BusyBeaverWiki - User contributions [en]

Champions

2026-03-13T16:15:41Z

50 ft lock: fixed 13 state champion to start on state A

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]]. Note that most champions above 6 states are self-reported and have not been independently verified.

{| class="wikitable"
|+
!
!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>> 2\uparrow\uparrow\uparrow 5</math>
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|mxdys in 2025
|See mxdys's analysis on the TM page
|-
|[[BB(7)]]
|<math>> 2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF|halt}}
|[https://discord.com/channels/960643023006490684/1369339127652159509/1370678203395604562 Pavel Kropitz in 2025]
|Analyzed by Shawn Ligocki (see TM page)
|-
|[[BB(8)]]
|
|
|
|
|-
|BB(9)
|<math>> f_\omega(f_9(2))</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|Jacobzheng in 2024
|
|-
|BB(10)
|<math>> f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|Racheline in 2024
|
|-
|BB(11)
|<math>> f_\omega^2(2 \uparrow\uparrow 12) > f_\omega^2(f_3(9))</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RZ0LI_0LD1LE|halt}}
|Racheline in 2024
|
|-
|BB(12)
|<math>> f_\omega^4(2 \uparrow\uparrow\uparrow 4-3) > f_\omega^4(f_4(2))</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_1RZ1LA_1RF1LL|halt}}
|Racheline in 2024
|
|-
|BB(13)
|<math>> f_{\omega + 1}(2047) > g_{64}</math>
|{{TM|1RB1RA_1LC1RD_1LA1LC_1LG0RE_1LC1RB_0RL1LG_0LM0RH_1RI1RH_1LK0RI_---0LK_1LF1LK_1LJ1RL_1RZ1RH|halt}}
|[https://discord.com/channels/960643023006490684/1331570843829932063/1481871400640839691 50_ft_lock in 2026]
|
|-
|BB(14)
|<math>> f_{\omega + 1}(65\,536) > g_{64}</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ---|halt}}
|[https://discord.com/channels/960643023006490684/960643023530762341/1274366178529120287 Racheline in 2024]
|
|-
|BB(15)
|<math>> f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_1RZ0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|Jacobzheng in 2025
|
|-
|BB(16)
|<math>> f_{\omega + 1}^2(10^{10^{57}})</math>
|[[User:Jacobzheng/BB(16)]]
|Jacobzheng in 2025
|
|-
|BB(17)
|
|
|
|
|-
|BB(18)
|<math>> f_{\omega + 2}(f_{\omega + 1}^3(f_{\omega}^2(60)))</math>
|[[User:Jacobzheng/BB(18)]]
|Jacobzheng in 2025
|
|-
|BB(19)
|
|
|
|
|-
|BB(20)
|<math>> f_{\omega + 2}^2(21)</math>
|
|[https://discord.com/channels/960643023006490684/1026577255754903572/1274414683331366924 Racheline in 2024]
|
|-
|BB(21)
|<math>> f_{\omega^2}^2(4 \uparrow\uparrow 341)</math>
|
|[https://discord.com/channels/960643023006490684/1026577255754903572/1274471360206344213 Racheline in 2024]
|
|-
|BB(40)
|<math>> f_{\omega^\omega}(75\,500)</math>
|[[User:Jacobzheng/BB(40)]]
|Jacobzheng in 2024
|
|-
|BB(41)
|<math>> f_{\omega^\omega}^4(32)</math>
|[[User:Jacobzheng/BB(41)]]
|Jacobzheng in 2024
|
|-
|BB(51)
|<math>> f_{\varepsilon_0 + 1}(8)</math>
|
|[https://discord.com/channels/960643023006490684/1026577255754903572/1276881449685094495 Racheline in 2024]
|
|-
|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>
|
|}

== 3-Symbol TMs ==

{| class="wikitable"
|+
!
!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
|
|}

== 4-Symbol TMs ==
{| class="wikitable"
|+
!
!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]
|}

== 5-Symbol TMs ==
{| class="wikitable"
!
!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
|
|}

== 6-Symbol TMs ==
{| class="wikitable"
!
!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]
|}

== Zoology ==
{| class="wikitable"
|+
!Classification
!Description
!Examples
!Scale
|-
|Trivial
|The simplest champions that can exist. They mostly appear in some BB-adjacent functions like [[Fractran|BBf]].
|
|<math>O(n)</math>
|-
|Chaotic
|Have a chaotic behavior with often repeating patterns that go back and forth.
|
* {{TM|1RB1LB_1LA---|halt}}
* {{TM|1RB---_1LB0RC_1LC1LA|halt}}
* {{TM|1RB1LB_1LA0LC_---1LD_1RD0RA|halt}}
|<math>O(n^2)</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|1RB2LB---_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_1LB1LA3RB---|halt}}
* {{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_---0LA|halt}}
|<math>O(2^n)</math>
|}

== References ==

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

Champions

2026-03-13T16:13:45Z

50 ft lock: fixed 13 state champion to start on state A

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]]. Note that most champions above 6 states are self-reported and have not been independently verified.

{| class="wikitable"
|+
!
!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>> 2\uparrow\uparrow\uparrow 5</math>
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|mxdys in 2025
|See mxdys's analysis on the TM page
|-
|[[BB(7)]]
|<math>> 2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF|halt}}
|[https://discord.com/channels/960643023006490684/1369339127652159509/1370678203395604562 Pavel Kropitz in 2025]
|Analyzed by Shawn Ligocki (see TM page)
|-
|[[BB(8)]]
|
|
|
|
|-
|BB(9)
|<math>> f_\omega(f_9(2))</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|Jacobzheng in 2024
|
|-
|BB(10)
|<math>> f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|Racheline in 2024
|
|-
|BB(11)
|<math>> f_\omega^2(2 \uparrow\uparrow 12) > f_\omega^2(f_3(9))</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RZ0LI_0LD1LE|halt}}
|Racheline in 2024
|
|-
|BB(12)
|<math>> f_\omega^4(2 \uparrow\uparrow\uparrow 4-3) > f_\omega^4(f_4(2))</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_1RZ1LA_1RF1LL|halt}}
|Racheline in 2024
|
|-
|BB(13)
|<math>> f_{\omega + 1}(2047) > g_{64}</math>
|{{TM|0RL1LG_0LM0RH_1RI1RH_1LK0RI_---0LK_1LF1LK_1LJ1RL_1RZ1RH_1RB1RA_1LC1RD_1LA1LC_1LG0RE_1LC1RB|halt}}
|[https://discord.com/channels/960643023006490684/1331570843829932063/1481871400640839691 50_ft_lock in 2026]
|
|-
|BB(14)
|<math>> f_{\omega + 1}(65\,536) > g_{64}</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ---|halt}}
|[https://discord.com/channels/960643023006490684/960643023530762341/1274366178529120287 Racheline in 2024]
|
|-
|BB(15)
|<math>> f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_1RZ0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|Jacobzheng in 2025
|
|-
|BB(16)
|<math>> f_{\omega + 1}^2(10^{10^{57}})</math>
|[[User:Jacobzheng/BB(16)]]
|Jacobzheng in 2025
|
|-
|BB(17)
|
|
|
|
|-
|BB(18)
|<math>> f_{\omega + 2}(f_{\omega + 1}^3(f_{\omega}^2(60)))</math>
|[[User:Jacobzheng/BB(18)]]
|Jacobzheng in 2025
|
|-
|BB(19)
|
|
|
|
|-
|BB(20)
|<math>> f_{\omega + 2}^2(21)</math>
|
|[https://discord.com/channels/960643023006490684/1026577255754903572/1274414683331366924 Racheline in 2024]
|
|-
|BB(21)
|<math>> f_{\omega^2}^2(4 \uparrow\uparrow 341)</math>
|
|[https://discord.com/channels/960643023006490684/1026577255754903572/1274471360206344213 Racheline in 2024]
|
|-
|BB(40)
|<math>> f_{\omega^\omega}(75\,500)</math>
|[[User:Jacobzheng/BB(40)]]
|Jacobzheng in 2024
|
|-
|BB(41)
|<math>> f_{\omega^\omega}^4(32)</math>
|[[User:Jacobzheng/BB(41)]]
|Jacobzheng in 2024
|
|-
|BB(51)
|<math>> f_{\varepsilon_0 + 1}(8)</math>
|
|[https://discord.com/channels/960643023006490684/1026577255754903572/1276881449685094495 Racheline in 2024]
|
|-
|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>
|
|}

== 3-Symbol TMs ==

{| class="wikitable"
|+
!
!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
|
|}

== 4-Symbol TMs ==
{| class="wikitable"
|+
!
!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]
|}

== 5-Symbol TMs ==
{| class="wikitable"
!
!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
|
|}

== 6-Symbol TMs ==
{| class="wikitable"
!
!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]
|}

== Zoology ==
{| class="wikitable"
|+
!Classification
!Description
!Examples
!Scale
|-
|Trivial
|The simplest champions that can exist. They mostly appear in some BB-adjacent functions like [[Fractran|BBf]].
|
|<math>O(n)</math>
|-
|Chaotic
|Have a chaotic behavior with often repeating patterns that go back and forth.
|
* {{TM|1RB1LB_1LA---|halt}}
* {{TM|1RB---_1LB0RC_1LC1LA|halt}}
* {{TM|1RB1LB_1LA0LC_---1LD_1RD0RA|halt}}
|<math>O(n^2)</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|1RB2LB---_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_1LB1LA3RB---|halt}}
* {{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_---0LA|halt}}
|<math>O(2^n)</math>
|}

== References ==

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

Champions

2026-03-13T16:00:06Z

50 ft lock: added new 13 state champion

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]]. Note that most champions above 6 states are self-reported and have not been independently verified.

{| class="wikitable"
|+
!
!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>> 2\uparrow\uparrow\uparrow 5</math>
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|mxdys in 2025
|See mxdys's analysis on the TM page
|-
|[[BB(7)]]
|<math>> 2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF|halt}}
|[https://discord.com/channels/960643023006490684/1369339127652159509/1370678203395604562 Pavel Kropitz in 2025]
|Analyzed by Shawn Ligocki (see TM page)
|-
|[[BB(8)]]
|
|
|
|
|-
|BB(9)
|<math>> f_\omega(f_9(2))</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|Jacobzheng in 2024
|
|-
|BB(10)
|<math>> f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|Racheline in 2024
|
|-
|BB(11)
|<math>> f_\omega^2(2 \uparrow\uparrow 12) > f_\omega^2(f_3(9))</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RZ0LI_0LD1LE|halt}}
|Racheline in 2024
|
|-
|BB(12)
|<math>> f_\omega^4(2 \uparrow\uparrow\uparrow 4-3) > f_\omega^4(f_4(2))</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_1RZ1LA_1RF1LL|halt}}
|Racheline in 2024
|
|-
|BB(13)
|<math>> f_{\omega + 1}(2047) > g_{64}</math>
|{{TM|0RG1LB_0LH0RC_1RD1RC_1LF0RD_---0LF_1LA1LF_1LE1RG_1RZ1RC_1RJ1RI_1LK1RL_1LI1LK_1LB0RM_1LK1RJ|halt}}
|[https://discord.com/channels/960643023006490684/1331570843829932063/1481871400640839691 50_ft_lock in 2026]
|
|-
|BB(14)
|<math>> f_{\omega + 1}(65\,536) > g_{64}</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ---|halt}}
|[https://discord.com/channels/960643023006490684/960643023530762341/1274366178529120287 Racheline in 2024]
|
|-
|BB(15)
|<math>> f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_1RZ0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|Jacobzheng in 2025
|
|-
|BB(16)
|<math>> f_{\omega + 1}^2(10^{10^{57}})</math>
|[[User:Jacobzheng/BB(16)]]
|Jacobzheng in 2025
|
|-
|BB(17)
|
|
|
|
|-
|BB(18)
|<math>> f_{\omega + 2}(f_{\omega + 1}^3(f_{\omega}^2(60)))</math>
|[[User:Jacobzheng/BB(18)]]
|Jacobzheng in 2025
|
|-
|BB(19)
|
|
|
|
|-
|BB(20)
|<math>> f_{\omega + 2}^2(21)</math>
|
|[https://discord.com/channels/960643023006490684/1026577255754903572/1274414683331366924 Racheline in 2024]
|
|-
|BB(21)
|<math>> f_{\omega^2}^2(4 \uparrow\uparrow 341)</math>
|
|[https://discord.com/channels/960643023006490684/1026577255754903572/1274471360206344213 Racheline in 2024]
|
|-
|BB(40)
|<math>> f_{\omega^\omega}(75\,500)</math>
|[[User:Jacobzheng/BB(40)]]
|Jacobzheng in 2024
|
|-
|BB(41)
|<math>> f_{\omega^\omega}^4(32)</math>
|[[User:Jacobzheng/BB(41)]]
|Jacobzheng in 2024
|
|-
|BB(51)
|<math>> f_{\varepsilon_0 + 1}(8)</math>
|
|[https://discord.com/channels/960643023006490684/1026577255754903572/1276881449685094495 Racheline in 2024]
|
|-
|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>
|
|}

== 3-Symbol TMs ==

{| class="wikitable"
|+
!
!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
|
|}

== 4-Symbol TMs ==
{| class="wikitable"
|+
!
!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]
|}

== 5-Symbol TMs ==
{| class="wikitable"
!
!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
|
|}

== 6-Symbol TMs ==
{| class="wikitable"
!
!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]
|}

== Zoology ==
{| class="wikitable"
|+
!Classification
!Description
!Examples
!Scale
|-
|Trivial
|The simplest champions that can exist. They mostly appear in some BB-adjacent functions like [[Fractran|BBf]].
|
|<math>O(n)</math>
|-
|Chaotic
|Have a chaotic behavior with often repeating patterns that go back and forth.
|
* {{TM|1RB1LB_1LA---|halt}}
* {{TM|1RB---_1LB0RC_1LC1LA|halt}}
* {{TM|1RB1LB_1LA0LC_---1LD_1RD0RA|halt}}
|<math>O(n^2)</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|1RB2LB---_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_1LB1LA3RB---|halt}}
* {{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_---0LA|halt}}
|<math>O(2^n)</math>
|}

== References ==

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