Champions: Difference between revisions

Revision as of 18:27, 14 August 2025

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.

2-Symbol TMs

Rows are blank if no champion has been found which surpasses a smaller size problem. Take also note that the $f_{x} (n)$ used in the lower bounds represent the Fast-Growing Hierarchy. Note that most champions above 6 states are self-reported and have not been independently verified.


	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)	$> 2 ↑ ↑ ↑ 5$	`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)	$> f_{ω}^{2} (2 ↑ ↑ 12) > f_{ω}^{2} (f_{3} (9))$	`1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RZ0LI_0LD1LE` (bbch)	Racheline in 2024
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)
BB(14)	$> f_{ω + 1} (65 536) > g_{64}$	`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)	$> f_{ω + 1}^{2} (1 0^{1 0^{57}})$	User:Jacobzheng/BB(16)	Jacobzheng in 2025
BB(17)
BB(18)	$> f_{ω + 2} (f_{ω + 1}^{3} (f_{ω}^{2} (60)))$	User:Jacobzheng/BB(18)	Jacobzheng in 2025
BB(19)
BB(20)	$> f_{ω + 2}^{2} (21)$		Racheline in 2024
BB(21)	$> f_{ω^{2}}^{2} (4 ↑ ↑ 341)$		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)	$> f_{ε_{0} + 1} (8)$		Racheline in 2024

3-Symbol TMs


	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)	$> 2 ↑ ↑ ↑ 2^{2^{32}}$	`0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD` (bbch)	Pavel Kropitz in 2024

4-Symbol TMs


	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

5-Symbol TMs

	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 Coq
BB(3,5)	$> f_{ω} (2 ↑^{15} 5) > f_{ω}^{2} (15)$	`1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC` (bbch)	Racheline in 2024

6-Symbol TMs

	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

@@ Line 13: / Line 13: @@
 |-
 |[[BB(2)]]
-|<math> 6 </math>
+|<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
@@ Line 19: / Line 19: @@
 |-
 |[[BB(3)]]
-|<math> 21 </math>
+|<math>21</math>
 |{{TM|1RB1RZ_1LB0RC_1LC1LA|halt}}
 |Proven by Shen Lin
@@ Line 25: / Line 25: @@
 |-
 |[[BB(4)]]
-|<math> 107 </math>
+|<math>107</math>
 |{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}}
 |Allen Brady
@@ Line 31: / Line 31: @@
 |-
 |[[BB(5)]]
-|<math> 47\,176\,870 </math>
+|<math>47\,176\,870</math>
 |{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}
 |Heiner Marxen & Jürgen Buntrock in 1989
@@ Line 37: / Line 37: @@
 |-
 |[[BB(6)]]
-|<math> > 2\uparrow\uparrow\uparrow 5 </math>
+|<math>> 2\uparrow\uparrow\uparrow 5</math>
 |{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
 |mxdys in 2025
@@ Line 55: / Line 55: @@
 |-
 |BB(9)
-|<math> > f_\omega(f_9(2)) </math>
+|<math>> f_\omega(f_9(2))</math>
 |{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
 |Jacobzheng in 2024
@@ Line 61: / Line 61: @@
 |-
 |BB(10)
-|<math> > f_\omega^2(25) </math>
+|<math>> f_\omega^2(25)</math>
 |{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
 |Racheline in 2024
@@ Line 67: / Line 67: @@
 |-
 |BB(11)
-|<math> > f_\omega^2(2 \uparrow\uparrow 12) > f_\omega^2(f_3(9)) </math>
+|<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
@@ Line 73: / Line 73: @@
 |-
 |BB(12)
-|<math> > f_\omega^4(2 \uparrow\uparrow\uparrow 4-3) > f_\omega^4(f_4(2)) </math>
+|<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
@@ Line 85: / Line 85: @@
 |-
 |BB(14)
-|<math> > f_{\omega + 1}(65\,536) > g_{64} </math>
+|<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]
@@ Line 91: / Line 91: @@
 |-
 |BB(15)
-|<math> > f_{\omega + 1}(f_\omega(10^{57})) </math>
+|<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
@@ Line 97: / Line 97: @@
 |-
 |BB(16)
-|<math> > f_{\omega + 1}^2(10^{10^{57}}) </math>
+|<math>> f_{\omega + 1}^2(10^{10^{57}})</math>
 |[[User:Jacobzheng/BB(16)]]
 |Jacobzheng in 2025
@@ Line 109: / Line 109: @@
 |-
 |BB(18)
-|<math> > f_{\omega + 2}(f_{\omega + 1}^3(f_{\omega}^2(60))) </math>
+|<math>> f_{\omega + 2}(f_{\omega + 1}^3(f_{\omega}^2(60)))</math>
 |[[User:Jacobzheng/BB(18)]]
 |Jacobzheng in 2025
@@ Line 121: / Line 121: @@
 |-
 |BB(20)
-|<math> > f_{\omega + 2}^2(21) </math>
+|<math>> f_{\omega + 2}^2(21)</math>
 |
 |[https://discord.com/channels/960643023006490684/1026577255754903572/1274414683331366924 Racheline in 2024]
@@ Line 127: / Line 127: @@
 |-
 |BB(21)
-|<math> > f_{\omega^2}^2(4 \uparrow\uparrow 341) </math>
+|<math>> f_{\omega^2}^2(4 \uparrow\uparrow 341)</math>
 |
 |[https://discord.com/channels/960643023006490684/1026577255754903572/1274471360206344213 Racheline in 2024]
@@ Line 133: / Line 133: @@
 |-
 |BB(40)
-|<math> > f_{\omega^\omega}(75\,500) </math>
+|<math>> f_{\omega^\omega}(75\,500)</math>
 |[[User:Jacobzheng/BB(40)]]
 |Jacobzheng in 2024
@@ Line 139: / Line 139: @@
 |-
 |BB(41)
-|<math> > f_{\omega^\omega}^4(32) </math>
+|<math>> f_{\omega^\omega}^4(32)</math>
 |[[User:Jacobzheng/BB(41)]]
 |Jacobzheng in 2024
@@ Line 145: / Line 145: @@
 |-
 |BB(51)
-|<math> > f_{\varepsilon_0 + 1}(8) </math>
+|<math>> f_{\varepsilon_0 + 1}(8)</math>
 |
 |[https://discord.com/channels/960643023006490684/1026577255754903572/1276881449685094495 Racheline in 2024]
@@ Line 162: / Line 162: @@
 |-
 |[[BB(2,3)]]
-|<math> 38 </math>
+|<math>38</math>
 |{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
 |Allen Brady in 1988
@@ Line 168: / Line 168: @@
 |-
 |[[BB(3,3)]]
-|<math> > 10^{17} </math>
+|<math>> 10^{17}</math>
 |{{TM|0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC|halt}}
 |Terry & Shawn Ligocki in 2007
@@ Line 174: / Line 174: @@
 |-
 |[[BB(4,3)]]
-|<math> > 2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
+|<math>> 2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
 |{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
 |Pavel Kropitz in 2024
@@ Line 190: / Line 190: @@
 |-
 |[[BB(2,4)]]
-|<math> 3\,932\,964 </math>
+|<math>3\,932\,964</math>
 |{{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
 |Terry & Shawn Ligocki in 2005
@@ Line 196: / Line 196: @@
 |-
 |[[BB(3,4)]]
-|<math> > 2 \uparrow^{15} 5 </math>
+|<math>> 2 \uparrow^{15} 5</math>
 |{{TM|1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
 |Pavel Kropitz in 2024
@@ Line 211: / Line 211: @@
 |-
 |[[BB(2,5)]]
-|<math> > 10^{10^{10^{3\,314\,360}}} </math>
+|<math>> 10^{10^{10^{3\,314\,360}}}</math>
 |{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}
 |Daniel Yuan in 2024
@@ Line 217: / Line 217: @@
 |-
 |[[BB(3,5)]]
-|<math> > f_\omega(2 \uparrow^{15} 5) > f_\omega^2(15) </math>
+|<math>> f_\omega(2 \uparrow^{15} 5) > f_\omega^2(15)</math>
 |{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC|halt}}
 |Racheline in 2024
@@ Line 232: / Line 232: @@
 |-
 |[[BB(2,6)]]
-|<math> > 10 \uparrow\uparrow 10 \uparrow\uparrow 10^{10^{115}} </math>
+|<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]
 |}

Champions: Difference between revisions

Revision as of 18:27, 14 August 2025

Contents

2-Symbol TMs

3-Symbol TMs

4-Symbol TMs

5-Symbol TMs

6-Symbol TMs

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Champions: Difference between revisions

Revision as of 18:27, 14 August 2025

2-Symbol TMs

3-Symbol TMs

4-Symbol TMs

5-Symbol TMs

6-Symbol TMs

Navigation menu

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