Champions

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 $\uparrow$ represents Knuth's up-arrow notation. Machines 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)	$47176870$	1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA (bbch)	Heiner Marxen & Jürgen Buntrock in 1989	Direct Simulation
BB(6)	$>2\uparrow \uparrow \uparrow 5$	1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE (bbch)	mxdys in 2025	See mxdys's analysis on the TM page
BB(7)	$>2{\uparrow }^{11}2{\uparrow }^{11}3$	1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF (bbch)	Pavel Kropitz in 2025	Analyzed by Shawn Ligocki (see TM page)
BB(8)
BB(9)	$>f_{\omega }(f_9(2))$	1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH (bbch)	Jacobzheng in 2024
BB(10)	$>f_{\omega }^2(25)$	1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ (bbch)	Racheline in 2024
BB(11)	$>f_{\omega }^2(2\uparrow \uparrow 12)>f_{\omega }^2(f_3(9))$	1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RZ0LI_0LD1LE (bbch)	Racheline in 2024
BB(12)	$>f_{\omega }^4(2\uparrow \uparrow \uparrow 4-3)>f_{\omega }^4(f_4(2))$	0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_1RZ1LA_1RF1LL (bbch)	Racheline in 2024
BB(13)	$>f_{\omega +1}(2046)>g_{64}$	1RB1RA_1LC1RD_1LA1LC_1LG0RE_1LC1RB_0RL1LG_0LM0RH_1RI1RH_1LK0RI_---0LK_1LF1LK_1LJ1RL_1RZ1RH (bbch)	50_ft_lock in 2026
BB(14)	$>f_{\omega +1}(65536)$	1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ--- (bbch)	Racheline in 2024
BB(15)	$>f_{\omega +1}(f_{\omega }(10^{57}))$	0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_1RZ0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN (bbch)	Jacobzheng in 2025
BB(16)	$>f_{\omega +1}^2(10^{10^{57}})$	User:Jacobzheng/BB(16)	Jacobzheng in 2025
BB(17)
BB(18)	$>f_{\omega +2}(f_{\omega +1}^3(f_{\omega }^2(60)))$	User:Jacobzheng/BB(18)	Jacobzheng in 2025
BB(19)
BB(20)	$>f_{\omega +2}^2(21)$		Racheline in 2024
BB(21)	$>f_{{\omega }^2}^2(4\uparrow \uparrow 341)$		Racheline in 2024
BB(40)	$>f_{{\omega }^{\omega }}(75500)$	User:Jacobzheng/BB(40)	Jacobzheng in 2024
BB(41)	$>f_{{\omega }^{\omega }}^4(32)$	User:Jacobzheng/BB(41)	Jacobzheng in 2024
BB(51)	$>f_{{\epsilon }_0+1}(8)$		Racheline in 2024
BB(150)	$>f_{lim(BMS)}(10\uparrow \uparrow 15)$	too large to show	Patcail in 2025

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

	Runtime	Champions	Discovered By	Verification
BB(2,3)	$38$	1RB2LB1RZ_2LA2RB1LB (bbch)	Allen Brady in 1988	Direct Simulation
BB(3,3)	$>10^{17}$	0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC (bbch)	Terry & Shawn Ligocki in 2007	Analysis by Pascal Michel
BB(4,3)	$>10{\uparrow }^{4}4$	1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD (bbch)	Pavel Kropitz in 2024

4-Symbol TMs

	Runtime	Champions	Discovered By	Verification
BB(2,4)	$3932964$	1RB2LA1RA1RA_1LB1LA3RB1RZ (bbch)	Terry & Shawn Ligocki in 2005	Pascal Michel, Heiner Marxen, Allen Brady
BB(3,4)	$>2{\uparrow }^{15}5$	1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC (bbch)	Pavel Kropitz in 2024	Analysis by Shawn Ligocki

5-Symbol TMs

	Runtime	Champions	Discovered By	Verification
BB(2,5)	$>10^{10^{10^{3314360}}}$	1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ (bbch)	Daniel Yuan in 2024	mxdys in Rocq
BB(3,5)	$>f_{\omega }(2{\uparrow }^{15}5)>f_{\omega }^2(15)$	1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC (bbch)	Racheline in 2024

6-Symbol TMs

	Runtime	Champions	Discovered By	Verification
BB(2,6)	$>10\uparrow \uparrow 10\uparrow \uparrow 10^{10^{115}}$	1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA (bbch)	Pavel Kropitz in 2023	Analysis by Shawn Ligocki

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			$O(2\uparrow \uparrow n)$
Ackermannian			$O(2{\uparrow }^{n}3)$