BB(8)

The 8-state, 2-symbol busy beaver problem BB(8) is unsolved and largely unexplored. It is assumed that the longest running machine will run for more than Graham's number steps.

The current BB(8) champion is 1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LH0LC_1RG1LD_0RG0RF_1RG1RZ (bbch), which takes over $f_{\omega }^3(12)$ steps to halt. It was constructed by Jeff Ketchersid in 2026 by modifying the Pavel Kropitz's former BB(7) champion.

In November 2025, an enumeration of a test sample of BB(8) TMs has shown the enumeration of one of the 1 million subtasks to take 7.5 hours and produce 35,000 holdouts. These values are 225 times longer and 350 times larger than for BB(7). This makes BB(8) even harder to enumerate than BB(2,7). It is expected that an enumeration of this domain will leave about 147,000,000,000 holdouts.

History

Timeline of bounds established for Σ(8) and S(8)
Time of discovery	Machine	S(8)	Σ(8)	Discoverer	Notes
November 1964	1LB1RZ_0LC1LC_0LD0LC_1LE1RA_0LF0LE_1LG1RD_0LH0LG_1RH1RF (bbch)		$\frac{7\cdot 3^{93}-3}{2}>8.247\times 10^{44}$	Green, M. W.	This is the $n=8$ Green's machine.
July 2000	1RB0RC_0LA0RD_1RD1RZ_1LE0LD_1RF1LB_1RA1RE (bbch)	$>5.3\times 10^{42}$	$>2.5\times 10^{21}$	Marxen, H., Buntrock, J.
August 2000	1RB0LC_1LA1RC_1RA0LD_1LE1LC_1RF1RZ_1RA1RE (bbch)	$>6.1\times 10^{119}$	$>1.4\times 10^{60}$	Marxen, H., Buntrock, J.
	1RB0LB_0RC1LB_1RD0LA_1LE1LF_1LA0LD_1RH1LE (bbch)	$>6.1\times 10^{925}$	$>6.4\times 10^{60}$	Marxen, H., Buntrock, J.
February 2001	1RB0LF_0RC0RD_1LD1RE_0LE0LD_0RA1RC_1LA1RZ (bbch)	$>3.00\times 10^{1,730}$	$>1.29\times 10^{865}$
November 2007	1RB0RF_0LB1LC_1LD0RC_1LE1RZ_1LF0LD_1RA0LE (bbch)	$>8.92\times 10^{1,762}$	$>2.50\times 10^{881}$	Ligocki, T., Ligocki, S.
December 2007	1RB0LE_1LC0RA_1LD0RC_1LE0LF_1LA1LC_1LE1RZ (bbch)	$>2.58\times 10^{2,879}$	$>4.64\times 10^{1,439}$
May 2010	1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LA0RB_0RC0RE (bbch)	$>3.80\times 10^{21,132}$	$>3.18\times 10^{10,566}$	Kropitz, P.
June 2010	1RB1LE_1RC1RF_1LD0RB_1RE0LC_1LA0RD_1RZ1RC (bbch)	$>7.41\times 10^{36,534}$	$>3.51\times 10^{18,267}$
13 May 2022	1RB1RC_1LC0RF_1RA0LD_0LC0LE_1LD0RA_1RE1RZ (bbch)	$>9.66\times 10^{78,913}$	$>6.02\times 10^{39,456}$	Ligocki, S.
15 May 2022	1RB1RZ_1RC1RA_1RD0RB_1LE0RC_0LF0LD_0LB1LA (bbch)	$>5.42\times 10^{197,282}$	$>2.01\times 10^{98,641}$	Kropitz, P.
22 May 2022	1RB1RZ_0LC0LD_1LD1LC_1RE1LB_1RF1RD_0LD0RA (bbch)	$>8.27\times 10^{1,292,913,985}$	$>1.76\times 10^{646,456,993}$
27 May 2022	1RB0LA_1LC1LF_0LD0LC_0LE0LB_1RE0RA_1RZ1LD (bbch)	$>10\uparrow \uparrow 5.63$		Ligocki, S.
30 May 2022	1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE (bbch)	$>10\uparrow \uparrow 15.60$		Kropitz, P.
2024	1RH1RF_0LC0LH_0RD1LC_0RE1RA_1RB1RE_1RZ1RG_1RF0RE_1LB1LH (bbch)	$>2{\uparrow }^{5}4$		racheline
10 May 2025	1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF (bbch)	$>2{\uparrow }^{11}2{\uparrow }^{11}3$		Kropitz, P.	This was the first example of a 7-state, 2-symbol Ackermann-level machine.
4 October 2026	1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_1LC0RH_1RG1LF (bbch)	$>2{\uparrow }^{37}3$		Ketchersid, J., Claude Opus 5.5, and Gemini 4 Argon
19 October 2026	1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LH0LC_1RG1LD_0RG0RF_1RG1RZ (bbch)	$>f_{\omega }^3(12)$		Ketchersid, J., Claude Opus 5.5, and Gemini 4 Argon

Machines highlighted in blue were inherited from the hunt for smaller machines.