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.

1-Symbol TMs

BB(n,1) = n

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. 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)	$47176870$	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)
BB(14)	$>f_{\omega +1}(65536)>g_{64}$	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

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)	$>2\uparrow \uparrow \uparrow 2^{2^{32}}$	0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD (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