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:

$$S(6)>\mathrm{\Sigma }(6)>2\uparrow \uparrow \uparrow 5$$

History

• In 1964, Green established Σ(5) ≥ 35.
• In 1972, Lynn established S(6) ≥ 522 and Σ(6) ≥ 42.
• In 1983, Brady established S(6) ≥ 13,488 and Σ(6) ≥ 117.
• In 1982, Schult established S(6) ≥ 4,208,824 and Σ(6) ≥ 2,075, although these bounds were only published later.
• In 1990, Heiner Marxen and Jürgen Buntrock established first S(6) ≥ 13,122,572,797 and Σ(6) ≥ 136,612, then S(6) ≥ 8,690,333,381,690,951 and Σ(6) ≥ 95,524,079.
• In July 2000, Heiner Marxen and Jürgen Buntrock established S(6) > 5.3 × 10⁴² and Σ(6) > 2.5 × 10²¹.
• In August 2000, Heiner Marxen and Jürgen Buntrock established first S(6) > 6.1 × 10¹¹⁹ and Σ(6) > 1.4 × 10⁶⁰, then S(6) > 6.1 × 10⁹²⁵ and Σ(6) > 6.4 × 10⁴⁶².
• In February 2001, Heiner Marxen and Jürgen Buntrock established S(6) > 3.0 × 10¹⁷³⁰ and Σ(6) > 1.2 × 10⁸⁶⁵.
• In November 2007, Terry and Shawn Ligocki established S(6) > 8.9 × 10¹⁷⁶² and Σ(6) > 2.5 × 10⁸⁸¹.
• In December 2007, Terry and Shawn Ligocki established S(6) > 2.5 × 10²⁸⁷⁹ and Σ(6) > 4.6 × 10¹⁴³⁹.
• In May 2010, Pavel Kropitz established S(6) > 3.8 × 10²¹¹³² and Σ(6) > 3.1 × 10¹⁰⁵⁶⁶.
• In June 2010, Pavel Kropitz established S(6) > 7.4 × 10³⁶⁵³⁴ and Σ(6) > 3.5 × 10¹⁸²⁶⁷.

Number of BB(6) holdouts over time.

@mxdys's informal holdouts list has 1003 machines up to equivalence and 2190 machines not considering equivalence as of August 2026. Partial Rocq proof is available on Github.

Always up-to-date annotated spreadsheet, with links to Discord discussions: Spreadsheet. The informal holdout count is 1101.

All machines have been simulated out to 1e13 steps. ~150 machines remain to be simulated to 1e14, and ~230 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

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)

Top Halters

Below is a table of the machines with the 20 highest known runtimes. Their sigma scores are expressed using an extension of Knuth's up-arrow notation.

Top Known BB(6) Halters

Standard format	(approximate) Σ	Discoverer
1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE (bbch)	2 ↑↑↑ 5	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
1RB0LF_1RC1RB_1LD0RA_1RF0LE_1RZ0LC_1LA1LF (bbch)	10 ↑↑ 7.52390
1RB0LF_1RC1RB_1LD0RA_1LF0LE_1RZ0LC_1LA1LF (bbch)	10 ↑↑ 7.52390
1RB1RC_1LC1RE_1LD0LB_1RE1LC_1LE0RF_1RZ1RA (bbch)	10 ↑↑ 7.23619
1RB1RA_1LC1LE_1RE0LD_1LC0LF_1RZ0RA_0RA0LB (bbch)	10 ↑↑ 6.96745	poppuncher
1RB0RF_1LC0RA_1RZ0LD_1LE1LD_1RB1RC_0LD0RE (bbch)	10 ↑↑ 5.77573	poppuncher
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 ↑↑ 5.12468
1RB0RF_1LC1LB_0RE0LD_0LC0LB_0RA1RE_0RD1RZ (bbch)	10 ↑↑ 5.03230
1RB1LA_1LC0RF_1LD1LC_1LE0RE_0RB0LC_1RZ1RA (bbch)	10 ↑↑ 4.91072
1RB0LE_1LC1RA_1RE0LD_1LC1LF_1LA0RC_1RZ1LC (bbch)	10 ↑↑ 3.33186
1RB1RF_1LC1RE_0LD1LB_1LA0RA_0RA0RB_1RZ0RD (bbch)	10 ↑↑ 3.31128
1RB0LF_1LC0RA_1RD0LB_1LE1RC_1RZ1LA_1LA1LE (bbch)	10 ↑↑ 3.18855
1RB0RF_1LC1RB_0RD0LB_1RZ0LE_1RE0RA_1RD1RE (bbch)	10 ↑↑ 3.16005
1RB0RB_0RC0LF_0RD0RA_0LE---_1LE0LA_1LF1RA (bbch)	$10^{1400000000}$(runtime)	Racheline
1RB1RZ_0LC0LD_1LD1LC_1RE1LB_1RF1RD_0LD0RA (bbch)	$10^{646456993}$	Pavel Kropitz

The runtimes are presumed to be about ${\text{score}}^2$ 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.