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.

	Runtime	Champions	Comment
BB(2)	$6$	1RB1LB_1LA1RZ (bbch) 1RB0LB_1LA1RZ (bbch) 1RB1RZ_1LB1LA (bbch) 1RB1RZ_0LB1LA (bbch) 0RB1RZ_1LA1RB (bbch)	Discovered and proven by hand by Tibor Radó
BB(3)	$21$	1RB1RZ_1LB0RC_1LC1LA (bbch)	Proven by Shen Lin
BB(4)	$107$	1RB1LB_1LA0LC_1RZ1LD_1RD0RA (bbch)	Discovered and proven by Allen Brady
BB(5)	$47\,176\,870$	1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA (bbch)	Discovered by Heiner Marxen & Jürgen Buntrock in 1989 Proven by bbchallenge.org in 2024
BB(6)	$>10\uparrow \uparrow 15$	1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE (bbch)	Discovered by Pavel Kropitz in 2022
BB(7)
BB(8)	$>2\uparrow ^{5}4>f_{6}(2)$	1RH1RF_0LC0LH_0RD1LC_0RE1RA_1RB1RE_1RZ1RG_1RF0RE_1LB1LH (bbch)	Discovered by Racheline in 2024
BB(9)
BB(10)	$>f_{\omega }^{2}(25)$	1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ (bbch)	Discovered by 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)	Discovered by 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)	Discovered by 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)	Discovered by Racheline in 2024
BB(15)
BB(16)	$>f_{\omega +1}(2\uparrow \uparrow \uparrow \uparrow 2\uparrow \uparrow \uparrow \uparrow 9)$		Designed by Daniel Nagaj in 2021
BB(20)	$>f_{\omega +2}^{2}(21)$		Discovered by Racheline in 2024
BB(21)	$>f_{\omega ^{2}}^{2}(4\uparrow \uparrow 341)$		Discovered by Racheline in 2024
BB(51)	$>f_{\varepsilon _{0}+1}(8)$		Designed by Racheline in 2024

3-Symbol TMs

	Runtime	Champions	Comment
BB(2,3)	$38$	1RB2LB1RZ_2LA2RB1LB (bbch)
BB(3,3)	$>10^{17}$	0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC (bbch)
BB(4,3)	$>10^{14072}$	1RB1RZ2RC_2LC2RD0LC_1RA2RB0LB_1LB0LD2RC (bbch)

4-Symbol TMs

	Runtime	Champions	Comment
BB(2,4)	$3\,932\,964$	1RB2LA1RA1RA_1LB1LA3RB1RZ (bbch)
BB(3,4)	$>2\uparrow ^{15}5$	1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC (bbch)

5-Symbol TMs

	Runtime	Champions	Comment
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)