''50,000''' hours.[https://discord.com/channels/960643023006490684/1369339127652159509/1459719965396566078]
**The holdout count was further reduced to '''19,879,953''' (a 1.57% reduction), which breaks the 20 million barrier.[https://discord.com/channels/960643023006490684/1369339127652159509/1460718041577951365]
**Another reduction brought the holdout count down to '''19,781,295''', a 0.5% reduction, then '''19,303,801''', a 2.41% reduction.
*[[BB(2,5)]]:
**Andrew Ducharme '''[https://discord.com/channels/960643023006490684/1259770421046411285/1466208979511414885 found] a machine to be nonhalting via @mxdys's FAR decider''', which was [https://discord.com/channels/960643023006490684/1259770421046411285/1466331107279769736 confirmed in Rocq].
*[[BB(2,6)]]:
**Andrew Ducharme reduced the number of holdouts from 870,085 to '''867,008''', a 0.35% reduction, with more application of Enumerate.py.
**Using the newly released mxdys FAR decider, the holdout count was brought down to '''558,039''', a 35.6% reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1467385975704391897]

[[Category:This Month in Beaver Research|2026-01]]

User:BB fan weird

2026-03-06T14:25:45Z

BB fan weird:

Welcome to my user page!Hi everyone! I am bb weird fan. I am 8 years old and I am super excited about the Busy Beaver challenge! Even though I am still a beginner in theoretical computer science and I'm learning how Turing Machines work, I love making predictions about giant numbers. My current predictions: BB(6) > \(2↑↑↑6 BB(7) >2[11]5[11]3 (I predict this involves about 13 to 15 arrows!) I have already created 17 pages on this wiki. I might be young, but I'm committed to helping the experts to solve BB(7)!

I have much happy, but i a time wrong the real number, thanks

User:BB fan weird

2026-03-06T14:22:17Z

BB fan weird:

User:BB fan weird

2026-03-06T14:21:57Z

BB fan weird:

TMBR: March 2026

2026-03-06T14:17:25Z

BB fan weird: Ho precisato perche se c'è una cosa che e differente da un altra ma e vera, allora preciso ĺ altra

{{TMBRnav|February 2026|April 2026}}

''This edition of TMBR is in progress and has not yet been released. Please add any notes you think may be relevant (including in the form a of a TODO with a link to any relevant Discord discussion).''

[[:Category:This Month in Beaver Research|This Month in Beaver Research]] for February 2026. TODO: Write a proper introductory paragraph.

== Holdouts ==

* [[BB(6)]]: 1e14 machines: '''175'''. 1e15 machines: '''239'''. '''10 solved machines'''.
** Holdouts lists for machines not simulated to '''1e14 and 1e15''' steps were created. The holdouts list count is '''175 and 239''' respectively. See [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 Spreadsheet for BB6].
** Later, [https://discord.com/channels/960643023006490684/1477591686514212894/1478401591307407554 prurq found 10 more machines] in the holdouts list that had previously been simulated to 1e15: thus the new 1e15 holdout count is '''239.'''
** Andrew Ducharme [https://discord.com/channels/960643023006490684/1239205785913790465/1478626634729914450 solved two machines] using FAR.
** prurq found two machines [https://discord.com/channels/960643023006490684/1239205785913790465/1478518151393312822][https://discord.com/channels/960643023006490684/1239205785913790465/1478525451826888775] to be [[Translated Cycler|Translated Cyclers]]. Shawn Ligocki [https://discord.com/channels/960643023006490684/1239205785913790465/1478781268664778989 verified one of them] and discovered its preperiod to be over <math>10^{12}</math> steps. mxdys verified the other. Both were 1e14 and 1e15 holdouts, thus reducing those holdout counts by 2.
** mxdys found four more [https://discord.com/channels/960643023006490684/1239205785913790465/1478734570672095333][https://discord.com/channels/960643023006490684/1239205785913790465/1479045001089384600] Translated Cyclers in the remaining holdouts and solved [https://discord.com/channels/960643023006490684/1239205785913790465/1478828100707025151 one more TM] using FAR.
** Andrew Ducharme [https://discord.com/channels/960643023006490684/1239205785913790465/1479313219003748484 solved a machine] using FAR. This machine was a 1e14 and 1e15 holdout, thus reducing those holdout counts by 1.

[[Category:This Month in Beaver Research|2026-03]]

TMBR: March 2026

2026-03-05T22:01:03Z

BB fan weird: Ho ri fatto di precisare

{{TMBRnav|February 2026|April 2026}}

''This edition of TMBR is in progress and has not yet been released. Please add any notes you think may be relevant (including in the form a of a TODO with a link to any relevant Discord discussion).''

[[:Category:This Month in Beaver Research|This Month in Beaver Research]] for February 2026. TODO: Write a proper introductory paragraph.

== Holdouts ==

* [[BB(6)]]: 1e14 machines: '''176'''. 1e15 machines: '''240'''. '''9 solved machines'''.
** Holdouts lists for machines not simulated to '''1e14 and 1e15''' steps were created. The holdouts list count is '''176 and 240''' respectively. See [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 Spreadsheet for BB6].
** Later, [https://discord.com/channels/960643023006490684/1477591686514212894/1478401591307407554 prurq found 10 more machines] in the holdouts list that had previously been simulated to 1e15: thus the new 1e15 holdout count is '''240'''.
** Andrew Ducharme [https://discord.com/channels/960643023006490684/1239205785913790465/1478626634729914450 solved two machines] using FAR.
** prurq found two machines [https://discord.com/channels/960643023006490684/1239205785913790465/1478518151393312822][https://discord.com/channels/960643023006490684/1239205785913790465/1478525451826888775] to be [[Translated Cycler|Translated Cyclers]]. Shawn Ligocki [https://discord.com/channels/960643023006490684/1239205785913790465/1478781268664778989 verified one of them] and discovered its preperiod to be over <math>10^{12}</math> steps. mxdys verified the other. Both were 1e14 and 1e15 holdouts, thus reducing those holdout counts by 2.
** mxdys found four more [https://discord.com/channels/960643023006490684/1239205785913790465/1478734570672095333][https://discord.com/channels/960643023006490684/1239205785913790465/1479045001089384600] Translated Cyclers in the remaining holdouts and solved [https://discord.com/channels/960643023006490684/1239205785913790465/1478828100707025151 one more TM] using FAR.

[[Category:This Month in Beaver Research|2026-03]]

TMBR: March 2026

2026-03-05T22:00:17Z

BB fan weird: Ho precisato di nuovo

{{TMBRnav|February 2026|April 2026}}

''This edition of TMBR is in progress and has not yet been released. Please add any notes you think may be relevant (including in the form a of a TODO with a link to any relevant Discord discussion).''

[[:Category:This Month in Beaver Research|This Month in Beaver Research]] for February 2026. TODO: Write a proper introductory paragraph.

== Holdouts ==

* [[BB(6)]]: 1e14 machines: '''176'''. 1e15 machines: '''240'''. '''9 solved machines'''.
** Holdouts lists for machines not simulated to '''1e14 and 1e15''' steps were created. The holdouts list count is '''178 and 240''' respectively. See [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 Spreadsheet for BB6].
** Later, [https://discord.com/channels/960643023006490684/1477591686514212894/1478401591307407554 prurq found 10 more machines] in the holdouts list that had previously been simulated to 1e15: thus the new 1e15 holdout count is '''240'''.
** Andrew Ducharme [https://discord.com/channels/960643023006490684/1239205785913790465/1478626634729914450 solved two machines] using FAR.
** prurq found two machines [https://discord.com/channels/960643023006490684/1239205785913790465/1478518151393312822][https://discord.com/channels/960643023006490684/1239205785913790465/1478525451826888775] to be [[Translated Cycler|Translated Cyclers]]. Shawn Ligocki [https://discord.com/channels/960643023006490684/1239205785913790465/1478781268664778989 verified one of them] and discovered its preperiod to be over <math>10^{12}</math> steps. mxdys verified the other. Both were 1e14 and 1e15 holdouts, thus reducing those holdout counts by 2.
** mxdys found four more [https://discord.com/channels/960643023006490684/1239205785913790465/1478734570672095333][https://discord.com/channels/960643023006490684/1239205785913790465/1479045001089384600] Translated Cyclers in the remaining holdouts and solved [https://discord.com/channels/960643023006490684/1239205785913790465/1478828100707025151 one more TM] using FAR.

[[Category:This Month in Beaver Research|2026-03]]

TMBR: March 2026

2026-03-04T20:00:11Z

BB fan weird: Ho cercato di precisare al meglio possibile, mi piace la precisione!

TMBR: March 2026

2026-03-04T19:59:11Z

BB fan weird: Ho cercato di migliorare come era, se c'era scritto 242 (che e vero), allora io scrivo 242 per precisare!

User:BB fan weird

2026-03-03T17:15:45Z

BB fan weird:

Main Page

2026-03-03T13:33:59Z

BB fan weird: Ho cercato di aggiungere il TMBR del marzo 2026 perche e appena uscito

The [[Busy Beaver function]] BB (called ''S'' originally) was introduced by [[Tibor Radó]] in 1962 for 2-symbol [[Turing machines]] and later generalised to ''m''-symbol Turing machines:<ref>Radó, T. (1962), On Non-Computable Functions. Bell System Technical Journal, 41: 877-884. https://doi.org/10.1002/j.1538-7305.1962.tb00480.x</ref><ref>Brady, Allen H, and the Meaning of Life, 'The Busy Beaver Game and the Meaning of Life', in Rolf Herken (ed.), The Universal Turing Machine: A Half-Century Survey (Oxford, 1990; online edn, Oxford Academic, 31 Oct. 2023), https://doi.org/10.1093/oso/9780198537748.003.0009, accessed 8 June 2024.</ref>

{| class="wikitable"
|-
| BB(''n'', ''m'') = Maximum number of steps taken by a halting ''n''-state, ''m''-symbol Turing machine starting from a blank (all 0) tape
|}

The 2-symbol case BB(''n'', 2) is abbreviated as BB(''n''). The busy beaver function is not computable, but a few of its values are known:

{| class="wikitable"
|+ Small busy beaver values<ref>P. Michel, "[https://bbchallenge.org/~pascal.michel/ha.html Historical survey of Busy Beavers]".</ref>
! !!2-state!!3-state !!4-state!!5-state!!6-state
!7-state
|-
! 2-symbol
| [[BB(2)]] = 6
| [[BB(3)]] = 21
| [[BB(4)]] = 107
| [[BB(5)]] = 47,176,870
| style="background: orange;" | [[BB(6)]] > <math>2 \uparrow \uparrow \uparrow 5</math>
| style="background: orange;" | [[BB(7)]] > <math>2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|-
! 3-symbol
| [[BB(2,3)]] = 38
| style="background: orange;" | [[BB(3,3)]] > <math>10^{17}</math>
| style="background: #ffe4b2;" | [[BB(4,3)]] > <math>10 \uparrow^{4} 4</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 4-symbol
| [[BB(2,4)]] = 3,932,964
| style="background: #ffe4b2;" | [[BB(3,4)]] > <math>2 \uparrow^{15} 5</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 5-symbol
| style="background: orange;" | [[BB(2,5)]] > <math>10\uparrow\uparrow 4</math>
| style="background: #ffe4b2;" | [[BB(3,5)]] > <math> f_\omega(2 \uparrow^{15} 5)</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|-
! 6-symbol
| style="background: #ffe4b2;" | [[BB(2,6)]] > <math>10 \uparrow\uparrow\uparrow 3</math>
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
| style="background: #ffe4b2;" |
|}

In the above table, cells are highlighted in orange when there are known [[Cryptids]] (mathematically-hard machines) in that class, and cells are highlighted in light orange when the existence of a Cryptid is given by using a known one with less states or symbols.

1-state domains and 1-symbol domains are omitted in the table as [[BB(1,m)]] = 1 and [[BB(n,1)]] = n.

== About bbchallenge ==
[[bbchallenge]] is a massively collaborative research project whose general goal is to obtain more knowledge on the [[Busy Beaver function]]. In practice, it mainly consists in collaboratively building [[Deciders]], programs that automatically prove that some Turing machines do not halt. Other efforts also include:

* Formalising results using theorem provers (such as [https://en.wikipedia.org/wiki/Rocq Rocq])
* Maintaining [[Holdouts lists]] for small busy beaver values
* [[Analysis Tools and Techniques|Proving]] the behavior of [[:Category:Individual Machines|Individual machines]]
* Finding [[Cryptids]] (mathematically-hard machines)
* Searching for new [[Champions]]
* Building [[Accelerated Simulator]]s to simulate halting machines faster
* Writing papers and giving talks about busy beaver, see [[Papers & Talks]]

In June 2024, bbchallenge achieved a significant milestone by proving in [https://en.wikipedia.org/wiki/Rocq Rocq] (previously known as Coq) that the 5th busy beaver value, [[BB(5)]], is equal to the lower bound found in 1989: 47,176,870.<ref>H. Marxen and J. Buntrock. Attacking the Busy Beaver 5.
Bulletin of the EATCS, 40, pages 247-251, February 1990. https://turbotm.de/~heiner/BB/mabu90.html</ref>

== This Month in Beaver Research (TMBR) ==
[[:Category:This Month in Beaver Research|This Month in Beaver Research]] (TMBR, pronounced "timber") is a monthly summary of Busy Beaver research progress. Here are the three most recent released entries:

* [[TMBR: December 2025|TMBR December 2025]]
* [[TMBR: November 2025|TMBR November 2025]]
* [[TMBR: October 2025|TMBR October 2025]]

[[TMBR: January 2026|TMBR January 2026]],[[TMBR: February 2026|TMBR February 2026]] and [[TMBR: March 2026]] are currently work in progress.

This Year in Beaver Research (TYBR) is a yearly summary of Busy Beaver research progress. Its first edition, [[TYBR: 2025]] is currently work in progress.

==Notes==
<references />

BB(12)

2026-02-27T19:34:15Z

BB fan weird: Created page with "il problema a 12 stati e 2 simboli e inesplorato al 100% (per ora), io prevedo che bb(12) sarà praticamente impossibile da enumerare"

il problema a 12 stati e 2 simboli e inesplorato al 100% (per ora), io prevedo che bb(12) sarà praticamente impossibile da enumerare

User:BB fan weird

2026-02-19T18:58:58Z

BB fan weird:

Welcome to my user page!Hi everyone! I am bb weird fan. I am 8 years old and I am super excited about the Busy Beaver challenge! Even though I am still a beginner in theoretical computer science and I'm learning how Turing Machines work, I love making predictions about giant numbers. My current predictions: BB(6) > \(2↑↑↑6 BB(7) >2[11]5[11]3 (I predict this involves about 13 to 15 arrows!) I have already created 12 pages on this wiki. I might be young, but I'm committed to helping the experts solve BB(7)!

BB(6)

2026-02-18T20:23:49Z

BB fan weird:

The 6-state, 2-symbol Busy Beaver problem, '''BB(6)''', refers to the unsolved 6th value of the [[Busy Beaver function]]. With the discovery of the [[Cryptid]] machine [[Antihydra]] in June 2024, we now know that we must solve a [[Collatz-like]] problem in order to solve BB(6) and thus [https://www.sligocki.com/2024/07/06/bb-6-2-is-hard.html BB(6) is Hard].

The current BB(6) [[champion]] {{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}} was discovered by mxdys in June 2025, proving the lower bound:

<math display="block">S(6) > \Sigma(6) > 2 \uparrow\uparrow\uparrow 5</math>

== History ==
[[File:BB(6) holdouts decrease over time.png|alt=BB(6) Holdouts count decrease overtime.|thumb|Number of BB(6) holdouts over time.]]
@mxdys's informal [[Holdouts lists|holdouts list]] has 1314 machines up to equivalence as of January 2026. Partial Rocq proof is [https://github.com/ccz181078/busycoq/tree/BB6 available on Github].

Always up-to-date annotated spreadsheet, with links to Discord discussions: [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=1330361301#gid=1330361301 Spreadsheet].

All holdouts have been simulated up to at least 1e13 steps. 210 holdouts have not been simulated out to 1e13 steps. (The list can be found in the spreadsheet)
== Cryptids ==
Several [[Turing machines]] have been found that are [[Cryptids]], considered so because each of them have a [[Collatz-like]] halting problem, a type of problem that is generally difficult to solve. However, probabilistic arguments have allowed all but one of them to be categorized as [[probviously]] halting or probviously non-halting.

Probviously non-halting Cryptids:

* {{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA}}, [[Antihydra]]
* {{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}, a variant of [[Hydra]] and Antihydra
* {{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}, similar to Antihydra
* {{TM|1RB0LD_1RC1RF_1LA0RA_0LA0LE_1LD1LA_0RB---|undecided}}, similar to Antihydra
* {{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}, similar to Antihydra
* {{TM|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|undecided}}

Probviously halting Cryptids:

* {{TM|1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA}}, [[Lucy's Moonlight]]
* {{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}, a family of 16 related TMs
* {{TM|1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF}}
* {{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB}}
* {{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA}}
* {{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE}}
* {{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}}
* {{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE}}
* {{TM|1RB1LD_1RC0LE_1LA1RE_0LF1LA_1RB0RB_---0LB}}
* {{TM|1RB0RE_1LC0RA_1LA1LD_1LC1LF_0LC0LB_1LE---}}

Although {{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE}} behaves similarly to the probviously halting Cryptids, it is estimated to have a 3/5 chance of becoming a [[translated cycler]] and a 2/5 chance of halting.

There are a few machines considered notable for their chaotic behaviour, but which have not been classified as Cryptids due to seemingly lacking a connection to any known open mathematical problems, such as Collatz-like problems.

Potential Cryptids:

* {{TM|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}}
* {{TM|1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE|undecided}}
* {{TM|1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC|undecided}}
* {{TM|1RB1RF_1LC1LF_0RE1LD_0LB1LD_---1RC_1RA0RD|undecided}}

== Top Halters ==
Below is a table of the machines with the 20 highest known runtimes.<ref>Shawn Ligocki's list of 6-state, 2-symbol machines with large runtimes ([https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/6x2.txt Link])</ref> Their sigma scores are expressed using an extension of [[wikipedia:Knuth's_up-arrow_notation|Knuth's up-arrow notation]].<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/06/25/ext-up-notation.html "Extending Up-arrow Notation"]</ref>
{| class="wikitable"
|+Top Known BB(6) Halters
!Standard format
!(approximate) Σ
!Discoverer
|-
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|2 ↑↑↑ 5
|mxdys
|-
|{{TM|1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB|halt}}
|10 ↑↑ 11010000
|mxdys
|-
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}
|10 ↑↑ 15.60465
|Pavel Kropitz
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LB0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1RF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB1RC_1LC1RE_1LD0LB_1RE1LC_1LE0RF_1RZ1RA|halt}}
|10 ↑↑ 7.23619
|
|-
|{{TM|1RB1RA_1LC1LE_1RE0LD_1LC0LF_1RZ0RA_0RA0LB|halt}}
|10 ↑↑ 6.96745
|poppuncher
|-
|{{TM|1RB0RF_1LC0RA_1RZ0LD_1LE1LD_1RB1RC_0LD0RE|halt}}
|10 ↑↑ 5.77573
|poppuncher
|-
|{{TM|1RB0LA_1LC1LF_0LD0LC_0LE0LB_1RE0RA_1RZ1LD|halt}}
|10 ↑↑ 5.63534
|Shawn Ligocki
|-
|{{TM|1RB1RE_1LC1LF_1RD0LB_1LE0RC_1RA0LD_1RZ1LC|halt}}
|10 ↑↑ 5.56344
|
|-
|{{TM|1RB0LE_0RC1RA_0LD1RF_1RE0RB_1LA0LC_0RD1RZ|halt}}
|10 ↑↑ 5.12468
|
|-
|{{TM|1RB0RF_1LC1LB_0RE0LD_0LC0LB_0RA1RE_0RD1RZ|halt}}
|10 ↑↑ 5.03230
|
|-
|{{TM|1RB1LA_1LC0RF_1LD1LC_1LE0RE_0RB0LC_1RZ1RA|halt}}
|10 ↑↑ 4.91072
|
|-
|{{TM|1RB0LE_1LC1RA_1RE0LD_1LC1LF_1LA0RC_1RZ1LC|halt}}
|10 ↑↑ 3.33186
|
|-
|{{TM|1RB1RF_1LC1RE_0LD1LB_1LA0RA_0RA0RB_1RZ0RD|halt}}
|10 ↑↑ 3.31128
|
|-
|{{TM|1RB0LF_1LC0RA_1RD0LB_1LE1RC_1RZ1LA_1LA1LE|halt}}
|10 ↑↑ 3.18855
|
|-
|{{TM|1RB0RF_1LC1RB_0RD0LB_1RZ0LE_1RE0RA_1RD1RE|halt}}
|10 ↑↑ 3.16005
|
|-
|{{TM|1RB1RZ_0LC0LD_1LD1LC_1RE1LB_1RF1RD_0LD0RA|halt}}
|<math>10^{646\,456\,993}</math>
|Pavel Kropitz
|-
|{{TM|1RB0RC_0LC0LB_0LD1LC_0LE1LA_0LF---_1RF1RA|halt}}
|<math>> 10^{16\,780\,816}</math>
|Racheline
|}
The runtimes are presumed to be about <math>\text{score}^2</math> which is roughly indistinguishable in tetration notation.

== Techniques ==
Simulating tetrational machines, such as the former champion {{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}, requires [[Accelerated simulator|accelerated simulation]] that can handle Collatz Level 2 [[Inductive rule|inductive rules]]. In other words, it requires a simulator that can prove the rules:

<math display="block">\begin{array}{lcl}
C(4k) & \to & {\operatorname{Halt}}\Big(\frac{3^{k+3} - 11}{2}\Big) \\
C(4k+1) & \to & C\Big(\frac{3^{k+3} - 11}{2}\Big) \\
C(4k+2) & \to & C\Big(\frac{3^{k+3} - 11}{2}\Big) \\
C(4k+3) & \to & C\Big(\frac{3^{k+3} + 1}{2}\Big) \\
\end{array}</math>

and also compute the remainder mod 3 of numbers produced by applying these rules 15 times (which requires some fancy math related to [[wikipedia:Euler's_totient_function|Euler's totient function]]).

We are also applying existing automatic deciders on current holdout lists with more extreme choices of parameters (more computational resources). [[User:XnoobSpeakable|XnoobSpeakable]] was able to solve 11 of the final 2728 holdouts using higher order parameters with the Ligockis' Enumerate.py. An example command line entry is:
<syntaxhighlight lang="bash">
python3 Code/Enumerate.py --infile "bb6in/bb6tm{i}.txt" --outfile "bb6out/t{i}.pb" -r --no-steps --exp-linear-rules --max-loops=50_000_000 --block-mult=3 --max-block-size=100 --time=500 --force --save-freq=1
</syntaxhighlight>
XnoobSpeakable ran Enumerate.py on all TMs in the 2728 holdout list with the above max-loops and max-block-size parameters using <code>--block-mult=1</code> ,<code>--block-mult=2</code> , and <code>--block-mult=3</code>. For context, during the Stage 2 BB(7) enumeration, where speed was more important due to the tens of millions of known holdouts, parameters of <code>--max-loops=100_000 --block-mult=2 --time=30 --save-freq=100</code> were used.

@Iijil's [[MITMWFAR|MITMWFAR decider]] is likely too weak to be of any assistance: running the decider on 2650 BB(6) holdouts, using parameters not strong enough to solve BB(5) TMs, took prohibitively long to compute. Instead, [https://discord.com/channels/960643023006490684/1028746861395316776/1442964185599447152 a new FAR method] by mxdys was able to decide 113 of the 1534 holdouts ([https://github.com/ccz181078/TM/tree/FAR code] on GitHub).

== References ==
<references />

[[Category:BB Domains]][[Category:BB(6)]]

User:BB fan weird

2026-02-18T20:15:06Z

BB fan weird:

Welcome to my user page!Hi everyone! I am bb weird fan. I am 8 years old and I am super excited about the Busy Beaver challenge! Even though I am still a beginner in theoretical computer science and I'm learning how Turing Machines work, I love making predictions about giant numbers. My current predictions: BB(6) > \(2\uparrow \uparrow \uparrow 6\)BB(7) > \(2[11]5[11]3\) (I predict this involves about 13 to 15 arrows!) I have already created 12 pages on this wiki. I might be young, but I'm committed to helping the experts solve BB(7)!

BB(3,6)

2026-02-18T20:13:45Z

BB fan weird: Created page with "Il problema a 3 stati e 6 simboli e inesplorato, confermo (quasi perche ho meno di 2 cifre di anni e sto ancora imparando a usare le TuringMachine) che bb(3,6)>f omega+100 (3)"

Il problema a 3 stati e 6 simboli e inesplorato, confermo (quasi perche ho meno di 2 cifre di anni e sto ancora imparando a usare le TuringMachine) che bb(3,6)>f omega+100 (3)

BB(11)

2026-02-17T14:55:17Z

BB fan weird: Created page with "Il problema di bb(11) e inesplorato ed e quasi impossibile per ora, NESSUN LIMITE INFERIORE"

Il problema di bb(11) e inesplorato ed e quasi impossibile per ora, NESSUN LIMITE INFERIORE

BB(2,9)

2026-02-12T13:36:26Z

BB fan weird: Created page with "Il problema di bb(2,9) (2 stati e 9 simboli) e inesplorato da tutti, cio implica che bb(2,9) e difficilissimo, ho ipotizzato che bb(2,9)≥bb(7)"

Il problema di bb(2,9) (2 stati e 9 simboli) e inesplorato da tutti, cio implica che bb(2,9) e difficilissimo, ho ipotizzato che bb(2,9)≥bb(7)

BB(2,8)

2026-02-11T20:00:22Z

BB fan weird: Created page with "Il problema di bb(2,8) (macchina di Turing con 2 stati e 8 simboli) e inesplorato, cio significa che bb(2,8) e molto difficile, non ci sono limiti inferiori (lower bound) per bb(2,8)"

Il problema di bb(2,8) (macchina di Turing con 2 stati e 8 simboli) e inesplorato, cio significa che bb(2,8) e molto difficile, non ci sono limiti inferiori (lower bound) per bb(2,8)

User talk:BB fan weird

2026-02-11T12:57:55Z

BB fan weird:

User:BB fan weird

2026-02-10T17:07:34Z

BB fan weird: Created page with "Welcome to my user page!Hi everyone! I am bb weird fan. I am 8 years old and I am super excited about the Busy Beaver challenge! Even though I am still a beginner in theoretical computer science and I'm learning how Turing Machines work, I love making predictions about giant numbers. My current predictions: BB(6) > \(2\uparrow \uparrow \uparrow 5\)BB(7) > \(2[11]5[11]3\) (I predict this involves about 13 to 15 arrows!) I have already created 4 pages on this wiki. I m..."

Welcome to my user page!Hi everyone! I am bb weird fan. I am 8 years old and I am super excited about the Busy Beaver challenge! Even though I am still a beginner in theoretical computer science and I'm learning how Turing Machines work, I love making predictions about giant numbers. My current predictions: BB(6) > \(2\uparrow \uparrow \uparrow 5\)BB(7) > \(2[11]5[11]3\) (I predict this involves about 13 to 15 arrows!) I have already created 4 pages on this wiki. I might be young, but I'm committed to helping the experts solve BB(7)!

User talk:BB fan weird

2026-02-10T16:48:12Z

BB fan weird:

Talk:Main Page

2026-01-25T11:47:35Z

BB fan weird: /* Approssimazioni */ new section

Can you improve the change frequency of this wiki? --[[User:Jacobzheng|Jacobzheng]] ([[User talk:Jacobzheng|talk]]) 22:14, 20 September 2024 (UTC)

== Approssimazioni ==

Secondo me, bb(7)~2^[13 frecce] 6, io non lo sò, ma dico questo, bb(6)~2^[4 frecce] 4, ovviamente non ho una sfreda magica per vedere i valori, ma approssimo, bb(3,5)~f omega (2^[15 frecce] 6]

User talk:BB fan weird

2026-01-25T09:35:15Z

BB fan weird: Created page with "La busy beaver challenge è hard, ma bb(n,m) e per creare numeri ancora piu grandi, bb(3,5)~f omega (2^[15 volte ^] 5, bb(6) e un atomo confrontato con bb(3,5), bb(7) e un pochino pochino piu grande ma bb(3,5) e ĺ intero multiverso!"