Champions

Busy Beaver Champions are the current record holding 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 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 lowerbounds 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)	$>10\uparrow \uparrow 15$	1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE (bbch)	Pavel Kropitz in 2022	Analysis by Shawn Ligocki
BB(7)	$>10\uparrow \uparrow 286$	1RB1RF_0RC1RG_1LD1LE_0LE1LD_0RF0LC_1RA0LC_0RF1RZ (bbch)	Terry Ligocki in 2025
BB(8)	$>2\uparrow ^{5}4>f_{6}(2)$	1RH1RF_0LC0LH_0RD1LC_0RE1RA_1RB1RE_1RZ1RG_1RF0RE_1LB1LH (bbch)	Racheline in 2024
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)	$>f_{\omega +1}(65\,536)>g_{64}$	1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ--- (bbch)	Racheline in 2024
BB(15)
BB(16)	$>f_{\omega +1}(2\uparrow \uparrow \uparrow \uparrow 2\uparrow \uparrow \uparrow \uparrow 9)$		Daniel Nagaj in 2021	Analysis by Shawn Ligocki
BB(17)	$>f_{\omega +1}(f_{\omega }(60))$	User:Jacobzheng/BB(17)	Jacobzheng in 2024
BB(18)	$>f_{\omega +1}(f_{\omega }^{2}(60))$	User:Jacobzheng/BB(18)	Jacobzheng in 2024
BB(19)	$>f_{\omega +1}^{3}(f_{\omega }(60))$	User:Jacobzheng/BB(19)	Jacobzheng in 2024
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)	$>f_{\omega ^{\omega }}(75\,500)$	User:Jacobzheng/BB(40)	Jacobzheng in 2024
BB(41)	$>f_{\omega ^{\omega }}^{4}(32)$	User:Jacobzheng/BB(41)	Jacobzheng in 2024
BB(51)	$>f_{\varepsilon _{0}+1}(8)$		Racheline in 2024

3-Symbol TMs

	Runtime	Champions	Discovered By	Verification
BB(2,3)	$38$	1RB2LB1RZ_2LA2RB1LB (bbch)
BB(3,3)	$>10^{17}$	0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC (bbch)
BB(4,3)	$>2\uparrow \uparrow \uparrow 2^{2^{32}}$	0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD (bbch)

4-Symbol TMs

	Runtime	Champions	Discovered By	Verification
BB(2,4)	$3\,932\,964$	1RB2LA1RA1RA_1LB1LA3RB1RZ (bbch)	Shawn & Terry Ligocki in 2005	Pascal Michel, Hiener Marxen, Allen Brady
BB(3,4)	$>2\uparrow ^{15}5$	1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC (bbch)

5-Symbol TMs

	Runtime	Champions	Discovered By	Verification
BB(2,5)	$>10^{10^{10^{3\,314\,360}}}$	1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ (bbch)
BB(3,5)	$>f_{\omega }(2\uparrow ^{15}5)>f_{\omega }^{2}(15)$	1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC (bbch)

6-Symbol TMs

	Runtime	Champions	Discovered By	Verification
BB(2,6)	$>10^{10^{10^{3\,314\,360}}}$