BusyBeaverWiki - User contributions [en]

Champions

2026-11-05T06:28:28Z

Jacobzheng:

Busy Beaver '''Champions''' are the current record holding [[Turing machine|Turing machines]] which maximize a [[Busy Beaver function]]. In this article we focus specifically on the longest running TMs. Some have been proven to be the longest running of all (and so are the ultimate champion) while others are only current champions and may be usurped in the future. For smaller domains, Pascal Michel's website is the canonical source for [https://bbchallenge.org/~pascal.michel/bbc Busy Beaver champions] and the [https://bbchallenge.org/~pascal.michel/ha History of Previous Champions]. 1-state domains are omitted as [[BB(1,m)]] = 1 for m > 1.

== Trivial Champions ==
[[BB(n,1)]] = n

[[BB(1,m)]] = 1

== 2-Symbol TMs ==
Rows are blank if no champion has been found which surpasses a smaller size problem. Also take note that the <math>f_{x}(n)</math> used in the lower bounds represent the [[Fast-Growing Hierarchy]] while <math>\uparrow</math> represents [[wikipedia:Knuth's_up-arrow_notation|Knuth's up-arrow notation]]; see [[measuring large numbers]] for more information. Machines below 6 states are listed in [[Tree Normal Form|TNF]] format, where machines which tie for champion are all listed. Note that most champions above 6 states are self-reported and have not been independently verified.



{| class="wikitable"
|+
!Domain
!Runtime
!Champions
!Discovered by
!Verification
|-
|[[BB(2)]]
|<math>6</math>
|{{TM|1RB1LB_1LA1RZ|halt}} {{TM|1RB0LB_1LA1RZ|halt}} {{TM|1RB1RZ_1LB1LA|halt}} {{TM|1RB1RZ_0LB1LA|halt}} {{TM|0RB1RZ_1LA1RB|halt}}
|[[Tibor Radó]]
|Direct Simulation
|-
|[[BB(3)]]
|<math>21</math>
|{{TM|1RB1RZ_1LB0RC_1LC1LA|halt}}
|Proven by [[Shen Lin]]
|Direct Simulation
|-
|[[BB(4)]]
|<math>107</math>
|{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}}
|Allen Brady
|Direct Simulation
|-
|[[BB(5)]]
|<math>47\,176\,870</math>
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}
|Heiner Marxen & Jürgen Buntrock in 1989
|Direct Simulation
|-
|[[BB(6)]]
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|mxdys in 2025
|See mxdys's analysis on the TM page
|-
|[[BB(7)]]
|<math>> 3 \uparrow^{20} 3 > f_\omega(20)</math>
|{{TM|1RB0RG_1RC1RB_0RD1RZ_0LE0LD_0LF1LE_1RA1LF_1LC0RB|halt}}
|Jeff Ketchersid and Gemini in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/blob/main/BB7/1RB0RG_1RC1RB_0RD---_0LE0LD_0LF1LE_1RA1LF_0LD0RB.md Claimed analysis by Claude] never verified.
|-
|[[BB(8)]]
|<math>> f_\omega^3(12)</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LH0LC_1RG1LD_0RG0RF_1RG1RZ|halt}}
|Jeff Ketchersid and Gemini in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/blob/main/BB8/1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LH0LC_1RG1LD_0RG0RF_1RG---.md Claimed analysis by Claude] never verified.
|-
|BB(9)
|<math>> f_\omega^7(12)</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_0LH0LC_1RG1LD_0RG0RF_0LI1RZ_1RI1RC|halt}}
|Jeff Ketchersid and Gemini in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/blob/main/BB9/1RB0RA_1LC1LF_1RD0LB_1RA1LE_0LH0LC_1RG1LD_0RG0RF_0LI---_1RI1RC.md Claimed analysis by Claude] never verified.
|-
|BB(10)
|<math>> f_{\omega+1}(2 \uparrow^{36} 3) > g_{64}</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LI0LC_0RG1LD_1RH1LG_1LC0RG_0LJ0LI_1RZ0LC|halt}}
|[https://discord.com/channels/960643023006490684/1331570843829932063/1557806386073702622 Jeff Ketchersid and Claude in 10/2026 (claimed)]
|[https://github.com/jeff87654/busy-beaver/tree/main/BB10 Claimed analysis by Claude] never verified.
|-
|BB(11)
|<math>> f_{\omega+1}(f_\omega^2(5))</math>
|{{TM|1RG0LH_1RK1RC_0LD1LA_0RE1LD_0LI1RF_1RZ0RG_1RB1RG_1LA1LH_0LK0LJ_1LH1LI_1LJ1RE|halt}}
|Jeff Ketchersid and Gemini in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB11 Claimed analysis by Claude] never verified.
|-
|BB(12)
|<math>> f_{\omega+2}(f_{\omega+1}^2(3))</math>
|{{TM|1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LK1RF_1RG0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_1RZ0LL_0LH1LK|halt}}
|Jeff Ketchersid and Gemini in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB12 Claimed analysis by Claude] never verified.
|-
|BB(13)
|<math>> f_{\omega+2}^6(f_{\omega+1}^2(3))</math>
|{{TM|1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LK1RF_1RM0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_0LL0LL_0LH1LK_1RZ0LH|halt}}
|Jeff Ketchersid, Claude and Gemini in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB13 Claimed analysis by Claude] never verified.
|-
|BB(14)
|<math>> f_{\omega+3}(17)</math>
|{{TM|1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LK1RF_1RM0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_0LL0LL_0LN1LK_1RZ0LH_0LG1LJ|halt}}
|Jeff Ketchersid, Claude and Gemini in 11/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB14 Claimed analysis by Claude] never verified.
|-
|BB(15)
|
|
|
|
|-
|BB(16)
|
|
|
|
|-
|BB(17)
|<math>> f_{\omega+3}^3(9)</math>
|{{TM|1LB1LA_1RC0LA_1RD1RC_1RE1RK_1RF0LJ_1LG1RI_1RM0LM_1LG0RC_0RN0LE_1LA1LE_0LL1LB_0RM1LL_1LF1RH_1RQ0LO_1LP1LN_0RB1LI_1RZ1LO|halt}}
|Jeff Ketchersid, Claude and Gemini in 11/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB17 Claimed analysis by Claude] never verified.
|-
|BB(18)
|<math>> f_{\omega+4}(228)</math>
|{{TM|1LB1LA_1RC0LA_1RD1RC_1RE1RK_1RF0LJ_1LG1RI_1RM0LM_1LG0RC_0RN0LE_1LA1LE_0LL1LB_0RM1LL_1LF1RH_1RQ0LO_1LP1LR_0RB1LI_1RZ1RR_1RN0LO|halt}}
|Jeff Ketchersid, Claude and Gemini in 11/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB18 Claimed analysis by Claude] never verified.
|-
|BB(19)
|<math>> f_{\omega + 7}^2(5)</math>
|{{TM|1RQ1RL_0LI1RC_0LD1LF_0RA1LD_1RB1RE_1LH0LG_1LF1LG_1RE1RM_1LJ1LI_0LK1LI_1RL1RA_0RL0RE_0LN1RM_1LO0LO_1RE1LP_1LN1LN_1LR1RQ_1RZ1LS_0LE0LR|halt}}
|Jacobzheng in 2026
|Not independently verified.
|-
|BB(20)
|<math>> f_{\omega \times 2}(f_{\omega + 2}^2(f_{10}(2)))</math>
|{{TM|1LH0LG_0LI1RC_0LD1LA_0RE1LD_1RQ1RL_1RB1RF_1LA1LG_1RF1RM_1LJ1LI_0LK1LI_1RL1RE_0RL0RF_0LN1RM_0RF0LO_0RF1LP_0RF1LN_1LR1RQ_1RZ1LS_0LF1LT_0LF0LR|halt}}
|Jacobzheng in 2026
|Not independently verified.
|-
|BB(21)
|<math>> f_{\omega^2}^2(4 \uparrow\uparrow 341)</math>
|{{TM|0LI0LF_0RJ1RG_0RD1LC_1RH1RE_1LO0RH_1LA1LF_0LC1LA_1RB1RH_1RD0LF_1LP0RK_0LM1LL_0LL1LM_1LE1LN_0RQ0LM_0RP1LO_1LR0RH_1LF1RQ_1RZ0LS_1LH0LT_1LH1LU_0LE1LR|halt}}
|[https://discord.com/channels/960643023006490684/1026577255754903572/1274471360206344213 Racheline in 2024]
|Not independently verified.
|-
|colspan="5"|...
|-
|BB(29)
|<math>> f_{\omega^\omega}(3)</math>
|[https://github.com/jeff87654/busy-beaver/blob/main/BB29/hydra29.md]
|mxdys in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB29 Claimed analysis by Claude] never verified.
|-
|BB(31)
|<math>> f_{\omega \uparrow \uparrow 4}(3)</math>
|[https://github.com/jeff87654/busy-beaver/blob/main/BB31/hydra31.md]
|mxdys in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB31 Claimed analysis by Claude] never verified.
|-
|BB(33)
|<math>> f_{\omega \uparrow \uparrow 6}(3)</math>
|[https://github.com/jeff87654/busy-beaver/blob/main/BB33/hydra33.md]
|mxdys in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB33 Claimed analysis by Claude] never verified.
|-
|BB(48)
|<math>> f_{\varepsilon_1}(3)</math>
|[https://discord.com/channels/960643023006490684/1331570843829932063/1567585091172040726]
|[https://discord.com/channels/960643023006490684/1331570843829932063/1567585091172040726 roromachine14 and Gemini in 2026]
|Not independently verified.
|-
|BB(150)
|<math>> f_{lim(BMS)}(10\uparrow\uparrow 15)</math>
|[https://morphett.info/turing/turing.html?c95a199c8e8a3dd56452f8b7e28fabbf too large to show]
|Patcail in 2025<ref>https://discord.com/channels/960643023006490684/1026577255754903572/1328863966688182345</ref>
|Not independently verified.
|-
|BB(1015)
|<math>> D^{5}(99)</math> = Loader's Number
|
|CatsAreFluffy in 2024<ref>https://github.com/CatsAreFluffy/metamath-turing-machines/commit/85948b04fc4aeb983ca6d63d6aee5ad6ef308bfe</ref>
|Not independently verified.
|}

=== Conjectured ===

{| class="wikitable"
|+
!Domain
!Runtime
!Champions
!Discovered by
!Notes
|-
|BB(67)
|<math>> q^{81}(q(5)-2)</math>
|[https://wiki.bbchallenge.org/wiki/Talk:Champions]
|[https://discord.com/channels/960643023006490684/1331570843829932063/1339669138510970881 racheline in 2025]
|<math>q(n)</math> is derived from [https://en.wikipedia.org/wiki/Laver%20table Laver tables]. From [https://discord.com/channels/960643023006490684/960643023530762341/1525489299032641657 racheline]: "The known information about q(n) isn't enough to prove that this machine scores higher than the shown champions with fewer states, or that it scores lower than the shown champions with more states."
If q(5) doesn't exist (this can only happen if [[Logical_independence|ZFC+I3 is inconsistent]]), then this machine doesn't halt.
|}

== 3-Symbol TMs ==

{| class="wikitable"
|+
!Domain
!Runtime
!Champions
!Discovered by
!Verification
|-
|[[BB(2,3)]]
|<math>38</math>
|{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
|Allen Brady in 1988
|Direct Simulation
|-
|[[BB(3,3)]]
|<math>> 10^{17}</math>
|{{TM|0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC|halt}}
|Terry & Shawn Ligocki in 2007
|[https://bbchallenge.org/~pascal.michel/beh#tm33h Analysis by Pascal Michel]
|-
|[[BB(4,3)]]
|<math>> 10 \uparrow^4 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|Pavel Kropitz in 2024
|See Polygon's analysis on the TM page
|-
|BB(5,3)
|
|
|
|
|-
|BB(6,3)
|<math>> f_\omega^2(10\uparrow\uparrow\uparrow 6)</math>
|{{TM|1LE2LD2LC_2RB1RB1RA_2RB1LC0LC_1RB1LD2LD_2RD2LE2LF_1RZ---1LE|halt}}
|Jacobzheng in 2026
|Not independently verified.
|-
|BB(7,3)
|
|
|
|
|-
|BB(8,3)
|<math>> f_{\omega+1}(f_\omega^4(f_7(2))) > g_{64}</math>
|{{TM|1RD2RE2LA_2RC2LC2RB_1LH2RF1LA_2RC1RD2RB_---1LC2RE_1RZ0LF1LG_1RD2LG2LG_0LA0LD1LH|halt}}
|Jacobzheng in 2026
|Not independently verified.
|-
|BB(9,3)
|
|
|
|
|-
|BB(10,3)
|
|
|
|
|-
|BB(11,3)
|<math>> f_{\omega+3}^2(4)</math>
|{{TM|1LD2RG2LA_2LE2LH2RB_2LE1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RI---_0RC1LH2RG_1RC2LH1LA_1RI1RI1LJ_0LA2LK---_1RZ2LJ---|halt}}
|Jacobzheng in 2026
|Not independently verified.
|-
|colspan="5"|...
|-
|BB(36,3)
|<math>> f_{\varepsilon_0}(7)</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(36,3)</ref>
|Not independently verified.
|-
|BB(37,3)
|<math>> f_{\varepsilon_0}(373\,676\,378)</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(37,3)</ref>
|Not independently verified.
|}

== 4-Symbol TMs ==
{| class="wikitable"
|+
!Domain
!Runtime
!Champions
!Discovered by
!Verification
|-
|[[BB(2,4)]]
|<math>3\,932\,964</math>
|{{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
|Terry & Shawn Ligocki in 2005
|Pascal Michel, Heiner Marxen, Allen Brady
|-
|[[BB(3,4)]]
|<math>> 2 \uparrow^{15} 5</math>
|{{TM|1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|Pavel Kropitz in 2024
|[https://www.sligocki.com/2024/05/22/bb-3-4-a14.html Analysis by Shawn Ligocki]
|-
|[[BB(4,4)]]
|<math>> 2 \uparrow^{509} 3 > f_\omega(508)</math>
|{{TM|0LB3LB1RZ2RA_1LD3RB1LC2RA_3RB1LB3LC2RC_0RD2RC2LB0RA|halt}}
|[[User:Loader3229]] in 2026
|[https://github.com/loader3229/busy-beaver/tree/main/BB44 Claimed analysis by MiniMax] Not independently verified.
|-
|BB(5,4)
|<math>> f_\omega^2(10\uparrow^4 10\uparrow\uparrow 65\,533)</math>
|{{TM|1LD2RA3RB1LC_1LD2LA3RB---_2RA1LA3RC2LC_2RA---2LD2LE_------1RZ1LD|halt}}
|Jacobzheng in 2026
|Not independently verified.
|-
|BB(6,4)
|<math>> f_{\omega+1}(10\uparrow\uparrow 65\,533) > g_{64}</math>
|{{TM|1LF2RA3RB---_3RD2LB3RB1LC_2RA1LB3RC2LC_1LD2RE0LA1LD_1RZ0LE1LF---_2RA---2LF2LF|halt}}
|Jacobzheng in 2026
|Not independently verified.
|-
|BB(7,4)
|
|
|
|
|-
|BB(8,4)
|<math>> f_{\omega+2}(f_\omega^4(12))</math>
|{{TM|1RC3LB2RD2RA_1RF3RB1LC2RA_3RB1LB3LC2RC_0RF2LD1LD2LE_3RE3LB2RE2LE_0LG0LH1RZ3RB_1LG2RD1LG3RG_0LH0LC0LH3RD|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(9,4)
|<math>> f_{\omega+3}(16)</math>
|{{TM|2LI3LB2RD2RA_2LC3RB1LC2RA_3RB1LB3LC2RC_0RF2LD1LD2LE_3RE3LB2RE2LE_0LG0LH1RZ---_1LG2RD1LG3RG_0LH---0LH3RD_2LB---1LI3RF|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(10,4)
|
|
|
|
|-
|BB(11,4)
|<math>> f_{\omega \times 2}(2 \uparrow^{11} 6)</math>
|{{TM|1RK3LB1RZ2RA_0LG1LE1LD2RA_3RB1LB3LC2RC_3RE0LI3LC2RC_1RJ3RE3LH2LF_3RF2RF2LE1LF_3LC3LH2LJ2RG_0LE3RH1LI2RG_3RH1LH3LI2RI_0LB0LJ3LJ3RK_2LC0LD2RA3RK|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|colspan="5"|...
|-
|BB(31,4)
|<math>> f_{\varepsilon_0}(373\,676\,378)</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(31,4)</ref>
|Not independently verified.
|}

== 5-Symbol TMs ==
{| class="wikitable"
!Domain
!Runtime
!Champions
!Discovered by
!Verification
|-
|[[BB(2,5)]]
|<math>> 10^{10^{10^{3\,314\,360}}}</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}
|Daniel Yuan in 2024
|[https://discord.com/channels/960643023006490684/1259770421046411285/1379877629288644722 mxdys in Rocq]
|-
|[[BB(3,5)]]
|<math>> f_\omega(2 \uparrow^{15} 5) > f_\omega^2(15)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC|halt}}
|Racheline in 2024
|Not independently verified.
|-
|BB(4,5)
|<math>> f_{\omega+1}(27\,041) > g_{64}</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA4RD_3RB1LB3LC2RC4LC_2RC2RD1RZ4RB1LC|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(5,5)
|<math>> f_{\omega+1}(f_\omega^4(6))</math>
|{{TM|1LB3LB4LC2RA4LB_3LE3RB1LC2RA4RD_3RB1LB3LC2RC4LC_1RZ------4LD4LE_3RB------4LE3LC|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(6,5)
|<math>> f_{\omega+3}(11)</math>
|{{TM|2LE3LB4LC2RA4LB_4RA3RB1LC2RA4RD_3RB1LB3LC2RC4LC_1RZ3LF1LA4LD4LE_3RB---0RD4LE3LC_4LC------3LF3LF|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(7,5)
|<math>> f_{\omega+3}^2(10)</math>
|{{TM|2LE3LB4LC2RA4LB_4RA3RB1LC2RA4RD_3RB1LB3LC2RC4LC_1RZ3LF1LG4LD4LE_3RB---1LE4LE3LC_4LC------3LF3LF_3LF------1LG1LG|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(8,5)
|<math>> f_{\omega \times 2+1}(93)</math>
|{{TM|2RG3LB4LC2RA4LB_1RZ---1LD2RA4LD_3RB1LB3LC2RC4LC_3RE---3LC2RC3LC_---3RE4LE4LH4RB_4LE3LG3LB2RF4LG_2LH3RG1LH2RF2RA_3RG1LG3LH2RH4LH|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|colspan="5"|...
|-
|BB(19,5)
|<math>> f_{\varepsilon_0}(1.7\cdot10^{352})</math>
|{{TM|1RB2LA1RA2LB2LA_0LA2RB3RB4RA1LQ_1RZ1RC2LD------_0LK3RD2RD2RD4RD_------0RH1LS0RF_1LG1RF2RF---4RF_4LE1LG2LG---4LG_0LI1RH2RH---4RH_0RK0LJ------4RK_2LE1LJ2LJ---4LJ_4RP---0LE---0LK_2RQ---2LL---4LL_2LN---2RM3RM4RM_4RQ2RQ2LN3LN4LN_4LN---2RO3RO4RO_3LL1LP2LP1LQ4LP_1RC3RR1RM3LP0RO_3LL1RR2RR0RR4RR_1RC1LS2LS1LS4LS|halt}}
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(19,5)</ref>
|Not independently verified.
|-
|BB(94,5)
|<math>> f_{\psi_0(\Omega_2)}(1.7\cdot10^{352})</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User_blog:Wythagoras/Hydra_machines</ref>
|Not independently verified.
|}

== 6-Symbol TMs ==
{| class="wikitable"
!Domain
!Runtime
!Champions
!Discovered by
!Verification
|-
|[[BB(2,6)]]
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10^{10^{115}}</math>
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
|Pavel Kropitz in 2023
|[https://www.sligocki.com/2023/05/20/bb-2-6-p3.html Analysis by Shawn Ligocki]
|-
|BB(3,6)
|<math>> f_\omega^3(15)</math>
|{{TM|1RB3LB4LC2RA4LB5LB_2LC3RB1LC2RA5LC1RZ_3RB1LB3LC2RC4LC5LC|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(4,6)
|<math>> f_{\omega+2}(29)</math>
|{{TM|5LD3LB5LC2RA4LB4LC_1RZ3RB1LC2RA3RD0LA_3RB1LB3LC2RC4LC1RA_3RB0LA2LC4LD4LC4LD|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(5,6)
|<math>> f_{\omega+3}(31)</math>
|{{TM|1LE3LB2LA2RA4LB5RA_1RZ3RB1LC2RA3RD4RA_3RB1LB3LC2RC4LC2LB_3RB0LA5LE4LD4LC4LD_4LC5LC0RE5LE1RE2RE|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(6,6)
|<math>> f_{\omega \times 2}(89)</math>
|{{TM|4LF3LB1RZ2RA4LB---_2LC3RB1LD2RA3LF---_3RB1LB3LC2RC4LC0LE_3RB---3LC2RC---3RE_5RC3RE---4LC------_5RB0LF1RB1LF---5RB|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(7,6)
|<math>> f_{\omega^2}(63)</math>
|{{TM|2LE3LB1RZ2RA4LB2LD_0LA3RB1LD2RA3LF5RB_3RB1LB3LC2RC4LC1LE_3RB3RE3LC2RC---3RD_5RB3RE0LG4LC3LG5LE_5RB5RB---1LF---5LF_5LF---3RC1LG---5LG|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(8,6)
|<math>> f_{\omega^2+1}(7)</math>
|{{TM|0LC3LB2LH2RA4LB1RZ_3LH3RB1LD2RA3LF5RB_3RB1LB3LC2RC4LC1LE_3RB3RE3LC2RC2LD3RD_5RB3RE---4LC3LG5LE_5RB5RB---1LF5LF5LF_5LF4RG1LB1LG5LD5LG_3LE---1LH1LH---5RG|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|colspan="5"|...
|-
|BB(61,6)
|<math>> f_{\psi_0(\Omega_2)}(1.7\cdot10^{352})</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User_blog:Wythagoras/Hydra_machines</ref>
|Not independently verified.
|}

== 7-Symbol TMs ==
{| class="wikitable"
!Domain
!Runtime
!Champions
!Discovered by
!Verification
|-
|[[BB(2,7)]]
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 155</math>
|{{TM|1RB3RB4LA1LB5LA6LB5RA_2LA6RA4RB1RZ3LB6LA4RB|halt}}
|[https://discord.com/channels/960643023006490684/960643023530762341/1563057558011252761 roromachine14 in 2026]
|Not independently verified.
|-
|BB(3,7)
|<math>> f_{\omega+1}(117) > g_{64}</math>
|{{TM|6LC3LB6RA2RA4LB5LC5LA_1RZ3RB1LC2RA3RB4LC5RA_3RB1LB3LC2RC4LC1RA6LA|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(4,7)
|<math>> f_{\omega+2}(124)</math>
|{{TM|6LD3LB5LD2RA4LB3LC5LA_1RZ3RB1LC2RA3RD---2RA_3RB1LB3LC2RC4LC3LC5LD_3RB---6LC4LD4LC4LD---|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|colspan="5"|...
|-
|BB(18,7)
|<math>> f_{\varepsilon_0}(1.7\cdot10^{352})</math>
|{{TM|1RB2LA1RA2LB2LA------_0LA2RB3RB4RA1LP------_0RZ1RC2LD------------_0LJ3RD2RD2RD4RD------_4LE1LE0RG1LR0RF2LE4LE_1LE1RF5RF---6RF------_0LH1RG2RG---4RG------_0RJ0LI------4RJ------_2LE1LI2LI---4LI------_4RO---0LE---0LJ------_2RP---2LK---4LK------_2LM---2RL3RL4RL------_4RP2RP2LM3LM4LM------_4LM---2RN3RN4RN------_3LK1LO2LO1LP4LO------_1RC3RQ1RL3LO0RN------_3LK1RQ2RQ0RQ4RQ------_1RC1LR2LR1LR4LR------|halt}}
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(18,7)</ref>
|Not independently verified.
|-
|BB(46,7)
|<math>> f_{\psi_0(\Omega_2)}(1.7\cdot10^{352})</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User_blog:Wythagoras/Hydra_machines</ref>
|Not independently verified.
|-
|BB(134,7)
|<math>> f_{\psi_0(\Omega_\omega)}(2\,050)</math>
|
|LittlePeng9+Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/Analysis_of_LittlePeng9%27s_work/Hydra_machines</ref>
|Not independently verified.
|}

== 8-Symbol TMs ==
{| class="wikitable"
!Domain
!Runtime
!Champions
!Discovered by
!Verification
|-
|BB(2,8)
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 155</math>
|{{TM|1RB3RB4LA1LB5LA6LB5RA1RZ_2LA6RA4RB7RA3LB6LA4RB4RA|halt}}
|[https://discord.com/channels/960643023006490684/1084047886494470185/1563070360956698804 roromachine14 in 2026]
|Not independently verified.
|-
|colspan="5"|...
|-
|BB(17,8)
|<math>> f_{\varepsilon_0}(1.7\cdot10^{352})</math>
|{{TM|1RB2LA1RA2LB2LA---------_0LA2RB3RB4RA1LO---------_0RZ1RC2LD---------------_0LJ3RD2RD2RD4RD---------_4LE1LE0RG1LQ0RF2LE4LE---_1LE1RF5RF---6RF---------_0LH1RG2RG---4RG---------_0RJ0LI------4RJ---------_2LE1LI2LI---4LI---------_4RN---0LE---0LJ---------_2RO2RO2LK3LK4LK2LK4LK4RO_2LK---5RL3RL6RL---------_4LK---5RM3RM6RM---------_3LK1LN2LN1LO4LN---------_1RC3RP1RL3LN7RM---------_3LK1RP2RP0RP4RP---------_1RC1LQ2LQ1LQ4LQ---------|halt}}
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(17,8)</ref>
|Not independently verified.
|}

== 9-Symbol TMs ==
{| class="wikitable"
!Domain
!Runtime
!Champions
!Discovered by
!Verification
|-
|BB(2,9)
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 155</math>
|{{TM|1RB3RB4LA1LB5LA6LB5RA8RA1RZ_2LA6RA4RB7RA3LB6LA4RB4RA4RA|halt}}
|[https://discord.com/channels/960643023006490684/1084047886494470185/1563070360956698804 roromachine14 in 2026]
|Not independently verified.
|-
|colspan="5"|...
|-
|BB(42,9)
|<math>> f_{\psi_0(\Omega_2)}(1.7\cdot10^{352})</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User_blog:Wythagoras/Hydra_machines</ref>
|Not independently verified.
|}

== Zoology ==
There are many types of champions.

Note: the scale here is not formally defined. It mostly serves to estimate the average values produced by a type of champions.
{| class="wikitable"
|+
!Classification
!Description
!Examples
!Scale
|-
|Trivial
|The simplest champions that can exist. They mostly appear in some BB-adjacent functions like [[Fractran|BBf]] or [[CounterScript|BBCS]].
|
* {{TM|1RZ---|halt}}
* {{TM|0RA_0RZ|halt}}
* {{TM|0RA_0RB_0RZ|halt}}
|<math>O(n)</math>
|-
|Chaotic
|Have a chaotic behavior with repeating patterns that go back and forth.
|
* {{TM|1RB1LB_1LA1RZ|halt}}
* {{TM|1RB1RZ_1LB0RC_1LC1LA|halt}}
* {{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}}
|<math>O(2^n)</math>
|-
|Countdown
|Compute a number then "count down" (usually while bouncing) until reaching 0. They are common in some BB-adjacent functions like [[Fractran|BBf]].
|
* {{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
|<math>O(n^2)</math>
|-
|Collatz-like
|Compute a [[Collatz-like]] function. Repeatedly multiply and add a number depending of its modulo until reaching a number with a certain modulo.
|
* {{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
* {{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}
|<math>O(2^n)</math>
|-
|Random walk
|Compute a [[Collatz-like]] function with a random walk.
|
|<math>O(2^{2^n})</math>
|-
|Tetrational
|
|
|<math>O(2 \uparrow\uparrow n)</math>
|-
|Ackermannian
|
|
|<math>O(2 \uparrow^n 3)</math>
|}

== References ==

[[Category:Individual machines]]
[[Category:Zoology]]

User:Jacobzheng/test

2026-11-05T06:19:39Z

Jacobzheng: Undo revision 10262 by Jacobzheng (talk)

User:Jacobzheng/test

2026-11-05T06:19:16Z

Jacobzheng: Replaced content with "~!@#$%^&*()_+QWERTYUIOP{}|ASDFGHJKL:"ZXCVBNM<>?`1234567890-=qwertyuiop[]\asdfghjkl;'zxcvbnm,./"

~!@#$%^&*()_+QWERTYUIOP{}|ASDFGHJKL:"ZXCVBNM<>?`1234567890-=qwertyuiop[]\asdfghjkl;'zxcvbnm,./

Main Page

2026-11-05T06:15:47Z

Jacobzheng:

The [[Busy Beaver function]] BB (called ''S'' originally) was introduced by [[Tibor Radó]] in 1962 for 2-symbol [[Turing machines]] and later generalized 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 [[wikipedia:Computability|computable]], but a few of its values are known:

{| class="wikitable"
|+ Small busy beaver values (up to 7 states and symbols)<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;" title="Cryptids: Antihydra and some other cryptids" | [[BB(6)]] > <math>10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
| style="background: #ddc290;" | [[BB(7)]] > <math>3 \uparrow^{20} 3</math>
|-
!3-symbol
| [[BB(2,3)]] = 38
| style="background: orange;" title="Cryptid: Bigfoot" | [[BB(3,3)]] <math>\ge</math> 119,112,334,170,342,541
| style="background: #ddc290;" | [[BB(4,3)]] > <math>10 \uparrow^{4} 4</math>
| style="background: #ddc290;" |
| style="background: #ddc290;" | BB(6,3) > <math> f_\omega^2(10\uparrow\uparrow\uparrow 6)</math>
| style="background: #ddc290;" |
|-
!4-symbol
| [[BB(2,4)]] = 3,932,964
| style="background: #ddc290;" | [[BB(3,4)]] > <math>2 \uparrow^{15} 5</math>
| style="background: #ddc290;" | [[BB(4,4)]] > <math>2 \uparrow^{509} 3</math>
| style="background: #ddc290;" | BB(5,4) > <math>f_\omega^2(10\uparrow^4 10\uparrow\uparrow 65\,533)</math>
| style="background: yellow;" | BB(6,4) > <math> f_{\omega+1}(10\uparrow\uparrow 65\,533)</math>
| style="background: yellow;" |
|-
!5-symbol
| style="background: orange;" title="Cryptids: Hydra and Bonus Cryptid" | [[BB(2,5)]] > <math>10^{10^{10^{3\,314\,360}}}</math>
| style="background: #ddc290;" | [[BB(3,5)]] > <math> f_\omega(2 \uparrow^{15} 5)</math>
| style="background: yellow;" | BB(4,5) > <math> f_{\omega+1}(27\,041)</math>
| style="background: yellow;" |BB(5,5) > <math>f_{\omega+1}(f_\omega^4(6))</math>
| style="background: yellow;" | BB(6,5) > <math>f_{\omega+3}(11)</math>
| style="background: yellow;" |BB(7,5) > <math>f_{\omega+3}^2(10)</math>
|-
!6-symbol
| style="background: #ddc290;" | [[BB(2,6)]] > <math>10 \uparrow \uparrow 10 \uparrow\uparrow 10^{10^{115}}</math>
| style="background: #ddc290;" | BB(3,6) > <math> f_\omega^3(15)</math>
| style="background: yellow;" | BB(4,6) > <math> f_{\omega+2}(29)</math>
| style="background: yellow;" | BB(5,6) > <math> f_{\omega+3}(31)</math>
| style="background: yellow;" | BB(6,6) > <math> f_{\omega \times 2}(89)</math>
| style="background: yellow;" | BB(7,6) > <math> f_{\omega^2}(63)</math>
|-
!7-symbol
| style="background: #ddc290;" | [[BB(2,7)]] > <math>10 \uparrow \uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 155</math>
| style="background: yellow;" | BB(3,7) > <math> f_{\omega+1}(117)</math>
| style="background: yellow;" | BB(4,7) > <math> f_{\omega+2}(124)</math>
| style="background: yellow;" |
| style="background: yellow;" |
| style="background: yellow;" |
|}

'''Table key:'''

* Orange cell: smallest known [[Cryptid]] (a mathematically-hard machine) is in this class
* Tan cell: a Cryptid is known using fewer states or symbols than this class.
* Yellow cell: the Busy Beaver number is known to be more than [[Graham's number]].
* Blank cell: no known champion that surpasses a champion with fewer states or symbols by a non-trivial amount.

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:

* Formalizing 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|Accelerated simulator]]s to simulate halting machines faster
* Writing papers and giving talks about the Busy Beaver function, see [[Papers & Talks]]
* Exploring different [[:Category:Functions|Busy Beaver-style functions]] built from different systems of computation

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

[[TMBR: June 2026]], [[TMBR: July 2026]], [[TMBR: August 2026]], [[TMBR: September 2026]], [[TMBR: October 2026]], and [[TMBR: November 2026]] are currently works in progress.

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

==Notes==
<references />

2026-10-31T14:50:26Z

Jacobzheng: Jacobzheng moved page File:Statistics 10000edits.png to File:Statistics 10000 edits.png

2026-10-27T05:48:21Z

Jacobzheng: Created page with "I hope the bbchallenge website and the bbchallenge wiki can grow faster. I hope every user on the bbchallenge wiki is kind and friendly. Jacobzheng (talk) 05:48, 27 October 2026 (UTC)"

BusyBeaverWiki talk:About

2026-10-27T05:47:36Z

User talk:Jacobzheng/Bbchallenge wiki's unverified formula

2026-10-27T05:41:07Z

Jacobzheng:

Champions

2026-10-27T02:31:06Z

Jacobzheng:

Champions

2026-10-27T02:29:42Z

Jacobzheng:

Champions

2026-10-27T02:29:17Z

Jacobzheng:

User talk:Loader3229

2026-10-26T09:53:00Z

Jacobzheng:

What are the answers for stage 1 to 20 of this [https://loader3229.github.io/ldrchallenge.html ldrchallenge]? [[User:Jacobzheng|Jacobzheng]] ([[User talk:Jacobzheng|talk]]) 23:59, 23 September 2026 (UTC)
* I won't tell you. --[[User:Loader3229|Loader3229]] ([[User talk:Loader3229|talk]]) 01:32, 24 September 2026 (UTC)
Why don't you tell me? [[User:Jacobzheng|Jacobzheng]] ([[User talk:Jacobzheng|talk]]) 04:05, 24 September 2026 (UTC)

Can you analyze and evaluate this formula?

[[User:Jacobzheng/Bbchallenge wiki's unverified formula|link]] [[User:Jacobzheng/m values of some notable users on the bbchallenge wiki|link]] [[User:Jacobzheng|Jacobzheng]] ([[User talk:Jacobzheng|talk]]) 09:52, 26 October 2026 (UTC)

User:Jacobzheng/m values of some notable users on the bbchallenge wiki

2026-10-26T09:35:20Z

Jacobzheng:

[[User:Jacobzheng/Bbchallenge wiki's unverified formula|See this page]].

If x=160.

Cosmo: {{#expr:1.00625^(({{#time:U|now}}-1716905509)/86400)*(({{#time:U|now}}-1716905509)/86400)}}

Sligocki: {{#expr:1.00625^(({{#time:U|now}}-1716905509)/86400)*(({{#time:U|now}}-1716916165)/86400)}}

Racheline: {{#expr:1.00625^(({{#time:U|now}}-1716905509)/86400)*(({{#time:U|now}}-1719597093)/86400)}}

C7X: {{#expr:1.00625^(({{#time:U|now}}-1716905509)/86400)*(({{#time:U|now}}-1721618168)/86400)}}

Jacobzheng: {{#expr:1.00625^(({{#time:U|now}}-1716905509)/86400)*(({{#time:U|now}}-1723125376)/86400)}}

Azerty: {{#expr:1.00625^(({{#time:U|now}}-1716905509)/86400)*(({{#time:U|now}}-1723727505)/86400)}}

Icy: {{#expr:1.00625^(({{#time:U|now}}-1716905509)/86400)*(({{#time:U|now}}-1730480236)/86400)}}

MrSolis: {{#expr:1.00625^(({{#time:U|now}}-1716905509)/86400)*(({{#time:U|now}}-1735042918)/86400)}}

Polygon: {{#expr:1.00625^(({{#time:U|now}}-1716905509)/86400)*(({{#time:U|now}}-1754389647)/86400)}}

Autumn-Pan: {{#expr:1.00625^(({{#time:U|now}}-1716905509)/86400)*(({{#time:U|now}}-1762128233)/86400)}}

Δ⁵: {{#expr:1.00625^(({{#time:U|now}}-1716905509)/86400)*(({{#time:U|now}}-1778124532)/86400)}}

HelpMe: {{#expr:1.00625^(({{#time:U|now}}-1716905509)/86400)*(({{#time:U|now}}-1780914841)/86400)}}

Loader3229: {{#expr:1.00625^(({{#time:U|now}}-1716905509)/86400)*(({{#time:U|now}}-1789970785)/86400)}}

[https://wiki.bbchallenge.org/w/index.php?title=User:Jacobzheng/m_values_of_some_notable_users_on_the_bbchallenge_wiki&action=purge update]