Champions: Difference between revisions

Latest revision as of 12:03, 25 September 2026

Busy Beaver Champions are the current record holding 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 Busy Beaver champions and the 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 $f_{x} (n)$ used in the lower bounds represent the Fast-Growing Hierarchy while $↑$ represents Knuth's up-arrow notation. Machines below 6 states are listed in TNF-1RB 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.


Domain	Runtime	Champions	Discovered by	Verification
BB(2)	$6$	`1RB1LB_1LA1RZ` (bbch) `1RB0LB_1LA1RZ` (bbch) `1RB1RZ_1LB1LA` (bbch) `1RB1RZ_0LB1LA` (bbch) `0RB1RZ_1LA1RB` (bbch)	Tibor Radó	Direct Simulation
BB(3)	$21$	`1RB1RZ_1LB0RC_1LC1LA` (bbch)	Proven by Shen Lin	Direct Simulation
BB(4)	$107$	`1RB1LB_1LA0LC_1RZ1LD_1RD0RA` (bbch)	Allen Brady	Direct Simulation
BB(5)	$47 176 870$	`1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA` (bbch)	Heiner Marxen & Jürgen Buntrock in 1989	Direct Simulation
BB(6)	$> 10 ↑ ↑ 10 ↑ ↑ 10 ↑ ↑ 8$	`1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE` (bbch)	mxdys in 2025	See mxdys's analysis on the TM page
BB(7)	$> 2 ↑^{11} 2 ↑^{11} 3$	`1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF` (bbch)	Pavel Kropitz in 2025	Analyzed by Shawn Ligocki (see TM page)
BB(8)
BB(9)	$> f_{ω} (f_{9} (2))$	`1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH` (bbch)	Jacobzheng in 2024
BB(10)	$> f_{ω}^{2} (25)$	`1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ` (bbch)	Racheline in 2024
BB(11)	$> f_{ω}^{2} (10 ↑^{4} 4)$	`1RB1RA_1LH1RC_0LD0LG_0RE1LD_1RA1RF_---0RA_1LC1LG_1LJ1LI_1LK1LJ_1RZ0LI_1RA1LH` (bbch)	Jacobzheng in 2026
BB(12)	$> f_{ω}^{4} (2 ↑ ↑ ↑ 4 - 3) > f_{ω}^{4} (f_{4} (2))$	`0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_1RZ1LA_1RF1LL` (bbch)	Racheline in 2024
BB(13)	$> f_{ω + 1} (2046) > g_{64}$	`1RB1RA_1LC1RD_1LA1LC_1LG0RE_1LC1RB_0RL1LG_0LM0RH_1RI1RH_1LK0RI_---0LK_1LF1LK_1LJ1RL_1RZ1RH` (bbch)	50_ft_lock in 2026
BB(14)	$> f_{ω + 1} (65 536)$	`1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ---` (bbch)	Racheline in 2024
BB(15)	$> f_{ω + 1} (f_{ω} (1 0^{57}))$	`0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_1RZ0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN` (bbch)	Jacobzheng in 2025
BB(16)	$> f_{ω + 1}^{2} (1 0^{1 0^{57}})$	`0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_1RZ0LJ_1LH1LO_1RB1RP` (bbch)	Jacobzheng in 2025
BB(17)
BB(18)	$> f_{ω + 2} (f_{ω + 1}^{3} (f_{ω}^{2} (60)))$	`0RH1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RE1RA_1RM0LJ_0LJ1LK_1LL1LL_1RG0LK_0LQ1RN_1RZ0LO_1LP1LO_0LG1LO_1LR1LQ_0LP1LQ` (bbch) User:Jacobzheng/BB(18)	Jacobzheng in 2025
BB(19)
BB(20)	$> f_{ω + 6}^{2} (4)$	`1RR1RG_0LJ0RC_1RB1LD_0LD1LE_1LF1RG_1LI0LE_0RA1LD_1RB1RH_1RH1RN_1LK1LJ_0LL1LJ_1RM1RA_0RM0RH_0LO1RN_1RH0LP_1RH1LQ_1RH1LO_1LS1RR_1RZ1LT_0LH0LS` (bbch) User:Jacobzheng/BB(20)	Jacobzheng in 2026
BB(21)	$> f_{ω^{2}}^{2} (4 ↑ ↑ 341)$	`0LI0LF_0RJ1RG_0RD1LC_1RH1RE_1LO0RH_1LA1LF_0LC1LA_1RB1RH_1RD0LF_1LP0RK_0LM1LL_0LL1LM_1LE1LN_0RQ0LM_0RP1LO_1LR0RH_1LF1RQ_1RZ0LS_1LH0LT_1LH1LU_0LE1LR` (bbch) See talk page	Racheline in 2024
...
BB(40)	$> f_{ω^{ω}} (75 500)$	User:Jacobzheng/BB(40)	Jacobzheng in 2024
BB(41)	$> f_{ω^{ω}}^{4} (32)$	User:Jacobzheng/BB(41)	Jacobzheng in 2024
BB(51)	$> f_{ε_{0} + 1} (8)$	See talk page	Racheline in 2024
BB(150)	$> f_{l i m (B M S)} (10 ↑ ↑ 15)$	too large to show	Patcail in 2025^[1]
BB(1015)	$> D^{5} (99)$ = Loader's Number		CatsAreFluffy in 2024^[2]

Conjectured


Domain	Runtime	Champions	Discovered by	Notes
BB(67)	$> q^{81} (q (5) - 2)$	[1]	racheline in 2025	$q (n)$ is derived from Laver tables. From 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."

3-Symbol TMs


Domain	Runtime	Champions	Discovered by	Verification
BB(2,3)	$38$	`1RB2LB1RZ_2LA2RB1LB` (bbch)	Allen Brady in 1988	Direct Simulation
BB(3,3)	$> 1 0^{17}$	`0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC` (bbch)	Terry & Shawn Ligocki in 2007	Analysis by Pascal Michel
BB(4,3)	$> 10 ↑^{4} 4$	`1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD` (bbch)	Pavel Kropitz in 2024
BB(5,3)
BB(6,3)	$> f_{ω}^{2} (10 ↑ ↑ ↑ 6)$	`1LE2LD2LC_2RB1RB1RA_2RB1LC0LC_1RB1LD2LD_2RD2LE2LF_1RZ---1LE` (bbch)	Jacobzheng in 2026
BB(7,3)
BB(8,3)	$> f_{ω + 1} (f_{ω} (f_{7} (2)))$	`1LD1LE2RA_1RE2LE2RB_1RD2RA2LC_1RB1RD2RB_1LH2RF1LC_1RZ0LF1LG_1RD2LG2LG_0LD1LH1LH` (bbch)	Jacobzheng in 2026
BB(9,3)
BB(10,3)
BB(11,3)	$> f_{ω + 3}^{2} (4)$	`1LD2RG2LA_2LE2LH2RB_2LE1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RI---_0RC1LH2RG_1RC2LH1LA_1RI1RI1LJ_0LA2LK---_1RZ2LJ---` (bbch)	Jacobzheng in 2026
...
BB(36,3)	$> f_{ε_{0}} (7)$		Wythagoras in 2014^[3]
BB(37,3)	$> f_{ε_{0}} (373 676 378)$		Wythagoras in 2014^[4]

4-Symbol TMs


Domain	Runtime	Champions	Discovered by	Verification
BB(2,4)	$3 932 964$	`1RB2LA1RA1RA_1LB1LA3RB1RZ` (bbch)	Terry & Shawn Ligocki in 2005	Pascal Michel, Heiner Marxen, Allen Brady
BB(3,4)	$> 2 ↑^{15} 5$	`1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC` (bbch)	Pavel Kropitz in 2024	Analysis by Shawn Ligocki
...
BB(31,4)	$> f_{ε_{0}} (373 676 378)$		Wythagoras in 2014^[5]

5-Symbol TMs

Domain	Runtime	Champions	Discovered by	Verification
BB(2,5)	$> 1 0^{1 0^{1 0^{3 314 360}}}$	`1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ` (bbch)	Daniel Yuan in 2024	mxdys in Rocq
BB(3,5)	$> f_{ω} (2 ↑^{15} 5) > f_{ω}^{2} (15)$	`1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC` (bbch)	Racheline in 2024
...
BB(19,5)	$> f_{ε_{0}} (1.7 \cdot 1 0^{352})$	`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` (bbch)	Wythagoras in 2014^[6]
BB(94,5)	$> f_{ψ_{0} (Ω_{2})} (1.7 \cdot 1 0^{352})$		Wythagoras in 2014^[7]

6-Symbol TMs

Domain	Runtime	Champions	Discovered by	Verification
BB(2,6)	$> 10 ↑ ↑ 10 ↑ ↑ 1 0^{1 0^{115}}$	`1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA` (bbch)	Pavel Kropitz in 2023	Analysis by Shawn Ligocki
...
BB(61,6)	$> f_{ψ_{0} (Ω_{2})} (1.7 \cdot 1 0^{352})$		Wythagoras in 2014^[8]

7-Symbol TMs

Domain	Runtime	Champions	Discovered by
BB(2,7)		(Enumerating Holdouts...)
...
BB(18,7)	$> f_{ε_{0}} (1.7 \cdot 1 0^{352})$	`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------` (bbch)	Wythagoras in 2014^[9]
BB(46,7)	$> f_{ψ_{0} (Ω_{2})} (1.7 \cdot 1 0^{352})$		Wythagoras in 2014^[10]
BB(134,7)	$> f_{ψ_{0} (Ω_{ω})} (2050)$		LittlePeng9+Wythagoras in 2014^[11]

8+-Symbol TMs

Domain	Runtime	Champions	Discovered by	Verification
BB(17,8)	$> f_{ε_{0}} (1.7 \cdot 1 0^{352})$	`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---------` (bbch)	Wythagoras in 2014^[12]
BB(42,9)	$> f_{ψ_{0} (Ω_{2})} (1.7 \cdot 1 0^{352})$		Wythagoras in 2014^[13]

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.


Classification	Description	Examples	Scale
Trivial	The simplest champions that can exist. They mostly appear in some BB-adjacent functions like BBf or BBCS.		$O (n)$
Chaotic	Have a chaotic behavior with repeating patterns that go back and forth.	`1RB1LB_1LA---` (bbch) `1RB---_1LB0RC_1LC1LA` (bbch) `1RB1LB_1LA0LC_---1LD_1RD0RA` (bbch)	$O (2^{n})$
Countdown	Compute a number then "count down" (usually while bouncing) until reaching 0. They are common in some BB-adjacent functions like BBf.	`1RB2LB---_2LA2RB1LB` (bbch)	$O (n^{2})$
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.	`1RB2LA1RA1RA_1LB1LA3RB---` (bbch) `1RB1LC_1RC1RB_1RD0LE_1LA1LD_---0LA` (bbch)	$O (2^{n})$
Random walk	Compute a Collatz-like function with a random walk.		$O (2^{2^{n}})$
Tetrational			$O (2 ↑ ↑ n)$
Ackermannian			$O (2 ↑^{n} 3)$

References

[1] ttps://discord.com/channels/960643023006490684/1026577255754903572/1328863966688182345

[2] ttps://github.com/CatsAreFluffy/metamath-turing-machines/commit/85948b04fc4aeb983ca6d63d6aee5ad6ef308bfe

[3] ttps://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(36,3)

[4] ttps://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(37,3)

[5] ttps://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(31,4)

[6] ttps://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(19,5)

[7] ttps://googology.fandom.com/wiki/User_blog:Wythagoras/Hydra_machines

[8] ttps://googology.fandom.com/wiki/User_blog:Wythagoras/Hydra_machines

[9] ttps://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(18,7)

[10] ttps://googology.fandom.com/wiki/User_blog:Wythagoras/Hydra_machines

[11] ttps://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/Analysis_of_LittlePeng9%27s_work/Hydra_machines

[12] ttps://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(17,8)

[13] ttps://googology.fandom.com/wiki/User_blog:Wythagoras/Hydra_machines

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

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Champions: Difference between revisions

Latest revision as of 12:03, 25 September 2026

Contents

Trivial Champions

2-Symbol TMs

Conjectured

3-Symbol TMs

4-Symbol TMs

5-Symbol TMs

6-Symbol TMs

7-Symbol TMs

8+-Symbol TMs

Zoology

References

Navigation menu

@@ Line 1: / Line 1: @@
-Busy Beaver '''Champions''' are the current record holding [[Turing machine|Turing machines]] who 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].
+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.
+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]]. Machines below 6 states are listed in [[Tree Normal Form#TNF-1RB|TNF-1RB]] 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
-!Comment
+!Discovered by
+!Verification
 |-
 |[[BB(2)]]
-|6
+|<math>6</math>
 |{{TM|1RB1LB_1LA1RZ|halt}} {{TM|1RB0LB_1LA1RZ|halt}} {{TM|1RB1RZ_1LB1LA|halt}} {{TM|1RB1RZ_0LB1LA|halt}} {{TM|0RB1RZ_1LA1RB|halt}}
-|Discovered and proven by hand by Tibor Radó
+|[[Tibor Radó]]
+|Direct Simulation
 |-
 |[[BB(3)]]
-|21
+|<math>21</math>
 |{{TM|1RB1RZ_1LB0RC_1LC1LA|halt}}
-|Proven by Shen Lin
+|Proven by [[Shen Lin]]
+|Direct Simulation
 |-
 |[[BB(4)]]
-|107
+|<math>107</math>
 |{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}}
-|Discovered and proven by Allen Brady
+|Allen Brady
+|Direct Simulation
 |-
 |[[BB(5)]]
-|47,176,870
+|<math>47\,176\,870</math>
 |{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}
-|Discovered by Heiner Marxen & Jürgen Buntrock in 1989
+|Heiner Marxen & Jürgen Buntrock in 1989
-Proven by [[bbchallenge.org]] in 2024
+|Direct Simulation
 |-
 |[[BB(6)]]
-|<math>> 10 \uparrow\uparrow 15</math>
+|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
-|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}
+|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
-|Discovered by Pavel Kropitz in 2022
+|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(8)
 |
 |
@@ Line 47: / Line 60: @@
 |-
 |BB(9)
-|<math>> 10 \uparrow\uparrow 30</math>
+|<math>> f_\omega(f_9(2))</math>
-|{{TM|1LD1LB_1LZ1LA_0LB1LD_0LE0LD_1LF1RC_0LG0LF_1LH1RE_0LI0LH_1RI1RG|halt}}
+|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
-|Designed by Milton Green in 1964 ([[Green's machines]])
+|Jacobzheng in 2024
+|
 |-
 |BB(10)
-|<math> > 10 \uparrow\uparrow 10^{10^{12}} </math>
+|<math>> f_\omega^2(25)</math>
-|{{TM|1LB1RZ_0LC1LC_0LD0LC_1LE1RA_0LF0LE_1LG1RD_0LH0LG_1LI1RF_0LJ0LI_1RJ1RH|halt}}
+|<span style="word-break:break-all">{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}</span>
-|[[Green's machines]]
+|Racheline in 2024
+|
 |-
 |BB(11)
-|<math> > 10 \uparrow\uparrow\uparrow 10^{12} </math>
+|<math>> f_\omega^2(10 \uparrow^4 4)</math>
-|{{TM|1LD1LB_1LZ1LA_0LB1LD_0LE0LD_1LF1RC_0LG0LF_1LH1RE_0LI0LH_1LJ1RG_0LK0LJ_1RK1RI|halt}}
+|<span style="word-break:break-all">{{TM|1RB1RA_1LH1RC_0LD0LG_0RE1LD_1RA1RF_---0RA_1LC1LG_1LJ1LI_1LK1LJ_1RZ0LI_1RA1LH|halt}}</span>
-|[[Green's machines]]
+|Jacobzheng in 2026
+|
 |-
 |BB(12)
-|<math> > 2 \uparrow^{12} 4 > Ack(11) </math>
+|<math>> f_\omega^4(2 \uparrow\uparrow\uparrow 4-3) > f_\omega^4(f_4(2))</math>
-|{{TM|1RB1LL_0RC1RC_1LD1LG_0RE1LC_1LD1RF_1RE1RI_1RH0LA_---0RF_0RB1LJ_---0LK_0LF1LF_1RZ0LA|halt}}
+|<span style="word-break:break-all">{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_1RZ1LA_1RF1LL|halt}}</span>
-|[https://github.com/sligocki/sligocki.github.io/issues/7#issuecomment-2143486164 Compilation] of a BB(3,4) TM by <code>@Iijil1</code> in 2024
+|Racheline in 2024
+|
 |-
 |BB(13)
-|<math> > Ack(2045) </math>
+|<math>> f_{\omega + 1}(2046) > g_{64}</math>
+|<span style="word-break:break-all">{{TM|1RB1RA_1LC1RD_1LA1LC_1LG0RE_1LC1RB_0RL1LG_0LM0RH_1RI1RH_1LK0RI_---0LK_1LF1LK_1LJ1RL_1RZ1RH|halt}}</span>
+|[https://discord.com/channels/960643023006490684/1331570843829932063/1481871400640839691 50_ft_lock in 2026]
 |
-|[https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(13) Designed] by <code>@Wythagoras</code> in 2016
 |-
 |BB(14)
-|<math> > Ack(10 \uparrow\uparrow 5) </math>
+|<math>> f_{\omega + 1}(65\,536)</math>
-|{{TM|1RB1RN_1RC0LG_1LD1RB_1LF1LE_1RH1LF_1RG0LD_1LB0RF_1RI1RH_1RZ0RJ_1RI1LK_0LK1LL_1LM1RN_1RH0LL_0RA1LK|halt}}
+|<span style="word-break:break-all">{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ---|halt}}</span>
-|[[User:Jacobzheng|Designed]] by <code>Jacobzheng</code> in 2024
+|[https://discord.com/channels/960643023006490684/960643023530762341/1274366178529120287 Racheline in 2024]
+|
 |-
 |BB(15)
+|<math>> f_{\omega + 1}(f_\omega(10^{57}))</math>
+|<span style="word-break:break-all">{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_1RZ0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}</span>
+|Jacobzheng in 2025
+|
+|-
+|BB(16)
+|<math>> f_{\omega + 1}^2(10^{10^{57}})</math>
+|<span style="word-break:break-all">{{TM|0RG1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RP0LE_1RD1RA_0LP1LO_1LO1RM_0LJ1LK_1LL1LL_1RP0LK_1LH0RN_1RZ0LJ_1LH1LO_1RB1RP|halt}}</span>
+|Jacobzheng in 2025
+|
+|-
+|BB(17)
+|
 |
 |
 |
 |-
-|BB(16)
+|BB(18)
-|<math> > f_{\omega + 1}(2 \uparrow\uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow\uparrow 9) > g_{64} </math>
+|<math>> f_{\omega + 2}(f_{\omega + 1}^3(f_{\omega}^2(60)))</math>
+|<span style="word-break:break-all">{{TM|0RH1LD_0RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RE1RA_1RM0LJ_0LJ1LK_1LL1LL_1RG0LK_0LQ1RN_1RZ0LO_1LP1LO_0LG1LO_1LR1LQ_0LP1LQ|halt}}</span>
+[[User:Jacobzheng/BB(18)]]
+|Jacobzheng in 2025
+|
+|-
+|BB(19)
+|
+|
+|
+|
+|-
+|BB(20)
+|<math>> f_{\omega + 6}^2(4)</math>
+|<span style="word-break:break-all">{{TM|1RR1RG_0LJ0RC_1RB1LD_0LD1LE_1LF1RG_1LI0LE_0RA1LD_1RB1RH_1RH1RN_1LK1LJ_0LL1LJ_1RM1RA_0RM0RH_0LO1RN_1RH0LP_1RH1LQ_1RH1LO_1LS1RR_1RZ1LT_0LH0LS|halt}}</span>
+[[User:Jacobzheng/BB(20)]]
+|Jacobzheng in 2026
+|
+|-
+|BB(21)
+|<math>> f_{\omega^2}^2(4 \uparrow\uparrow 341)</math>
+|<span style="word-break:break-all">{{TM|0LI0LF_0RJ1RG_0RD1LC_1RH1RE_1LO0RH_1LA1LF_0LC1LA_1RB1RH_1RD0LF_1LP0RK_0LM1LL_0LL1LM_1LE1LN_0RQ0LM_0RP1LO_1LR0RH_1LF1RQ_1RZ0LS_1LH0LT_1LH1LU_0LE1LR|halt}}</span>
+[[Talk:Champions|See talk page]]
+|[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>
+|[[Talk:Champions|See talk page]]
+|[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>
+|
+|-
+|BB(1015)
+|<math>> D^{5}(99)</math> = Loader's Number
+|
+|CatsAreFluffy in 2024<ref>https://github.com/CatsAreFluffy/metamath-turing-machines/commit/85948b04fc4aeb983ca6d63d6aee5ad6ef308bfe</ref>
+|
+|}
+=== 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."
+|}
+== 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
+|
+|-
+|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
+|
+|-
+|BB(7,3)
+|
+|
+|
+|
+|-
+|BB(8,3)
+|<math>> f_{\omega+1}(f_\omega(f_7(2)))</math>
+|<span style="word-break:break-all">{{TM|1LD1LE2RA_1RE2LE2RB_1RD2RA2LC_1RB1RD2RB_1LH2RF1LC_1RZ0LF1LG_1RD2LG2LG_0LD1LH1LH|halt}}</span>
+|Jacobzheng in 2026
+|
+|-
+|BB(9,3)
+|
+|
+|
+|
+|-
+|BB(10,3)
+|
+|
+|
+|
+|-
+|BB(11,3)
+|<math>> f_{\omega+3}^2(4)</math>
+|<span style="word-break:break-all">{{TM|1LD2RG2LA_2LE2LH2RB_2LE1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RI---_0RC1LH2RG_1RC2LH1LA_1RI1RI1LJ_0LA2LK---_1RZ2LJ---|halt}}</span>
+|Jacobzheng in 2026
+|
+|-
+|...
+|
+|
+|
+|
+|-
+|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>
+|
+|-
+|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>
+|
+|}
+== 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(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>
+|
+|}
+== 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
+|
+|-
+|...
+|
+|
+|
+|
+|-
+|BB(19,5)
+|<math>> f_{\varepsilon_0}(1.7\cdot10^{352})</math>
+|<span style="word-break:break-all">{{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}}</span>
+|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(19,5)</ref>
+|-
+|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>
+|
+|}
+== 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(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>
+|
+|}
+== 7-Symbol TMs ==
+{| class="wikitable"
+!Domain
+!Runtime
+!Champions
+!Discovered by
+!Verification
+|-
+|[[BB(2,7)]]
+|
+|(Enumerating Holdouts...)
+|
+|
+|-
+|...
+|
+|
+|
+|
+|-
+|BB(18,7)
+|<math>> f_{\varepsilon_0}(1.7\cdot10^{352})</math>
+|<span style="word-break:break-all">{{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}}</span>
+|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(18,7)</ref>
+|-
+|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>
+|
+|-
+|BB(134,7)
+|<math>> f_{\psi_0(\Omega_\omega)}(2050)</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>
+|
+|}
+== 8+-Symbol TMs ==
+{| class="wikitable"
+!Domain
+!Runtime
+!Champions
+!Discovered by
+!Verification
+|-
+|BB(17,8)
+|<math>> f_{\varepsilon_0}(1.7\cdot10^{352})</math>
+|<span style="word-break:break-all">{{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}}</span>
+|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(17,8)</ref>
+|-
+|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>
+|
+|}
+== 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]].
+|
+|<math>O(n)</math>
+|-
+|Chaotic
+|Have a chaotic behavior with repeating patterns that go back and forth.
+|
+* {{TM|1RB1LB_1LA---|halt}}
+* {{TM|1RB---_1LB0RC_1LC1LA|halt}}
+* {{TM|1RB1LB_1LA0LC_---1LD_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|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>
+|-
+|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
+|
 |
-|Designed by  Daniel Nagaj in 2021<ref>Shawn Ligocki. 2022. "B(16) > Graham's Number". https://www.sligocki.com/2022/07/11/bb-16-graham.html</ref>
+|<math>O(2 \uparrow^n 3)</math>
 |}
 == References ==
-<references />
+[[Category:Individual machines]]
+[[Category:Zoology]]

Champions: Difference between revisions

Latest revision as of 12:03, 25 September 2026

Trivial Champions

2-Symbol TMs

Conjectured

3-Symbol TMs

4-Symbol TMs

5-Symbol TMs

6-Symbol TMs

7-Symbol TMs

8+-Symbol TMs

Zoology

References

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