BB(6)

The 6-state, 2-symbol Busy Beaver problem, BB(6), refers to the unsolved 6^th value of the Busy Beaver function. With the discovery of the Cryptid machine Antihydra in June 2024, we now know that we must solve a Collatz-like problem in order to solve BB(6) and thus BB(6) is Hard.

The current BB(6) champion 1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE (bbch) was discovered by mxdys in June 2025, proving the lower bound:

$$S(6)>\mathrm{\Sigma }(6)>10\uparrow \uparrow 10\uparrow \uparrow 10\uparrow \uparrow 8$$

History

Timeline of lower bounds established for Σ(6) and S(6)
Month of discovery	Machine	S(6) lower bound	Σ(6) lower bound	Discoverer	Notes
November 1964	1LB1RZ_0LC1LC_0LD0LC_1LE1RA_0LF0LE_1RF1RD (bbch)	≥ 436	≥ 35	Green, M.	Part of an infinite family of machines now known as Green's machines.
August 1972	1RB0LB_1LC1RZ_0LD0LC_1LE0RA_0LF0LE_1RF1RD (bbch)	≥ 521	≥ 42	Lynn, D.	Lynn's stopping convention for these machines seems to return a stopping value one greater than the modern stopping convention.
November 1973	1RB0LD_1RC0RB_1LA0RA_0LC0LE_1LC1RZ (bbch)	≥ 992	≥ 23	Weimann, B.
1974	0RB1RZ_1RC0LD_1RD0RE_1LB0LD_1RE1RA (bbch)	≥ 7,707	≥ 88	Lynn, D.	These bounds were published simultaneously in April 1983.
	1RB0LE_1RC0RA_1LD1RZ_1LE1LD_1LA0LC (bbch)	≥ 6,147	≥ 112
August 1982	1RB0LC_1RC1RD_1LA0RB_0RE1RZ_1LC1RA (bbch)	≥ 134,467	≥ 501	Schult, U.	These bounds were published simultaneously in January 1983.
December 1982	1RB1RZ_1RC1RC_1RD1LF_1RE1RA_1RF0RD_1LB0LC (bbch)	≥ 4,208,824	≥ 2,075
January 1990	1RB1RA_1LC1LB_0LF1LD_1RA0LE_0RA1LC_1RE1RZ (bbch)	≥ 13,122,572,797	≥ 136,612	Marxen, H., Buntrock, J.
	1RB1RA_1LC1LB_0RF1LD_1RA0LE_1LZ1LF_0LA0LC (bbch)	≥ 8,690,333,381,690,951	≥ 95,524,079
July 2000	1RB0RC_0LA0RD_1RD1RZ_1LE0LD_1RF1LB_1RA1RE (bbch)	> 5.36 × 1042	> 2.53 × 1021
August 2000	1RB0LC_1LA1RC_1RA0LD_1LE1LC_1RF1RZ_1RA1RE (bbch)	> 6.12 × 10119	> 1.42 × 1060
	1RB0LB_0RC1LB_1RD0LA_1LE1LF_1LA0LD_1RH1LE (bbch)	> 6.19 × 10925	> 6.42 × 10462
February 2001	1RB0LF_0RC0RD_1LD1RE_0LE0LD_0RA1RC_1LA1RZ (bbch)	> 3.00 × 101,730	> 1.29 × 10865
November 2007	1RB0RF_0LB1LC_1LD0RC_1LE1RZ_1LF0LD_1RA0LE (bbch)	> 8.92 × 101,762	> 2.50 × 10881	Ligocki, T., Ligocki, S.
December 2007	1RB0LE_1LC0RA_1LD0RC_1LE0LF_1LA1LC_1LE1RZ (bbch)	> 2.58 × 102,879	> 4.64 × 101,439
May 2010	1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LA0RB_0RC0RE (bbch)	> 3.80 × 1021,132	> 3.18 × 1010,566	Kropitz, P.
June 2010	1RB1LE_1RC1RF_1LD0RB_1RE0LC_1LA0RD_1RZ1RC (bbch)	> 7.41 × 1036,534	> 3.51 × 1018,267
May 2022	1RB1RC_1LC0RF_1RA0LD_0LC0LE_1LD0RA_1RE1RZ (bbch)	> 9.66 × 1078,913	> 6.02 × 1039,456	Ligocki, S.
	1RB1RZ_1RC1RA_1RD0RB_1LE0RC_0LF0LD_0LB1LA (bbch)	> 5.42 × 10197,282	> 2.01 × 1098,641	Kropitz, P.
	1RB1RZ_0LC0LD_1LD1LC_1RE1LB_1RF1RD_0LD0RA (bbch)	> 8.27 × 101,292,913,985	> 1.76 × 10646,456,993
	1RB0LA_1LC1LF_0LD0LC_0LE0LB_1RE0RA_1RZ1LD (bbch)	> 10↑↑5.63		Ligocki, S.
	1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE (bbch)	> 10↑↑15.60		Kropitz, P.
June 2025	1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB (bbch)	> 10↑↑11,010,000		mxdys
	1RB1LE_1LC0RA_1RB1LD_1LC0LC_1RF0LB_1RZ1RE (bbch)	> 10↑↑11,010,000	> 10↑↑11,010,000		This machine's score exceeds the previous bound for S(6) by 2.
	1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE (bbch)	> 10↑↑10↑↑10↑↑8.10

Bounds highlighted in red indicate those which did not improve upon the best known at the time. Machines highlighted in blue were inherited from the hunt for BB(5).

Number of BB(6) holdouts over time.

In June 2024, the machine 1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA (bbch) (now known as Antihydra) was isolated from the BB(6) holdouts, and recognised as a Cryptid, implying that determining the exact value of S(6) would be, in a certain mathematical sense, hard.

@mxdys's holdouts list has 797 machines up to equivalence and 1630 machines not considering equivalence as of 30 September 2026. Partial Rocq proof is available on Github.

Often up-to-date annotated spreadsheet, with links to Discord discussions: Spreadsheet. The informal holdout count is 998.

All machines have been simulated out to 1e13 steps. ~140 machines remain to be simulated to 1e14, and ~190 to 1e15. See Spreadsheet

Cryptids

Several Turing machines have been found that are Cryptids, considered so because each of them have a Collatz-like halting problem, a type of problem that is generally difficult to solve. However, probabilistic arguments have allowed all but one of them to be categorized as probviously halting or probviously non-halting.

Probviously non-halting Cryptids:

• 1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA (bbch), Antihydra
• 1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA (bbch), a variant of Hydra and Antihydra
• 1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC (bbch), similar to Antihydra
• 1RB0LD_1RC1RF_1LA0RA_0LA0LE_1LD1LA_0RB--- (bbch), similar to Antihydra
• 1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB (bbch), similar to Antihydra
• 1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD (bbch), Space Needle
• 1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC (bbch), similar to Space Needle
• 1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE (bbch), similar to Space Needle
• 1RB0RF_1RC1RF_1LD0LE_---1LC_1RA1LE_1LC1RB (bbch), similar to Space Needle

Probviously halting Cryptids:

• 1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA (bbch), Lucy's Moonlight
• 1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC (bbch), a family of 16 related TMs
• 1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF (bbch)
• 1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB (bbch)
• 1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA (bbch)
• 1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE (bbch)
• 1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA (bbch)
• 1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE (bbch)
• 1RB1LD_1RC0LE_1LA1RE_0LF1LA_1RB0RB_---0LB (bbch)
• 1RB0RE_1LC0RA_1LA1LD_1LC1LF_0LC0LB_1LE--- (bbch)

Although 1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE (bbch) behaves similarly to the probviously halting Cryptids, it is estimated to have a 3/5 chance of becoming a translated cycler and a 2/5 chance of halting.

There are a few machines considered notable for their chaotic behaviour, but which have not been classified as Cryptids due to seemingly lacking a connection to any known open mathematical problems, such as Collatz-like problems.

Potential Cryptids:

• 1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE (bbch)
• 1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE (bbch)
• 1RB1RF_1LC1LF_0RE1LD_0LB1LD_---1RC_1RA0RD (bbch)
• 1RB1LA_1RC0RF_1RD---_0LE1RB_---0LA_1LD1RF (bbch)
• 1RB1RF_0LC0RF_1RD1LC_---0LE_0RC1LF_1RA0LE (bbch)
• 1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA (bbch) BMO8
• 1RB---_0RC0RD_1LD1RB_0LE0LC_1RA0LF_1LD1LE (bbch) BMO8-like

Top Halters

Below is a table of the machines in TNF-1RB format with the 20 highest known runtimes. Their runtimes are expressed using an extension of Knuth's up-arrow notation.

Top Known BB(6) Halters

Standard format	(approximate) runtime	Discoverer
1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE (bbch)	10 ↑↑ 10 ↑↑ 10 ↑↑ 8.10237	mxdys
1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB (bbch)	10 ↑↑ 11010000	mxdys
1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE (bbch)	10 ↑↑ 15.60465	Pavel Kropitz
1RB0LF_1RC1RB_1LD0RA_1LB0LE_1RZ0LC_1LA1LF (bbch)	10 ↑↑ 7.52390	Katelyn Doucette
1RB0LF_1RC1RB_1LD0RA_1RF0LE_1RZ0LC_1LA1LF (bbch)
1RB0LF_1RC1RB_1LD0RA_1LF0LE_1RZ0LC_1LA1LF (bbch)
1RB1RC_1LC1RE_1LD0LB_1RE1LC_1LE0RF_1RZ1RA (bbch)	10 ↑↑ 7.23637	racheline
1RB1RA_1LC1LE_1RE0LD_1LC0LF_1RZ0RA_0RA0LB (bbch)	10 ↑↑ 6.96745	poppuncher
1RB0RF_1LC0RA_1RZ0LD_1LE1LD_1RB1RC_0LD0RE (bbch)	10 ↑↑ 5.77573
1RB0LA_1LC1LF_0LD0LC_0LE0LB_1RE0RA_1RZ1LD (bbch)	10 ↑↑ 5.63534	Shawn Ligocki
1RB1RE_1LC1LF_1RD0LB_1LE0RC_1RA0LD_1RZ1LC (bbch)	10 ↑↑ 5.56344
1RB0LE_0RC1RA_0LD1RF_1RE0RB_1LA0LC_0RD1RZ (bbch)	10 ↑↑ 5.12467
1RB0RF_1LC1LB_0RE0LD_0LC0LB_0RA1RE_0RD1RZ (bbch)	10 ↑↑ 5.03230
1RB1LA_1LC0RF_1LD1LC_1LE0RE_0RB0LC_1RZ1RA (bbch)	10 ↑↑ 4.91072
1RB1LA_0RC1RD_1LC0LA_1LE0RB_0RD0RF_1RZ0RE (bbch)	$>10^{10^{11,347}}$ (score: 3)	Pavel Kropitz
1RB0LE_1LC1RA_1RE0LD_1LC1LF_1LA0RC_1RZ1LC (bbch)	$>10^{10^{140}}$
1RB1RF_1LC1RE_0LD1LB_1LA0RA_0RA0RB_1RZ0RD (bbch)	$>10^{10^{111}}$
1RB0LF_1LC0RA_1RD0LB_1LE1RC_1RZ1LA_1LA1LE (bbch)	$>10^{10^{34}}$
1RB0RF_1LC1RB_0RD0LB_1RZ0LE_1RE0RA_1RD1RE (bbch)	$>10^{10^{28}}$
1RB0RB_0RC0LF_0RD0RA_0LE1RZ_1LE0LA_1LF1RA (bbch)	$>10^{1400000000}$ (score: 13,964,326,472)	Racheline

The scores are presumed to be on the order of $\sqrt{\text{runtime}}$ (unless explicitly noted) which is roughly indistinguishable in tetration notation.

Techniques

Simulating tetrational machines, such as the former champion 1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE (bbch), requires accelerated simulation that can handle Collatz Level 2 inductive rules. In other words, it requires a simulator that can prove the rules:

$$\begin{array}{lcl}C(4k)& \to & \mathrm{Halt}\left(\frac{3^{k+3}-11}{2}\right)\\ C(4k+1)& \to & C\left(\frac{3^{k+3}-11}{2}\right)\\ C(4k+2)& \to & C\left(\frac{3^{k+3}-11}{2}\right)\\ C(4k+3)& \to & C\left(\frac{3^{k+3}+1}{2}\right)\\ \end{array}$$

and also compute the remainder mod 3 of numbers produced by applying these rules 15 times (which requires some fancy math related to Euler's totient function).

We are also applying existing automatic deciders on current holdout lists with more extreme choices of parameters (more computational resources). XnoobSpeakable was able to solve 11 of the final 2728 holdouts using higher order parameters with the Ligockis' Enumerate.py. An example command line entry is:

python3 Code/Enumerate.py --infile "bb6in/bb6tm{i}.txt" --outfile "bb6out/t{i}.pb" -r --no-steps --exp-linear-rules --max-loops=50_000_000 --block-mult=3 --max-block-size=100 --time=500 --force --save-freq=1

XnoobSpeakable ran Enumerate.py on all TMs in the 2728 holdout list with the above max-loops and max-block-size parameters using --block-mult=1 ,--block-mult=2 , and --block-mult=3. For context, during the Stage 2 BB(7) enumeration, where speed was more important due to the tens of millions of known holdouts, parameters of --max-loops=100_000 --block-mult=2 --time=30 --save-freq=100 were used.

@Iijil's MITMWFAR decider is likely too weak to be of any assistance: running the decider on 2650 BB(6) holdouts, using parameters not strong enough to solve BB(5) TMs, took prohibitively long to compute. Instead, a new FAR method by mxdys was able to decide 113 of the 1534 holdouts (code on GitHub) upon initial application.