BusyBeaverWiki - Recent changes [en]

Measuring large numbers

Sligocki — Thu, 05 Nov 2026 06:53:05 GMT

Fast-growing hierarchy: Tidy up fast-growing hierarchy

← Older revision		Revision as of 06:53, 5 November 2026
(One intermediate revision by the same user not shown)
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	<math>a \uparrow \uparrow b =\begin{cases}		<math>a \uparrow \uparrow b =\begin{cases}
	b + 1, & \text{if } b \leq 0,		b + 1, & \text{if } -1 < b \leq 0,
	\\ a^{a \uparrow \uparrow (b - 1)}, & \text{otherwise.}		\\ a^{a \uparrow \uparrow (b - 1)}, & \text{otherwise.}
	\end{cases}</math>		\end{cases}</math>
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	Note that when using this method, the function <math>10 \uparrow\uparrow x</math>, and other functions similar to it, are not analytic functions; however, that requirement is unnecessary when using this method to represent large numbers. Additionally, methods of obtaining analytic hyperoperators are complex, typically involving Abel functions, Riemann mappings, or Jabotinsky matrices; this method forgoes this, allowing for the ease of calculation.		Note that when using this method, the function <math>10 \uparrow\uparrow x</math>, and other functions similar to it, are not analytic functions; however, that requirement is unnecessary when using this method to represent large numbers. Additionally, methods of obtaining analytic hyperoperators are complex, typically involving Abel functions, Riemann mappings, or Jabotinsky matrices; this method forgoes this, allowing for the ease of calculation.

	In this format, the lower bound for <math>S(2,5)</math> can ~~instead~~ be written as <math>S(2,5) > 10 \uparrow\uparrow 4.8142</math>.		In this format, the lower bound for <math>S(2,5)</math> can be written more precisely as <math>S(2,5) > 10 \uparrow\uparrow 4.8142</math>.


	~~In popular mathematics<!-- This is an oxymoron, as mathematics is not popular; please find a better way to state this. -->,~~ Graham's number (<math>= g_{64}</math>) is ~~represented~~ using ~~the following recursive function involving~~ Knuth's up-arrow notation:		[[Graham's number]] (<math>g_{64}</math>) is a famously large number that is defined using Knuth's up-arrow notation:

	<math>g_n = \begin{cases}		<math>g_n = \begin{cases}
	3 \uparrow \uparrow \uparrow \uparrow 3, & \text{if }n = 1		3 \uparrow \uparrow \uparrow \uparrow 3, & \text{if }n = 1
	\\ 3 \uparrow^{g_{n - 1}} 3, & \text{~~otherwise.~~}		\\ 3 \uparrow^{g_{n - 1}} 3, & \text{if } n > 1
	\end{cases}</math>		\end{cases}</math>

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	== Fast-growing hierarchy ==		== Fast-growing hierarchy ==
	The fast-growing hierarchy uses ~~the fundamental sequences of ordinals to define diagonalizations~~ across previous functions in the ~~hierarchy~~. Functions in the fast-growing hierarchy are written in the form <math>f_\alpha(n)</math>, and are calculated according to three rules:		The [[fast-growing hierarchy]] uses function iteration and diagonalization across previous functions to extend far beyond the Ackermann function. Functions in the fast-growing hierarchy are written in the form <math>f_\alpha(n)</math>, and are calculated according to three rules:

	<math>f_0(n) = n + 1</math>		<math>f_0(n) = n + 1</math>

User:GregoryMorse

GregoryMorse — Thu, 05 Nov 2026 06:44:57 GMT

User account GregoryMorse was created

Measuring large numbers

Sligocki — Thu, 05 Nov 2026 06:42:25 GMT

Scientific notation: Tidy up sci notation section

← Older revision		Revision as of 06:42, 5 November 2026
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	== Scientific notation ==		== Scientific notation ==
	Scientific notation is one of the most common ways of representing ~~exceedingly~~ large numbers; numbers written in standard scientific notation symbolically consist of a multiplicand (taking a value between 1 and the base, though is sometimes omitted), a base (commonly 10) and an ~~index~~ (usually an integer, though sometimes takes non-integer values if the multiplicand is omitted).		Scientific notation is one of the most common ways of representing large numbers; numbers written in standard scientific notation symbolically consist of a multiplicand (taking a value between 1 and the base, though is sometimes omitted), a base (commonly 10) and an exponent (usually an integer, though sometimes takes non-integer values if the multiplicand is omitted).

	For example, <math>8.68 \cdot 10^6</math> represents the number 8,680,000, whereas <math>6.022 \cdot 10^{23}</math> represents the number 602,200,000,000,000,000,000,000.		For example, <math>8.68 \cdot 10^6</math> represents the number 8,680,000, whereas <math>6.022 \cdot 10^{23}</math> represents the number 602,200,000,000,000,000,000,000. These numbers are often approximations (to some number of digits) or lower/upper bounds on the precise integer.

	~~In cases where the index is not an integer, the representation usually represents an irrational number, though sometimes it can represent a rational number, or even an~~ integer.		Non-integer exponents are also often used to approximate. For example, <math>10^{7.67}</math> representing a lower bound for the number 46,773,515.

	For example, <math>10^{7.67}</math> ~~represents~~ the number 46,773,~~514.1287..., an irrational number, whereas <math>16^{6.75}</math> represents the number 134,217,728, an integer~~.


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	When representing ~~certain exceedingly large~~ numbers using scientific notation, the ~~index~~ itself may also be represented using scientific notation.		When representing even larger numbers using scientific notation, the exponent itself may also be represented using scientific notation. For example:

	<math>S(2,5) > 10^{10^{10^{3,314,360}}}</math>		<math>S(2,5) > 10^{10^{10^{3,314,360}}}</math>

Champions

Jacobzheng — Thu, 05 Nov 2026 06:28:28 GMT

← Older revision		Revision as of 06:28, 5 November 2026
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	\|-		\|-
	\|BB(14)		\|BB(14)
	\|<math>> f_{\omega+~~2}^9(f_{\omega+1~~}^2(3))</math>		\|<math>> f_{\omega+3}(17)</math>
	\|<span style="word-break:break-all">{{TM\|~~1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LK1RF_1RM0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_0LN0LL_0LH1LK_1RZ0LH_0RJ0LI~~\|halt}}</span>		\|<span style="word-break:break-all">{{TM\|1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LK1RF_1RM0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_0LL0LL_0LN1LK_1RZ0LH_0LG1LJ\|halt}}</span>
	\|Jeff Ketchersid, Claude and Gemini in 10/2026 (claimed)		\|Jeff Ketchersid, Claude and Gemini in 11/2026 (claimed)
	\|[https://github.com/jeff87654/busy-beaver/tree/main/BB14 Claimed analysis by Claude] never verified.		\|[https://github.com/jeff87654/busy-beaver/tree/main/BB14 Claimed analysis by Claude] never verified.
	\|-		\|-
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	\|-		\|-
	\|BB(16)		\|BB(16)
	\|~~<math>> f_{\omega+2}^{10}(9)</math>~~		\|
	\|~~<span style="word-break:break-all">{{TM\|1LB1LA_1RC0LA_1RD1RC_1RE1RK_1RF0LJ_1LG1RI_1RM0LM_1LG0RC_0RN0LE_1LA1LE_0LL1LB_0RM1LL_1LF1RH_0RO0LO_1LP1RM_1RZ1LI\|halt}}</span>~~		\|
	\|~~Jeff Ketchersid, Claude and Gemini in 10/2026 (claimed)~~		\|
	\|~~[https://github.com/jeff87654/busy-beaver/tree/main/BB16 Claimed analysis by Claude] never verified.~~		\|
	\|-		\|-
	\|BB(17)		\|BB(17)
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	\|-		\|-
	\|BB(18)		\|BB(18)
	\|<math>> f_{\omega+3}^4(9)</math>		\|<math>> f_{\omega+4}(228)</math>
	\|<span style="word-break:break-all">{{TM\|~~1LB1LA_1RC0LA_1RD1RC_1RE1RK_1RF0LJ_1LG1RI_1RM0LM_1LG0RC_0RN0LE_1LA1LE_0LL1LB_0RM1LL_1LF1RH_1RQ0LO_1LP1LN_0RB1LI_1RR0LO_0RO1RZ~~\|halt}}</span>		\|<span style="word-break:break-all">{{TM\|1LB1LA_1RC0LA_1RD1RC_1RE1RK_1RF0LJ_1LG1RI_1RM0LM_1LG0RC_0RN0LE_1LA1LE_0LL1LB_0RM1LL_1LF1RH_1RQ0LO_1LP1LR_0RB1LI_1RZ1RR_1RN0LO\|halt}}</span>
	\|Jeff Ketchersid, Claude and Gemini in 11/2026 (claimed)		\|Jeff Ketchersid, Claude and Gemini in 11/2026 (claimed)
	\|[https://github.com/jeff87654/busy-beaver/tree/main/BB18 Claimed analysis by Claude] never verified.		\|[https://github.com/jeff87654/busy-beaver/tree/main/BB18 Claimed analysis by Claude] never verified.

User:Jacobzheng/test

Jacobzheng — Thu, 05 Nov 2026 06:19:39 GMT

Undo revision 10262 by Jacobzheng (talk)

← Older revision		Revision as of 06:19, 5 November 2026
(One intermediate revision by the same user not shown)
(No difference)

Main Page

Jacobzheng — Thu, 05 Nov 2026 06:15:47 GMT

← Older revision		Revision as of 06:15, 5 November 2026
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	{\| class="wikitable"		{\| class="wikitable"
	\|+ Small busy beaver values (up to 7 states and symbols)<ref>P. Michel, "[https://bbchallenge.org/~pascal.michel/ha.html Historical survey of Busy Beavers]".</ref>		\|+ Small busy beaver values (up to 7 states and symbols)<ref>P. Michel, "[https://bbchallenge.org/~pascal.michel/ha.html Historical survey of Busy Beavers]".</ref>
	!~~state→ symbol↓~~ !!2 !!3 !!4 !!5 !!6 !!7		! !!2-state !!3-state !!4-state !!5-state !!6-state !!7-state
	\|-		\|-
	!2		!2-symbol
	\| [[BB(2)]] = 6		\| [[BB(2)]] = 6
	\| [[BB(3)]] = 21		\| [[BB(3)]] = 21
	\| [[BB(4)]] = 107		\| [[BB(4)]] = 107
	\| [[BB(5)]] = 47,176,870		\| [[BB(5)]] = 47,176,870
	\| style="background: orange;" title="Cryptids: Antihydra and some other cryptids" \| [[BB(6)]] > <math>10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>		\| style="background: orange;" title="Cryptids: Antihydra and some other cryptids" \| [[BB(6)]] > <math>10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
	\| style="background: #ddc290;" \| [[BB(7)]] > <math>3 \uparrow^{20} 3</math>		\| style="background: #ddc290;" \| [[BB(7)]] > <math>3 \uparrow^{20} 3</math>
	\|-		\|-
	!3		!3-symbol
	\| [[BB(2,3)]] = 38		\| [[BB(2,3)]] = 38
	\| style="background: orange;" title="Cryptid: Bigfoot" \| [[BB(3,3)]] <math>\ge</math> 119,112,334,170,342,541		\| style="background: orange;" title="Cryptid: Bigfoot" \| [[BB(3,3)]] <math>\ge</math> 119,112,334,170,342,541
	\| style="background: #ddc290;" \| [[BB(4,3)]] > <math>10 \uparrow^{4} 4</math>		\| style="background: #ddc290;" \| [[BB(4,3)]] > <math>10 \uparrow^{4} 4</math>
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	\| style="background: #ddc290;" \|		\| style="background: #ddc290;" \|
	\|-		\|-
	!4		!4-symbol
	\| [[BB(2,4)]] = 3,932,964		\| [[BB(2,4)]] = 3,932,964
	\| style="background: #ddc290;" \| [[BB(3,4)]] > <math>2 \uparrow^{15} 5</math>		\| style="background: #ddc290;" \| [[BB(3,4)]] > <math>2 \uparrow^{15} 5</math>
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	\| style="background: yellow;" \|		\| style="background: yellow;" \|
	\|-		\|-
	!5		!5-symbol
	\| style="background: orange;" title="Cryptids: Hydra and Bonus Cryptid" \| [[BB(2,5)]] > <math>10^{10^{10^{3\,314\,360}}}</math>		\| style="background: orange;" title="Cryptids: Hydra and Bonus Cryptid" \| [[BB(2,5)]] > <math>10^{10^{10^{3\,314\,360}}}</math>
	\| style="background: #ddc290;" \| [[BB(3,5)]] > <math> f_\omega(2 \uparrow^{15} 5)</math>		\| style="background: #ddc290;" \| [[BB(3,5)]] > <math> f_\omega(2 \uparrow^{15} 5)</math>
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	\| style="background: yellow;" \|BB(7,5) > <math>f_{\omega+3}^2(10)</math>		\| style="background: yellow;" \|BB(7,5) > <math>f_{\omega+3}^2(10)</math>
	\|-		\|-
	!6		!6-symbol
	\| style="background: #ddc290;" \| [[BB(2,6)]] > <math>10 \uparrow \uparrow 10 \uparrow\uparrow 10^{10^{115}}</math>		\| style="background: #ddc290;" \| [[BB(2,6)]] > <math>10 \uparrow \uparrow 10 \uparrow\uparrow 10^{10^{115}}</math>
	\| style="background: #ddc290;" \| BB(3,6) > <math> f_\omega^3(15)</math>		\| style="background: #ddc290;" \| BB(3,6) > <math> f_\omega^3(15)</math>
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	\| style="background: yellow;" \| BB(7,6) > <math> f_{\omega^2}(63)</math>		\| style="background: yellow;" \| BB(7,6) > <math> f_{\omega^2}(63)</math>
	\|-		\|-
	!7		!7-symbol
	\| style="background: #ddc290;" \| [[BB(2,7)]] > <math>10 \uparrow \uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 155</math>		\| style="background: #ddc290;" \| [[BB(2,7)]] > <math>10 \uparrow \uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 155</math>
	\| style="background: yellow;" \| BB(3,7) > <math> f_{\omega+1}(117)</math>		\| style="background: yellow;" \| BB(3,7) > <math> f_{\omega+1}(117)</math>

Champions

Loader3229 — Thu, 05 Nov 2026 06:08:44 GMT

4-Symbol TMs

← Older revision		Revision as of 06:08, 5 November 2026
(One intermediate revision by the same user not shown)
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	\|-		\|-
	\|BB(11,4)		\|BB(11,4)
	\|<math>> f_{\omega \times 2}(2 \uparrow^9 6)</math>		\|<math>> f_{\omega \times 2}(2 \uparrow^{11} 6)</math>
	\|<span style="word-break:break-all">{{TM\|~~1RK3LB1RZ2RA_0LG---1LD2RA_3RB1LB3LC2RC_3RE0LI3LC2RC_1LJ3RE3LH2LF_3RF2RF2LE1LF_3LC3LH2LJ2RG_0LE3RH1LI2RG_3RH1LH3LI2RI_2LB---3LJ3RK_2LC0LD2RA3RK~~\|halt}}</span>		\|<span style="word-break:break-all">{{TM\|1RK3LB1RZ2RA_0LG1LE1LD2RA_3RB1LB3LC2RC_3RE0LI3LC2RC_1RJ3RE3LH2LF_3RF2RF2LE1LF_3LC3LH2LJ2RG_0LE3RH1LI2RG_3RH1LH3LI2RI_0LB0LJ3LJ3RK_2LC0LD2RA3RK\|halt}}</span>
	\|[[User:Loader3229]] in 2026		\|[[User:Loader3229]] in 2026
	\|Not independently verified.		\|Not independently verified.

Main Page

JackzNkk06 — Thu, 05 Nov 2026 06:04:00 GMT

← Older revision		Revision as of 06:04, 5 November 2026
(4 intermediate revisions by the same user not shown)
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	{\| class="wikitable"		{\| class="wikitable"
	\|+ Small busy beaver values (up to 7 states and symbols)<ref>P. Michel, "[https://bbchallenge.org/~pascal.michel/ha.html Historical survey of Busy Beavers]".</ref>		\|+ Small busy beaver values (up to 7 states and symbols)<ref>P. Michel, "[https://bbchallenge.org/~pascal.michel/ha.html Historical survey of Busy Beavers]".</ref>
	! !!2~~-state~~!!3~~-state~~ !!4~~-state~~!!5~~-state~~!!6~~-state~~		!state→ symbol↓ !!2 !!3 !!4 !!5 !!6 !!7
	!7~~-state~~
	\|-		\|-
	! 2~~-symbol~~		!2
	\| [[BB(2)]] = 6		\| [[BB(2)]] = 6
	\| [[BB(3)]] = 21		\| [[BB(3)]] = 21
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	\| style="background: #ddc290;" \| [[BB(7)]] > <math>3 \uparrow^{20} 3</math>		\| style="background: #ddc290;" \| [[BB(7)]] > <math>3 \uparrow^{20} 3</math>
	\|-		\|-
	! 3~~-symbol~~		!3
	\| [[BB(2,3)]] = 38		\| [[BB(2,3)]] = 38
	\| style="background: orange;" title="Cryptid: Bigfoot" \| [[BB(3,3)]] <math>\ge</math> 119,112,334,170,342,541		\| style="background: orange;" title="Cryptid: Bigfoot" \| [[BB(3,3)]] <math>\ge</math> 119,112,334,170,342,541
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	\| style="background: #ddc290;" \|		\| style="background: #ddc290;" \|
	\|-		\|-
	! 4~~-symbol~~		!4
	\| [[BB(2,4)]] = 3,932,964		\| [[BB(2,4)]] = 3,932,964
	\| style="background: #ddc290;" \| [[BB(3,4)]] > <math>2 \uparrow^{15} 5</math>		\| style="background: #ddc290;" \| [[BB(3,4)]] > <math>2 \uparrow^{15} 5</math>
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	\| style="background: yellow;" \|		\| style="background: yellow;" \|
	\|-		\|-
	! 5~~-symbol~~		!5
	\| style="background: orange;" title="Cryptids: Hydra and Bonus Cryptid" \| [[BB(2,5)]] > <math>10^{10^{10^{3\,314\,360}}}</math>		\| style="background: orange;" title="Cryptids: Hydra and Bonus Cryptid" \| [[BB(2,5)]] > <math>10^{10^{10^{3\,314\,360}}}</math>
	\| style="background: #ddc290;" \| [[BB(3,5)]] > <math> f_\omega(2 \uparrow^{15} 5)</math>		\| style="background: #ddc290;" \| [[BB(3,5)]] > <math> f_\omega(2 \uparrow^{15} 5)</math>
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	\| style="background: yellow;" \|BB(7,5) > <math>f_{\omega+3}^2(10)</math>		\| style="background: yellow;" \|BB(7,5) > <math>f_{\omega+3}^2(10)</math>
	\|-		\|-
	! 6~~-symbol~~		!6
	\| style="background: #ddc290;" \| [[BB(2,6)]] > <math>10 \uparrow \uparrow 10 \uparrow\uparrow 10^{10^{115}}</math>		\| style="background: #ddc290;" \| [[BB(2,6)]] > <math>10 \uparrow \uparrow 10 \uparrow\uparrow 10^{10^{115}}</math>
	\| style="background: #ddc290;" \| BB(3,6) > <math> f_\omega^3(15)</math>		\| style="background: #ddc290;" \| BB(3,6) > <math> f_\omega^3(15)</math>
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	\| style="background: yellow;" \| BB(7,6) > <math> f_{\omega^2}(63)</math>		\| style="background: yellow;" \| BB(7,6) > <math> f_{\omega^2}(63)</math>
	\|-		\|-
	! 7~~-symbol~~		!7
	\| style="background: #ddc290;" \| [[BB(2,7)]] > <math>10 \uparrow \uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 155</math>		\| style="background: #ddc290;" \| [[BB(2,7)]] > <math>10 \uparrow \uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 155</math>
	\| style="background: yellow;" \| BB(3,7) > <math> f_{\omega+1}(117)</math>		\| style="background: yellow;" \| BB(3,7) > <math> f_{\omega+1}(117)</math>

User:JackzNkk06

JackzNkk06 — Thu, 05 Nov 2026 05:56:10 GMT

User account JackzNkk06 was created

Champions

Loader3229 — Thu, 05 Nov 2026 05:22:22 GMT

4-Symbol TMs

← Older revision		Revision as of 05:22, 5 November 2026
(One intermediate revision by the same user not shown)
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	\|-		\|-
	\|BB(11,4)		\|BB(11,4)
	\|<math>> f_{\omega \times 2}(98)</math>		\|<math>> f_{\omega \times 2}(2 \uparrow^9 6)</math>
	\|<span style="word-break:break-all">{{TM\|~~1RG3LB1RZ2RA_0LG~~---~~1LD2RA_3RB1LB3LC2RC_3RE0LA3LC2RC_1RI3RE3LH2LF_3RF2RF2LE1LF_2LC3LH2LJ2RG_0RG3RH1LI2RG_3RH1LH3LI2RI_~~---~~---3LJ3RK_------2RA3RK~~\|halt}}</span>		\|<span style="word-break:break-all">{{TM\|1RK3LB1RZ2RA_0LG---1LD2RA_3RB1LB3LC2RC_3RE0LI3LC2RC_1LJ3RE3LH2LF_3RF2RF2LE1LF_3LC3LH2LJ2RG_0LE3RH1LI2RG_3RH1LH3LI2RI_2LB---3LJ3RK_2LC0LD2RA3RK\|halt}}</span>
	\|[[User:Loader3229]] in 2026		\|[[User:Loader3229]] in 2026
	\|Not independently verified.		\|Not independently verified.

Large champions transitions

Jacobzheng — Thu, 05 Nov 2026 04:33:53 GMT

Created page with "{{Stub}} TODO"

New page

{{Stub}}
TODO

TMBR: November 2026

Jacobzheng — Thu, 05 Nov 2026 04:31:07 GMT

← Older revision		Revision as of 04:31, 5 November 2026
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	\|Unknown - pseudoquasihalts if the conjecture is false. [https://github.com/calr3/tinycryptids/tree/main/lean/TinyCryptids/Pillai Proof.]		\|Unknown - pseudoquasihalts if the conjecture is false. [https://github.com/calr3/tinycryptids/tree/main/lean/TinyCryptids/Pillai Proof.]
	\|}		\|}
	[[Category:This Month in Beaver Research]]		[[Category:This Month in Beaver Research\|2026-11]]

Main Page

Jacobzheng — Thu, 05 Nov 2026 04:28:15 GMT

Not yet released

← Older revision		Revision as of 04:28, 5 November 2026
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	[[:Category:This Month in Beaver Research\|This Month in Beaver Research]] (TMBR, pronounced "timber") is a monthly summary of Busy Beaver research progress. Here are the three most recent released entries:		[[:Category:This Month in Beaver Research\|This Month in Beaver Research]] (TMBR, pronounced "timber") is a monthly summary of Busy Beaver research progress. Here are the three most recent released entries:

	* [[TMBR: ~~November~~ 2026]]		* [[TMBR: May 2026]]
	* [[TMBR: ~~October~~ 2026]]		* [[TMBR: April 2026]]
	* [[TMBR: ~~September~~ 2026]]		* [[TMBR: March 2026]]

	, [[TMBR: June 2026]], [[TMBR: July 2026]], [[TMBR: August 2026]], [[TMBR: September 2026]], [[TMBR: October 2026]], and [[TMBR: November 2026]] are currently works in progress.		[[TMBR: June 2026]], [[TMBR: July 2026]], [[TMBR: August 2026]], [[TMBR: September 2026]], [[TMBR: October 2026]], and [[TMBR: November 2026]] are currently works in progress.

	This Year in Beaver Research (TYBR) is a yearly summary of Busy Beaver research progress. Its first edition, [[TYBR: 2025]] is currently a work in progress.		This Year in Beaver Research (TYBR) is a yearly summary of Busy Beaver research progress. Its first edition, [[TYBR: 2025]] is currently a work in progress.

Champions

Loader3229 — Thu, 05 Nov 2026 02:47:09 GMT

Conjectured

← Older revision		Revision as of 02:47, 5 November 2026
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	\|[https://discord.com/channels/960643023006490684/1331570843829932063/1339669138510970881 racheline in 2025]		\|[https://discord.com/channels/960643023006490684/1331570843829932063/1339669138510970881 racheline in 2025]
	\|<math>q(n)</math> is derived from [https://en.wikipedia.org/wiki/Laver%20table Laver tables]. From [https://discord.com/channels/960643023006490684/960643023530762341/1525489299032641657 racheline]: "The known information about q(n) isn't enough to prove that this machine scores higher than the shown champions with fewer states, or that it scores lower than the shown champions with more states."		\|<math>q(n)</math> is derived from [https://en.wikipedia.org/wiki/Laver%20table Laver tables]. From [https://discord.com/channels/960643023006490684/960643023530762341/1525489299032641657 racheline]: "The known information about q(n) isn't enough to prove that this machine scores higher than the shown champions with fewer states, or that it scores lower than the shown champions with more states."
			If q(5) doesn't exist (this can only happen if [[Logical_independence\|ZFC+I3 is inconsistent]]), then this machine doesn't halt.
	\|}		\|}

Instruction-Limited Busy Beaver

Loader3229 — Thu, 05 Nov 2026 01:13:57 GMT

Champions

← Older revision		Revision as of 01:13, 5 November 2026
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	\|		\|
	\|-		\|-
	\| style="text-align: right;" \| 95		\| style="text-align: right;" \| 94
	\| colspan="2" style="text-align: right;" \| <math>> f_{\varepsilon_1}(3)</math>		\| colspan="2" style="text-align: right;" \| <math>> f_{\varepsilon_1}(3)</math>
	\| colspan="2" \|[https://discord.com/channels/960643023006490684/1331570843829932063/1567585091172040726]		\| colspan="2" \|[https://discord.com/channels/960643023006490684/1331570843829932063/1567585091172040726]

TMBR: November 2026

Loader3229 — Thu, 05 Nov 2026 01:10:05 GMT

Champions

← Older revision		Revision as of 01:10, 5 November 2026
(One intermediate revision by the same user not shown)
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	** On November 2, BB(20) > <math>f_{\omega + 7}^2(5)</math>.		** On November 2, BB(20) > <math>f_{\omega + 7}^2(5)</math>.
	** On November 4, BB(19) > <math>f_{\omega + 7}^2(5)</math> and BB(20) > <math>f_{\omega * 2}(f_{\omega + 2}^2(f_{10}(2))</math>.		** On November 4, BB(19) > <math>f_{\omega + 7}^2(5)</math> and BB(20) > <math>f_{\omega * 2}(f_{\omega + 2}^2(f_{10}(2))</math>.
	* On November 4, Jeff Ketchersid used Claude and Gemini to discover the bounds BB(17) > <math>f_{\omega + 3}(9)</math> ~~and~~ BB(18) > <math>f_{\omega + 3}^4(9)</math>.		* On November 4, Jeff Ketchersid used Claude and Gemini to discover the bounds BB(17) > <math>f_{\omega + 3}(9)</math>, BB(18) > <math>f_{\omega + 3}^4(9)</math> and BB(19) > <math>f_{\omega + 4}(f_{\omega + 2}(114))</math>.

	== Holdouts ==		== Holdouts ==

Busy Bricklayer Problem

Azerty — Wed, 04 Nov 2026 21:43:00 GMT

created page

New page

The Busy Bricklayer Problem is a polyomino tiling problem for which its decidability is unknown.

== Definition ==
The order of a polyomino is the minimum number of copies required to tile a rectangle (of any finite size with no gaps or overlaps, allowing for rotation and reflection) or zero if it doesn't tile any rectangles. Let the sequence a(n) be the maximum over the orders of all polyominoes of size n.

For example, a(5) = 10. Pentominoes F, N, T, U, V, and W have order 0, pentomino I has order 1, pentominoes L and P have order 2, and the Y pentomino has order 10 (with the following tiling).

== Values ==
{| class="wikitable"
|+
!a(n)
!Value
!Champion
|-
|1
|1
|
|-
|2
|1
|
|-
|3
|2
|
|-
|4
|4
|
|-
|5
|10
|
|-
|6
|92
|
|-
|7
|76
|
|-
|8
|246
|
|-
|9
|4
|
|-
|10
|
|
|}

Pebble Automaton

Sligocki — Wed, 04 Nov 2026 20:56:13 GMT

I think we can be more direct about this

← Older revision		Revision as of 20:56, 4 November 2026
Line 2:		Line 2:
	A pebble automaton is a type of [[cellular automaton]] introduced by AlephSquirrel, based on a Codeforces problem, also called Pebble Automaton.<ref>https://codeforces.com/gym/106465/problem/B</ref> Each cell contains a non-negative number of pebbles, and the number of pebbles in the cell determines how many pebbles to push to each neighbor. Though in the original problem, the behavior of a cell also depended on the cell's location, the cellular automaton removes this dependence. Investigation so far has primarily focused on 1-dimensional pebble automata, but multi-dimensional pebble automata could be considered.		A pebble automaton is a type of [[cellular automaton]] introduced by AlephSquirrel, based on a Codeforces problem, also called Pebble Automaton.<ref>https://codeforces.com/gym/106465/problem/B</ref> Each cell contains a non-negative number of pebbles, and the number of pebbles in the cell determines how many pebbles to push to each neighbor. Though in the original problem, the behavior of a cell also depended on the cell's location, the cellular automaton removes this dependence. Investigation so far has primarily focused on 1-dimensional pebble automata, but multi-dimensional pebble automata could be considered.

	~~It is thought that pebble~~ automata ~~are~~ [[Turing-complete]]~~, as~~ a ~~program has been created that converts~~ 2-register [[Register machine\|Register machines]] ~~to pebble automata, though this has not been formally proven correct~~.<ref>https://discord.com/channels/960643023006490684/1535695107448381591/1567607590970458274</ref>		Pebble automata have been informally shown to be [[Turing-complete]] via a reduction to 2-register [[Register machine\|Register machines]].<ref>https://discord.com/channels/960643023006490684/1535695107448381591/1567607590970458274</ref>

	== Definition ==		== Definition ==

Pebble Automaton

AlephSquirrel — Wed, 04 Nov 2026 20:52:11 GMT

cite some stuff

← Older revision		Revision as of 20:52, 4 November 2026
(One intermediate revision by the same user not shown)
Line 2:		Line 2:
	A pebble automaton is a type of [[cellular automaton]] introduced by AlephSquirrel, based on a Codeforces problem, also called Pebble Automaton.<ref>https://codeforces.com/gym/106465/problem/B</ref> Each cell contains a non-negative number of pebbles, and the number of pebbles in the cell determines how many pebbles to push to each neighbor. Though in the original problem, the behavior of a cell also depended on the cell's location, the cellular automaton removes this dependence. Investigation so far has primarily focused on 1-dimensional pebble automata, but multi-dimensional pebble automata could be considered.		A pebble automaton is a type of [[cellular automaton]] introduced by AlephSquirrel, based on a Codeforces problem, also called Pebble Automaton.<ref>https://codeforces.com/gym/106465/problem/B</ref> Each cell contains a non-negative number of pebbles, and the number of pebbles in the cell determines how many pebbles to push to each neighbor. Though in the original problem, the behavior of a cell also depended on the cell's location, the cellular automaton removes this dependence. Investigation so far has primarily focused on 1-dimensional pebble automata, but multi-dimensional pebble automata could be considered.

	~~Though it~~ is ~~unknown whether~~ pebble automata are [[Turing-complete]], ~~it is highly suspected~~ that ~~they are, since certain automata have been found to compute~~ [[~~Collatz-like~~]] ~~functions~~.		It is thought that pebble automata are [[Turing-complete]], as a program has been created that converts 2-register [[Register machine\|Register machines]] to pebble automata, though this has not been formally proven correct.<ref>https://discord.com/channels/960643023006490684/1535695107448381591/1567607590970458274</ref>

	== Definition ==		== Definition ==
Line 309:		Line 309:
	== Other notable machines ==		== Other notable machines ==
	=== Golems ===		=== Golems ===
	The "golems" are a group of 22 holdouts in peBBle(11). They appear to be modeled by complex systems of equations, making them potential Cryptids. Discord user Sheep noted that they can be categorized into 7 unique "paths", though the details may differ on how they behave if they encounter an improbable position.		The "golems" are a group of 22 holdouts in peBBle(11). They appear to be modeled by complex systems of equations, making them potential Cryptids. Discord user Sheep noted that they can be categorized into 7 unique "paths"<ref>https://discord.com/channels/960643023006490684/1535695107448381591/1566012271719616573</ref>, though the details may differ on how they behave if they encounter an improbable position.<ref>https://discord.com/channels/960643023006490684/1535695107448381591/1566013254764073081</ref>
	{\| class="wikitable"		{\| class="wikitable"
	\|+ Golems text		\|+ Golems text

User:Polygon/Collection of BB Champions

Polygon — Wed, 04 Nov 2026 20:38:55 GMT

Maximum Shifts Function (S(n,m), also commonly called BB(n,m)): more precise BB(48) bound

← Older revision		Revision as of 20:38, 4 November 2026
Line 134:		Line 134:
	\|-		\|-
	\|BB(48)		\|BB(48)
	\|<math>> f_{\varepsilon_1}(3)</math>		\|<math>> f_{\varepsilon_0}(f_{\omega+1}(f_{\omega^{\varepsilon_0+1}}(f_{\omega^2}(f_{\omega^{\omega^{\varepsilon_0+1}}}(f_{\omega^3}(f_{\varepsilon_1}(3))))))) > f_{\varepsilon_1}(3)</math>
	\|[https://discord.com/channels/960643023006490684/1331570843829932063/1567585091172040726]		\|[https://discord.com/channels/960643023006490684/1331570843829932063/1567585091172040726]
	\|<ref>https://discord.com/channels/960643023006490684/1331570843829932063/1567585091172040726</ref>		\|<ref>https://discord.com/channels/960643023006490684/1331570843829932063/1567585091172040726</ref>

Catch-Em-Turing

Polygon — Wed, 04 Nov 2026 20:07:03 GMT

Added references section

← Older revision		Revision as of 20:07, 4 November 2026
Line 186:		Line 186:

	The finite search space makes exhaustive enumeration theoretically possible for every fixed pair (n,k), although the number of machines grows extremely rapidly.		The finite search space makes exhaustive enumeration theoretically possible for every fixed pair (n,k), although the number of machines grows extremely rapidly.

			== References ==

	[[Category:Functions]]		[[Category:Functions]]

Champions

Azerty — Wed, 04 Nov 2026 19:34:06 GMT

fixed typo

← Older revision		Revision as of 19:34, 4 November 2026
Line 70:		Line 70:
	\|<math>> f_{\omega+1}(2 \uparrow^{36} 3) > g_{64}</math>		\|<math>> f_{\omega+1}(2 \uparrow^{36} 3) > g_{64}</math>
	\|<span style="word-break:break-all">{{TM\|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LI0LC_0RG1LD_1RH1LG_1LC0RG_0LJ0LI_1RZ0LC\|halt}}</span>		\|<span style="word-break:break-all">{{TM\|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LI0LC_0RG1LD_1RH1LG_1LC0RG_0LJ0LI_1RZ0LC\|halt}}</span>
	\|[https://discord.com/channels/960643023006490684/1331570843829932063/1557806386073702622 Jeff Ketchersid and Claude in 10/2026 (~~clamied~~)]		\|[https://discord.com/channels/960643023006490684/1331570843829932063/1557806386073702622 Jeff Ketchersid and Claude in 10/2026 (claimed)]
	\|[https://github.com/jeff87654/busy-beaver/tree/main/BB10 Claimed analysis by Claude] never verified.		\|[https://github.com/jeff87654/busy-beaver/tree/main/BB10 Claimed analysis by Claude] never verified.
	\|-		\|-

Catch-Em-Turing

NathFR76 — Wed, 04 Nov 2026 19:23:40 GMT

i add some CET(n) champion and i corrected some text

← Older revision		Revision as of 19:23, 4 November 2026
Line 2:		Line 2:
	https://discord.com/channels/960643023006490684/1377330731222564904</ref> on bbwiki discord.		https://discord.com/channels/960643023006490684/1377330731222564904</ref> on bbwiki discord.

	It is a ~~different angle~~ of Busy ~~beaver because~~ in ~~the~~ tape ~~you have not one like Busy beaver but two agent~~.		Catch-Em-Turing is a variant of the Busy Beaver problem in which two independent agents operate simultaneously on a shared tape, rather than a single Turing machine head.

	== Definition ==		== Definition ==
Line 84:		Line 84:
	Exhaustive Research		Exhaustive Research
	\|}		\|}
			'''Notes:'''

			* Exact values are indicated explicitly.
			* A value preceded by ≥ represents a proven lower bound, not necessarily the exact value of the corresponding function.
			* Each champion is a machine that has been verified to reach the listed number of steps before satisfying its halting condition.
			* The discoverer column identifies the person or research effort responsible for finding the machine.

	== Extension ==		== Extension ==
Line 89:		Line 95:
	A extension of CET(n) by [[User:NathFR76\|NathFR76]] for n agent is called Multi Catch-Em-Turing, or MCET(n).		A extension of CET(n) by [[User:NathFR76\|NathFR76]] for n agent is called Multi Catch-Em-Turing, or MCET(n).

	~~Agent~~ position ~~in configuration~~ is:		For each integer n ≥ 1, an MCET(n) machine consists of:
	* Agent 1 ~~starts~~ at position 0.
	* Agent 2 ~~starts~~ at position 2.		* Exactly n agents.
	* Agent 3 ~~starts~~ at position 4.		* Each agent has exactly n states.
	* ~~Agent 4 starts at position 6~~.		* Each agent has its own transition table.
	* ...		* All agents operate on the same infinite tape.
	* Agent i starts at position 2(i-1).		* All agents position is for i index the position is 2(i-1) (so: Agent 1 is at position 0, Agent 2 is at position 2, Agent 3 is at position 4, ...)
			* The tape alphabet is binary, containing the symbols 0 and 1.
			* All agents begin in their initial state, state 0.

			At each step:

	If multiple agents attempt to write to the same cell during the same step, the agent with the lowest index has priority.		* Every agent reads the symbol in its current tape cell from the tape configuration at the beginning of that step.
			* Each agent independently determines its transition.
			* All writes are applied simultaneously. If multiple agents attempt to write to the same cell during the same step, the agent with the lowest index has priority.
			* All agents then move one cell left or right and update their states simultaneously.

	~~And the halt condition is~~ when all n agents occupy the same ~~position~~ at the same time. ~~So for MCET(3), all 3 agents must collide on the same cell~~.		An MCET(n) machine halts only when all n agents occupy the same tape cell at the same time. There is no individual HALT state.

	== MCET Champions ==		== MCET Champions ==
Line 117:		Line 130:
	Like CET(2)		Like CET(2)
	\|-		\|-
	\| MCET(3) \|\| ≥ 12,~~733~~ \|\| Agent 1: ~~1RC0RB_1RC1LB_1LA0LC~~ Agent 2: ~~0RA0RC_0LC1LA_1LB0RB~~ Agent 3: ~~1LA0RC_1LB1LA_0LA1RB~~		\| MCET(3) \|\| ≥ 4,239,504 \|\| Agent 1 : 1LC1LA_1RA1RB_0RB0RA
	\|\| ~~[[User:NathFR76\|NathFR76]]~~		Agent 2 : 1LC1LB_1RB0RA_0LC0RA
			Agent 3 : 0LA1RB_0RC1RB_0RA1RC
			\|\| roromachine14
	Exhaustive Research		Exhaustive Research
	\|}		\|}
Line 135:		Line 150:
	Exhaustive Research		Exhaustive Research
	\|}		\|}
			'''Notes:'''
			* Exact values are indicated explicitly.
			* A value preceded by ≥ represents a proven lower bound, not necessarily the exact value of the corresponding function.
			* Each champion is a machine that has been verified to reach the listed number of steps before satisfying its halting condition.
			* The discoverer column identifies the person or research effort responsible for finding the machine.

			== Computational complexity and finite search space ==
			For fixed values of n and k, the number of possible CET and MCET transition tables is finite.

			Each transition is determined by three choices:

			* The symbol to write: k possibilities.
			* The movement direction: 2 possibilities (Left or Right).
			* The next state: n possibilities.

			Therefore, each state-symbol pair has exactly 2kn possible transitions.

			Each agent has n states and k possible input symbols, resulting in nk transition entries per agent.

			=== CET(n,k) ===
			A CET machine contains exactly two agents, giving a total of 2nk transition entries.

			Therefore, the total number of possible transition tables is:

			<math>\mathrm{CET}(n,k) = (2kn)^{2nk}</math>

			=== MCET(n,k) ===
			An MCET machine contains exactly n agents, giving a total of n^2k transition entries.

			Therefore, the total number of possible transition tables is:

			<math>\mathrm{MCET}(n,k) = (2kn)^{n^2k}</math>

			These counts assume labelled agents, deterministic transition tables, and no individual halting state. Machines are counted as distinct whenever at least one transition entry differs.

			The finite search space makes exhaustive enumeration theoretically possible for every fixed pair (n,k), although the number of machines grows extremely rapidly.

	[[Category:Functions]]		[[Category:Functions]]

Instruction-Limited Busy Beaver

Polygon — Wed, 04 Nov 2026 18:06:07 GMT

Champions: BBi(101) --> BBi(95)

← Older revision		Revision as of 18:06, 4 November 2026
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	\|		\|
	\|-		\|-
	\| style="text-align: right;" \| ~~101~~		\| style="text-align: right;" \| 95
	\| colspan="2" style="text-align: right;" \| <math>> f_{\~~varepsilon_0 + 1~~}(8)</math>		\| colspan="2" style="text-align: right;" \| <math>> f_{\varepsilon_1}(3)</math>
	\| colspan="2" \|[~~[Talk~~:~~Champions\|See talk page]~~]		\| colspan="2" \|[https://discord.com/channels/960643023006490684/1331570843829932063/1567585091172040726]
	\|[https://discord.com/channels/960643023006490684/~~1026577255754903572~~/~~1276881449685094495 Racheline~~ in ~~2024~~]		\|[https://discord.com/channels/960643023006490684/1331570843829932063/1567585091172040726 roromachine14 and Gemini in 2026]
	\|Current BB(51) champion		\|Current BB(48) champion
	\|		\|
	\|-		\|-

User:Polygon/Collection of BB Champions

Polygon — Wed, 04 Nov 2026 18:04:17 GMT

Maximum Shifts Function (S(n,m), also commonly called BB(n,m)): BB(51) -> BB(48)

← Older revision		Revision as of 18:04, 4 November 2026
(One intermediate revision by the same user not shown)
Line 133:		Line 133:
	\|<ref name="KetchersidBB33wtet6">Jeff Ketchersid, "[https://github.com/jeff87654/busy-beaver/blob/main/BB33/hydra33.md BB33: a 33-state hydra machine above f_{ω^ω^ω^ω^ω^ω}(3)]". Github file, October 2026. Accessed 30 October 2026.</ref>		\|<ref name="KetchersidBB33wtet6">Jeff Ketchersid, "[https://github.com/jeff87654/busy-beaver/blob/main/BB33/hydra33.md BB33: a 33-state hydra machine above f_{ω^ω^ω^ω^ω^ω}(3)]". Github file, October 2026. Accessed 30 October 2026.</ref>
	\|-		\|-
	\|BB(51)		\|BB(48)
	\|<math>> f_{\~~varepsilon_0 + 1~~}(8)</math>		\|<math>> f_{\varepsilon_1}(3)</math>
	\|[~~[Talk~~:~~Champions\|See talk page]~~]		\|[https://discord.com/channels/960643023006490684/1331570843829932063/1567585091172040726]
	\|<ref ~~name="RachelineBB51eps0"~~>https://discord.com/channels/960643023006490684/~~1026577255754903572~~/~~1276881449685094495~~</ref>		\|<ref>https://discord.com/channels/960643023006490684/1331570843829932063/1567585091172040726</ref>
	\|-		\|-
	\|BB(150)		\|BB(150)
Line 1,442:		Line 1,442:
	\|		\|
	\|-		\|-
	\|BBi(~~101~~)		\|BBi(95)
	\|<math>> f_{\~~varepsilon_0 + 1~~}(8)</math>		\|<math>> f_{\varepsilon_1}(3)</math>
	\|[~~[Talk~~:~~Champions\|See talk page]~~]		\|[https://discord.com/channels/960643023006490684/1331570843829932063/1567585091172040726]
	\|BB(51)		\|BB(48)
	\|<ref ~~name="RachelineBB51eps0"~~ />		\|<ref>https://discord.com/channels/960643023006490684/1331570843829932063/1567585091172040726</ref>
	\|-		\|-
	\|BBi(237)		\|BBi(237)

Champions

Polygon — Wed, 04 Nov 2026 18:01:12 GMT

Updated BB(48)

← Older revision		Revision as of 18:01, 4 November 2026
Line 159:		Line 159:
	\|[https://github.com/jeff87654/busy-beaver/tree/main/BB33 Claimed analysis by Claude] never verified.		\|[https://github.com/jeff87654/busy-beaver/tree/main/BB33 Claimed analysis by Claude] never verified.
	\|-		\|-
	\|BB(51)		\|BB(48)
	\|<math>> f_{\~~varepsilon_0 + 1~~}(8)</math>		\|<math>> f_{\varepsilon_1}(3)</math>
	\|[~~[Talk~~:~~Champions\|See talk page]~~]		\|[https://discord.com/channels/960643023006490684/1331570843829932063/1567585091172040726]
	\|[https://discord.com/channels/960643023006490684/~~1026577255754903572~~/~~1276881449685094495 Racheline~~ in ~~2024~~]		\|[https://discord.com/channels/960643023006490684/1331570843829932063/1567585091172040726 roromachine14 and Gemini in 2026]
	\|Not independently verified.		\|Not independently verified.
	\|-		\|-

User:Polygon/Collection of BB Champions

Polygon — Wed, 04 Nov 2026 16:07:32 GMT

Blanking Busy Beaver (BLB(n,m)): referenced BLB(5)

← Older revision		Revision as of 16:07, 4 November 2026
Line 889:		Line 889:
	\|> 10<sup>14,006</sup>		\|> 10<sup>14,006</sup>
	\|{{TM\|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA\|non_halt}}		\|{{TM\|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA\|non_halt}}
	\|		\|<ref>Nick Drozd, "[https://nickdrozd.github.io/2026/10/20/blanking-beavers-results-and-open-problems.html Blanking Beavers: Results and Open Problems]". Blog post, October 2026. Accessed 4 November 2026.</ref>
	\|-		\|-
	\|BLB(6)		\|BLB(6)

Main Page

Loader3229 — Wed, 04 Nov 2026 15:48:32 GMT

← Older revision		Revision as of 15:48, 4 November 2026
Line 9:		Line 9:

	{\| class="wikitable"		{\| class="wikitable"
	\|+ Small busy beaver values<ref>P. Michel, "[https://bbchallenge.org/~pascal.michel/ha.html Historical survey of Busy Beavers]".</ref>		\|+ Small busy beaver values (up to 7 states and symbols)<ref>P. Michel, "[https://bbchallenge.org/~pascal.michel/ha.html Historical survey of Busy Beavers]".</ref>
	! !!2-state!!3-state !!4-state!!5-state!!6-state		! !!2-state!!3-state !!4-state!!5-state!!6-state
	!7-state		!7-state
Line 52:		Line 52:
	\| style="background: yellow;" \| BB(6,6) > <math> f_{\omega \times 2}(89)</math>		\| style="background: yellow;" \| BB(6,6) > <math> f_{\omega \times 2}(89)</math>
	\| style="background: yellow;" \| BB(7,6) > <math> f_{\omega^2}(63)</math>		\| style="background: yellow;" \| BB(7,6) > <math> f_{\omega^2}(63)</math>
			\|-
			! 7-symbol
			\| style="background: #ddc290;" \| [[BB(2,7)]] > <math>10 \uparrow \uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 155</math>
			\| style="background: yellow;" \| BB(3,7) > <math> f_{\omega+1}(117)</math>
			\| style="background: yellow;" \| BB(4,7) > <math> f_{\omega+2}(124)</math>
			\| style="background: yellow;" \|
			\| style="background: yellow;" \|
			\| style="background: yellow;" \|
	\|}		\|}

User:Polygon/Collection of BB Champions

Polygon — Wed, 04 Nov 2026 15:28:36 GMT

Maximum Shifts Function (S(n,m), also commonly called BB(n,m)): updated

← Older revision		Revision as of 15:28, 4 November 2026
Line 379:		Line 379:
	\|-		\|-
	\|BB(3,7)		\|BB(3,7)
	\|<math>> f_{\omega+1}(77) > g_{64}</math>		\|<math>> f_{\omega+1}(117) > g_{64}</math>
	\|{{TM\|~~6LC3LB5RA2RA4LB2LA5LC_1RZ3RB1LC2RA3RB4LC0LA_3RB1LB3LC2RC4LC5LC---~~\|halt}}		\|{{TM\|6LC3LB6RA2RA4LB5LC5LA_1RZ3RB1LC2RA3RB4LC5RA_3RB1LB3LC2RC4LC1RA6LA\|halt}}
	\|		\|<ref>https://discord.com/channels/960643023006490684/1084047886494470185/1567559567519522909</ref>
	\|-		\|-
	\|BB(4,7)		\|BB(4,7)
	\|<math>> f_{\omega+2}(124)</math>		\|<math>> f_{\omega+2}(124)</math>
	\|{{TM\|6LD3LB5LD2RA4LB3LC5LA_1RZ3RB1LC2RA3RD---2RA_3RB1LB3LC2RC4LC3LC5LD_3RB---6LC4LD4LC4LD---\|halt}}		\|{{TM\|6LD3LB5LD2RA4LB3LC5LA_1RZ3RB1LC2RA3RD---2RA_3RB1LB3LC2RC4LC3LC5LD_3RB---6LC4LD4LC4LD---\|halt}}
	\|		\|<ref>https://discord.com/channels/960643023006490684/1084047886494470185/1567559761904537684</ref>
	\|-		\|-
	\|BB(18,7)		\|BB(18,7)

Champions

Loader3229 — Wed, 04 Nov 2026 15:19:36 GMT

7-Symbol TMs

← Older revision		Revision as of 15:19, 4 November 2026
Line 497:		Line 497:
	\|-		\|-
	\|BB(3,7)		\|BB(3,7)
	\|<math>> f_{\omega+1}(77) > g_{64}</math>		\|<math>> f_{\omega+1}(117) > g_{64}</math>
	\|{{TM\|~~6LC3LB5RA2RA4LB2LA5LC_1RZ3RB1LC2RA3RB4LC0LA_3RB1LB3LC2RC4LC5LC---~~\|halt}}		\|{{TM\|6LC3LB6RA2RA4LB5LC5LA_1RZ3RB1LC2RA3RB4LC5RA_3RB1LB3LC2RC4LC1RA6LA\|halt}}
	\|[[User:Loader3229]] in 2026		\|[[User:Loader3229]] in 2026
	\|Not independently verified.		\|Not independently verified.

Measuring large numbers

Polygon — Wed, 04 Nov 2026 14:54:44 GMT

Comparing FGH and BB: Updated with Jacobzhengs new BB(20) champion

← Older revision		Revision as of 14:54, 4 November 2026
Line 120:		Line 120:
	\| <math>\omega+2</math>\|\| 10 \|\| 12 \|\| 12<ref>Jeff Ketchersid, "[https://github.com/jeff87654/busy-beaver/blob/main/BB12/1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LK1RF_1RG0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_1RZ0LL_0LH1LK.md M12: a 12-state machine at level f_(ω+2)]". Github file, October 2026. Accessed 27 October 2026.</ref>		\| <math>\omega+2</math>\|\| 10 \|\| 12 \|\| 12<ref>Jeff Ketchersid, "[https://github.com/jeff87654/busy-beaver/blob/main/BB12/1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LK1RF_1RG0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_1RZ0LL_0LH1LK.md M12: a 12-state machine at level f_(ω+2)]". Github file, October 2026. Accessed 27 October 2026.</ref>
	\|-		\|-
	\| <math>\omega\times 2</math>\|\| 10 \|\| 14 \|\| 21		\| <math>\omega\times 2</math>\|\| 10 \|\| 14 \|\| 20
	\|-		\|-
	\| <math>\omega^2</math>\|\| 10 \|\| 21 \|\| 21		\| <math>\omega^2</math>\|\| 10 \|\| 20 \|\| 21
	\|-		\|-
	\| <math>\omega^\omega</math>\|\| 10 \|\| 29 \|\| 31		\| <math>\omega^\omega</math>\|\| 10 \|\| 29 \|\| 31

Holdouts lists

Polygon — Wed, 04 Nov 2026 14:26:41 GMT

Busy Beaver for Lambda Calculus: updated BBlambda(42) holdout count

← Older revision		Revision as of 14:26, 4 November 2026
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	\|[https://discord.com/channels/960643023006490684/1355653587824283678/1517130491512492083 June 18, 2026]		\|[https://discord.com/channels/960643023006490684/1355653587824283678/1517130491512492083 June 18, 2026]
	\|@C++DSucker		\|@C++DSucker
	\|Variable (~~139~~)		\|Variable (138)
	\|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=1544675322#gid=1544675322 Spreadsheet]		\|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=1544675322#gid=1544675322 Spreadsheet]
	\|		\|

Cryptids

Polygon — Wed, 04 Nov 2026 14:19:55 GMT

Larger Cryptids: fixed doubled name

← Older revision		Revision as of 14:19, 4 November 2026
Line 194:		Line 194:
	\|style="white-space: nowrap; text-align: right;"\|Oct 2026		\|style="white-space: nowrap; text-align: right;"\|Oct 2026
	\|[[wikipedia:Cleo (mathematician)\|Vladimir Reshetnikov]]		\|[[wikipedia:Cleo (mathematician)\|Vladimir Reshetnikov]]
	\|Machine-checked in Peano-Arithmetic by [https://github.com/elliotglazer/k215-pa ~~elliotglazer~~ Elliot Glazer].		\|Machine-checked in Peano-Arithmetic by [https://github.com/elliotglazer/k215-pa Elliot Glazer].
	\|-		\|-
	\|'''BB(152,9)'''		\|'''BB(152,9)'''

Catch-Em-Turing

NathFR76 — Wed, 04 Nov 2026 13:50:06 GMT

Add The Catch Em Turing function

New page

The '''Catch-Em-Turing''' (CET) function is a variant of the Busy Beaver function proposed by [[User:NathFR76|NathFR76]] in the beginning of November 2026 in forum-general<ref>In forum-general: Catch Em Turing (CET(n)) (Busy Beaver with 2 agent) post
https://discord.com/channels/960643023006490684/1377330731222564904</ref> on bbwiki discord.

It is a different angle of Busy beaver because in the tape you have not one like Busy beaver but two agent.

== Definition ==

The Catch-Em-Turing system is a computational model inspired by [[Turing machine]] and the [[Busy Beaver]] problem.

For each integer n ≥ 1, a CET(n) machine consists of:
* Exactly 2 agents.
* Each agent has exactly n states.
* Each agent has its own transition table.
* Both agents execute their transitions simultaneously.

For Configuration:
* The tape is infinite in both directions.
* Initially, every cell contains the symbol 0.
* The agents are numbered Agent 1 and Agent 2.
* Agent 1 starts at position 0.
* Agent 2 starts at position 2.
* Both agents begin in their respective initial states.

At each step, each agent:
* Reads the symbol in the cell it currently occupies.
* Writes a symbol on that cell.
* Moves one cell to the left or right.
* Changes its state according to its transition table.
The agents perform these operations simultaneously.

A CET machine halts only when both agents occupy the same tape cell at the same time.
There is no individual HALT state. The collision between the two agents is the sole halting condition.
If the agents never collide, the machine does not halt.

The CET function is:

CET(n) = '''max number of steps before collision'''

CET(n) measures the maximum number of steps that two n-state agents can execute before meeting on the same tape cell.

== Growth rate ==

it's seem to see that CET(n) ≥ BB(n) by multiple agent with 2 different transition table before halt by collision.

And it's seem to see that for n states et k symbols, CET(n,k) ≥ BBB(n,k) if the champion is a translated cycler, which BBB is a reference to [[Beeping Busy Beaver]].

== CET Champions ==
This is the CET(n) table. for 2 symbols

{| class="wikitable"
|-
! Domain !! Max Steps !! Champions !! Discoverer
|-
| CET(1) || 1 || Agent 1: 1RA---
Agent 2: 1LA---
|| [[User:NathFR76|NathFR76]]
Brute Force Research
|-
| CET(2) || 97 || Agent 1: 1LB0LB_1RB0LA
Agent 2: 1RB1LA_1LA1RB
|| [[User:NathFR76|NathFR76]]
Brute Force Research
|-
| CET(3) || ≥ 3,746,207 || Agent 1: 1LC1LA_1LA0LC_1RC0LB
Agent 2: 0LB0RA_1RA1LB_0RA0LB
|| roromachine14
Exhaustive Research
|-
| CET(4) || ≥ 65,579,412 || Agent 1: 0LC0RA_0LD0RA_0LB0RA_0RA0RA
Agent 2: 1RB1LD_1RC1RB_1LC1LA_0RC0RD
|| roromachine14
Exhaustive Research
|}

For 3 symbols

{| class="wikitable"
|-
! Domain !! Max Steps !! Champions !! Discoverer
|-
| CET(2,3) || ≥ 20,350 || Agent 1: 2LB1LA2LA_2RB2LA0LB
Agent 2: 1RB0LB1RB_2LA2LA1RA
|| roromachine14
Exhaustive Research
|}

== Extension ==

A extension of CET(n) by [[User:NathFR76|NathFR76]] for n agent is called Multi Catch-Em-Turing, or MCET(n).

Agent position in configuration is:
* Agent 1 starts at position 0.
* Agent 2 starts at position 2.
* Agent 3 starts at position 4.
* Agent 4 starts at position 6.
* ...
* Agent i starts at position 2(i-1).

If multiple agents attempt to write to the same cell during the same step, the agent with the lowest index has priority.

And the halt condition is when all n agents occupy the same position at the same time. So for MCET(3), all 3 agents must collide on the same cell.

== MCET Champions ==
This is the MCET(n) table.

{| class="wikitable"
|-
! Domain !! Max Steps !! Champions !! Discoverer
|-
| MCET(1) || 1 || Agent 1: 1RA---
|| [[User:NathFR76|NathFR76]]
Is like BB(1)
|-
| MCET(2) || 97 || Agent 1: 1LB0LB_1RB0LA
Agent 2: 1RB1LA_1LA1RB
|| [[User:NathFR76|NathFR76]]
Like CET(2)
|-
| MCET(3) || ≥ 12,733 || Agent 1: 1RC0RB_1RC1LB_1LA0LC Agent 2: 0RA0RC_0LC1LA_1LB0RB Agent 3: 1LA0RC_1LB1LA_0LA1RB
|| [[User:NathFR76|NathFR76]]
Exhaustive Research
|}

For 3 symbols

{| class="wikitable"
|-
! Domain !! Max Steps !! Champions !! Discoverer
|-
| MCET(3,3) || ≥ 34,399 || Agent 1: 2RB2RC1LA_0LA2RA2RB_0LA2RA0LA
Agent 2: 2LC1RC0RA_1LA0RA1LA_0LA2RB

Agent 3: 2RB2LC2RC_0RC1RC2RB_1RC2RA2RC
|| [[User:NathFR76|NathFR76]]
Exhaustive Research
|}

[[Category:Functions]]

User:Polygon/Collection of BB Champions

Polygon — Wed, 04 Nov 2026 13:39:23 GMT

Instruction-Limited Maximum Shifts Function (BBi(n)): completed BBi section

← Older revision		Revision as of 13:39, 4 November 2026
(3 intermediate revisions by the same user not shown)
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	\|<math>> f_{lim(BMS)}(10\uparrow\uparrow 15)</math>		\|<math>> f_{lim(BMS)}(10\uparrow\uparrow 15)</math>
	\|[https://morphett.info/turing/turing.html?c95a199c8e8a3dd56452f8b7e28fabbf too large to show]		\|[https://morphett.info/turing/turing.html?c95a199c8e8a3dd56452f8b7e28fabbf too large to show]
	\|<ref>https://discord.com/channels/960643023006490684/1026577255754903572/1328863966688182345</ref>		\|<ref name="PatcailBB150BMS">https://discord.com/channels/960643023006490684/1026577255754903572/1328863966688182345</ref>
	\|-		\|-
	\|BB(1015)		\|BB(1015)
Line 1,446:		Line 1,446:
	\|[[Talk:Champions\|See talk page]]		\|[[Talk:Champions\|See talk page]]
	\|BB(51)		\|BB(51)
	\|~~TODO~~		\|<ref name="RachelineBB51eps0" />
			\|-
			\|BBi(237)
			\|<math>> f_{\psi_0(\Omega_2)}(1.7\cdot10^{352})</math>
			\|
			\|BB(42,9)
			\|
			\|-
			\|BBi(299)
			\|<math>> f_{lim(BMS)}(10\uparrow\uparrow 15)</math>
			\|[https://morphett.info/turing/turing.html?c95a199c8e8a3dd56452f8b7e28fabbf too large to show]
			\|BB(150)
			\|<ref name="PatcailBB150BMS" />
			\|-
			\|BBi(2029)
			\|> D<sup>5</sup>(99) = Loader's Number
			\|
			\|BB(1015)
			\|
	\|}		\|}