Revision as of 19:53, 14 August 2025

A collection of Busy Beaver Champions including Champions for BB-Adjacent functions.

Original Busy Beaver Functions

Maximum Shifts Function (BB)

2 Symbols:


	Runtime	Champions
BB(1)	$1$	`1RZ---` (bbch)
BB(2)	$6$	`1RB1LB_1LA1RZ` (bbch) `1RB0LB_1LA1RZ` (bbch) `1RB1RZ_1LB1LA` (bbch) `1RB1RZ_0LB1LA` (bbch) `0RB1RZ_1LA1RB` (bbch)
BB(3)	$21$	`1RB1RZ_1LB0RC_1LC1LA` (bbch)
BB(4)	$107$	`1RB1LB_1LA0LC_1RZ1LD_1RD0RA` (bbch)
BB(5)	$47\,176\,870$	`1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA` (bbch)
BB(6)	$>2\uparrow \uparrow 2\uparrow \uparrow 2\uparrow \uparrow 10$	`1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE` (bbch)
BB(7)	$>2\uparrow ^{11}2\uparrow ^{11}3$	`1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF` (bbch)
BB(8)
BB(9)	$>f_{\omega }(f_{9}(2))$	`1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH` (bbch)
BB(10)	$>f_{\omega }^{2}(25)$	`1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ` (bbch)
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)
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)
BB(14)	$>f_{\omega +1}(65\,536)>g_{64}$	`1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ---` (bbch)
BB(15)	$>f_{\omega +1}(f_{\omega }(10^{57}))$	`0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_1RZ0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN` (bbch)
BB(16)	$>f_{\omega +1}^{2}(10^{10^{57}})$
BB(18)	$>f_{\omega +2}(f_{\omega +1}^{3}(f_{\omega }^{2}(60)))$
BB(20)	$>f_{\omega +2}^{2}(21)$
BB(21)	$>f_{\omega ^{2}}^{2}(4\uparrow \uparrow 341)$
BB(40)	$>f_{\omega ^{\omega }}(75\,500)$
BB(41)	$>f_{\omega ^{\omega }}^{4}(32)$
BB(51)	$>f_{\varepsilon _{0}+1}(8)$

3 Symbols:


	Runtime	Champions
BB(1,3)	$1$	`1RZ------` (bbch)
BB(2,3)	$38$	`1RB2LB1RZ_2LA2RB1LB` (bbch)
BB(3,3)	$\geq 119\,112\,334\,170\,342\,541>10^{17}$	`0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC` (bbch)
BB(4,3)	$>2\uparrow \uparrow \uparrow (2^{2^{32}+1}-1)$	`0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD` (bbch)

4 Symbols:


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

5 Symbols:


	Runtime	Champions
BB(1,5)	$1$	`1RZ------------` (bbch)
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)

6 Symbols:


	Runtime	Champions
BB(1,6)	$1$	`1RZ---------------` (bbch)
BB(2,6)	$>10\uparrow \uparrow 10\uparrow \uparrow 10^{10^{115}}$	`1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA` (bbch)

Maximum Score Function (Σ)

2 Symbols:


	Score	Champions
Σ(1)	$1$	`1RZ---` (bbch)
Σ(2)	$4$	`1RB1LB_1LA1RZ` (bbch)
Σ(3)	$6$	`1RB1RZ_0RC1RB_1LC1LA` (bbch) `1RB1RC_1LC1RZ_1RA0LB` (bbch) `1RB1LC_1LA1RB_1LB1RZ` (bbch) `1RB1RA_1LC1RZ_1RA1LB` (bbch) `1RB1LC_1RC1RZ_1LA0LB` (bbch)
Σ(4)	$13$	`1RB1LB_1LA0LC_1RZ1LD_1RD0RA` (bbch) `1RB0RC_1LA1RA_1RZ1RD_1LD0LB` (bbch)
Σ(5)	$4098$	`1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA` (bbch) `1RB1RA_1LC1LB_1RA1LD_1RA1LE_1RZ0LC` (bbch)
Σ(6)	$>2\uparrow \uparrow 2\uparrow \uparrow 2\uparrow \uparrow 10$	`1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE` (bbch)
Σ(7)	$>2\uparrow ^{11}2\uparrow ^{11}3$	`1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF` (bbch)

3 Symbols:


	Score	Champions
Σ(1,3)	$1$	`1RZ------` (bbch)
Σ(2,3)	$9$	`1RB2LB1RZ_2LA2RB1LB` (bbch)
Σ(3,3)	$\geq 374\,676\,383$	`0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC` (bbch)
Σ(4,3)	$>2\uparrow \uparrow \uparrow 2^{2^{32}}$	`0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD` (bbch)

4 Symbols:


	Score	Champions
Σ(1,4)	$1$	`1RZ---------` (bbch)
Σ(2,4)	$2050$	`1RB2LA1RA1RA_1LB1LA3RB1RZ` (bbch)

5 Symbols:


	Score	Champions
Σ(1,5)	$1$	`1RZ------------` (bbch)
Σ(2,5)	$>10^{10^{10^{3\,314\,360}}}$	`1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ` (bbch)

Beeping Busy Beavers

Beeping Busy Beaver (BBB)

2 Symbols:


	Steps taken	Champions
BBB(1)	$1$
BBB(2)	$6$
BBB(3)	$55$	`1LB0RB_1RA0LC_1RC1RA` (bbch)
BBB(4)	$\geq 32\,779\,478$
BBB(5)	$\geq 10^{14006}$

3 Symbols:


	Steps taken	Champions
BBB(1,3)
BBB(2,3)
BBB(3,3)	$\geq 10\uparrow \uparrow 6$

Beeping Booping Busy Beaver (BBBB)

There are currently no known/available Champions for this function.

Maximum Consecutive Ones Function (Num)

2 Symbols:


	Number of Ones	Champions
num(1)	$1$	`1RZ---` (bbch)
num(2)	$4$	`1RB1LB_1LA1LZ` (bbch)
num(3)	$6$	`1RB1LC_1RC1LZ_1LA0LB` (bbch)
num(4)	$12$	`1RB0LA_1RC1LB_1LB1RD_1RZ0RA` (bbch)
num(5)	$165$	`1RB1LA_1RC1LE_1RD1RE_0LA1RC_1RZ0LB` (bbch) `0RB1LD_1LC1RB_1LD1RE_1LA1LE_1LZ0RC` (bbch)

Maximum Space Function (BB_space)

2 Symbols:


	Cells visited	Champions
BB_space(1,2)	$2$
BB_space(2,2)	$4$
BB_space(3,2)	$7$
BB_space(4,2)	$16$

Instruction-Limited Busy Beaver

Maximum amount of steps (BBi)


	Steps	Champions
BBi(1)	$1$	`0RH` (bbch) `1RH---` (bbch)
BBi(2)	$3$	`0RB---_1LA---` (bbch)
BBi(3)	$5$	`1RB1LB_1LA---` (bbch)
BBi(4)	$16$	`1RB---_0RC---_1LC0LA` (bbch)
BBi(5)	$37$	`1RB2LB---_2LA2RB1LB` (bbch)
BBi(6)	$123$	`1RB3LA1RA0LA_2LA------3RA` (bbch)
BBi(7)	$3\,932\,963$	`1RB2LA1RA1RA_1LB1LA3RB---` (bbch) `1RB2LA1RA_1LC1LA2RB_---1LA---` (bbch)
BBi(8)	$>6.889\times 10^{1565}$	`1RB1LA------_1RC3LB1RB---_2LA2LC---0LC` (bbch)

Maximum Score (Σi)


	Score	Champions
Σi(1)	$1$	`1RH---` (bbch)
Σi(2)	$2$	`1RB---_1LA---` (bbch)
Σi(3)	$4$	`1RB1LB_1LA---` (bbch)
Σi(4)	$5$	`1RB0LB---_1LA2RA---` (bbch)
Σi(5)	$9$	`1RB2LB---_2LA2RB1LB` (bbch)
Σi(6)	$14$	`1RB3LA1RA0LA_2LA------3RA` (bbch)
Σi(7)	$2050$	`1RB2LA1RA1RA_1LB1LA3RB---` (bbch) `1RB2LA1RA_1LC1LA2RB_---1LA---` (bbch)
Σi(8)	$>1.355\times 10^{783}$	`1RB1LA------_1RC3LB1RB---_2LA2LC---0LC` (bbch)

Reversible Turing Machines

Maximum Shifts Function (BB_rev)

2 Symbols:


	Steps	Champions
BB_rev(1)
BB_rev(2)	$6$	`0RB1RZ_1LA1RB` (bbch)
BB_rev(3)	$17$	`0RB1RZ_0LC1RA_1RB1LC` (bbch)
BB_rev(4)	$48$	`1RB0LD_0LC0RB_1LA1LD_1LC1RZ` (bbch)
BB_rev(5)	$388$	`1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA` (bbch)
BB_rev(6)	$\geq 537\,556$	`1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA` (bbch)
BB_rev(7)	$>10^{19}$	`1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB1RZ` (bbch)

Maximum Score Function (Σ_rev)

2 Symbols:


	Score	Champions
Σ_rev(1)
Σ_rev(2)	$\geq 2$	`0RB1RZ_1LA1RB` (bbch)
Σ_rev(3)	$\geq 4$	`0RB1RZ_0LC1RA_1RB1LC` (bbch)
Σ_rev(4)	$\geq 6$	`1RB0LD_0LC0RB_1LA1LD_1LC1RZ` (bbch)
Σ_rev(5)	$\geq 16$	`1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA` (bbch)
Σ_rev(6)	$\geq 1161$	`1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA` (bbch)

Blanking Busy Beaver (BLB)

There are currently no known/available Champions for this function.

Lazy Beaver

Shifts Function (LB)


	1 State	2 States	3 States	4 States	5 States	6 States
2 Symbols	$2$	$7$	$22$	$72$	$427$	$8407$
3 Symbols	$2$	$23$	$351$	$189\,270$
4 Symbols	$2$	$93$	$242\,789$
5 Symbols	$2$	$956$
6 Symbols	$2$	$33\,851$

Period-oriented Busy Beavers

Busy Preperiodic Beaver (BBS)

2 Symbols:


	Preperiod	Champions
BBS(1,2)
BBS(2,2)
BBS(3,2)	$101$	`1RB1LB_0RC0LA_1LC0LA` (bbch)
BBS(4,2)	$\geq 119\,120\,230\,102$	`1RB1LC_0LA1RD_0RB0LC_1LA0RD` (bbch)

3 Symbols: It seems that currently no information is available for this domain.

4 Symbols:


	Preperiod	Champions
BBS(1,4)
BBS(2,4)	$\geq 293\,225\,660\,896$	`1RB2LA0RA3LA_1LA1LB3RB1RA` (bbch)

Busy Periodic Beaver (BBP)

2 Symbols:


	Period	Champions
BBP(1,2)
BBP(2,2)
BBP(3,2)	$92$	`1RB0LA_0RC1LA_1LC0RB` (bbch)
BBP(4,2)	$\geq 212\,081\,736$	`1RB0LA_0RC1RD_1LD0RB_1LA1RB` (bbch)

3 Symbols: It seems that currently no information is available for this domain.

4 Symbols:


	Period	Champions
BBP(1,4)
BBP(2,4)	$\geq 33\,209\,131$	`1RB0RA3LB1RB_2LA0LB1RA2RB` (bbch)

Busy Beaver for Lambda Calculus (BBλ)

Regular Busy Beaver for Lambda Calculus

For n = 0,1,2,3,5 BBλ(n) is undefined, while for the rest of $20\geq n$ BBλ(n) = n.


	BBλ(n)	Champions
BBλ(21)	$22$	`\(\1 1) (1 (\2))`
BBλ(22)	$24$	`\(\1 1) (1 (\\1))\(\1 1 1) (1 1)`
BBλ(23)	$26$	`\(\1 1) (1 (\\2))`
BBλ(24)	$30$	`\(\1 1 1) (1 (\1))`
BBλ(25)	$42$	`\(\1 1) (\1 (2 1))`
BBλ(26)	$52$	`(\1 1) (\\2 (1 2))`
BBλ(27)	$44$	`\\(\1 1) (\1 (2 1))`
BBλ(28)	$58$	`\(\1 1) (\1 (2 (\2))))`
BBλ(29)	$223$	`\(\1 1) (\1 (1 (2 1)))`
BBλ(30)	$160$	`(\1 1 1) (\\2 (1 2))` and `(\1 (1 1)) (\\2 (1 2))`
BBλ(31)	$267$	`(\1 1) (\\2 (2 (1 2)))`
BBλ(32)	$298$	`\(\1 1) (\1 (1 (2 (\2))))`
BBλ(33)	$1812$	`\(\1 1) (\1 (1 (1 (2 1))))`
BBλ(34)	$327\,686$	`(\1 1 1 1) (\\2 (2 1))` and `(\1 (1 1) 1) (\\2 (2 1))`
BBλ(35)	$5\times 3^{3^{3}}+6>3.8\times 10^{13}$	`(\1 1 1) (\\2 (2 (2 1)))`
BBλ(36)	$5\times 2^{2^{2^{3}}}+6>5.7\times 10^{77}$	`(\1 1) (\1 (1 (\\2 (2 1))))`
BBλ(37)	$BB\lambda (35)+2=5\times 3^{3^{3}}+8>3.8\times 10^{13}$	`\(\1 1 1) (\\2 (2 (2 1)))`
BBλ(38)	$\geq 5\times 2^{2^{2^{2^{2}}}}+6>10^{10^{4}}$	`(\1 1 1 1 1) (\\2 (2 1))` and `(\1 (1 1) 1 1) (\\2 (2 1))`
BBλ(39)	$\geq 5\times 3^{3^{3^{3}}}+6>10^{10^{12}}$	`(\1 1 1 1) (\\2 (2 (2 1)))`
BBλ(40)	$>(2\uparrow \uparrow )^{15}33>10\uparrow \uparrow \uparrow 16$	`(\1 1 1) (\1 (\\2 (2 1)) 1)`
BBλ(41)	$\geq 5\times 3^{3^{85}}+6>10^{10^{40}}$	`(\1 (\1 1) 1) (\\2 (2 (2 1)))`
BBλ(42)	$\geq BB\lambda (40)+2>(2\uparrow \uparrow )^{15}33>10\uparrow \uparrow \uparrow 16$	`\(\1 1 1) (\1 (\\2 (2 1)) 1)`
BBλ(43)	$>2\uparrow \uparrow \uparrow 2\uparrow \uparrow \uparrow 2\uparrow \uparrow 8$	`(\1 1) (\1 (\1 (\\2 (2 1)) 2))`
BBλ(44)	$>10\uparrow \uparrow \uparrow 10\uparrow \uparrow \uparrow 16$	`(\1 1 1 1) (\1 (\\2 (2 1)) 1)`
BBλ(45)	$\geq BB\lambda (43)+2>2\uparrow \uparrow \uparrow 2\uparrow \uparrow \uparrow 2\uparrow \uparrow 8$	`\(\1 1) (\1 (\1 (\\2 (2 1)) 2))`
BBλ(46)	$\geq BB\lambda (44)+2>10\uparrow \uparrow \uparrow 10\uparrow \uparrow \uparrow 16$	`\(\1 1 1 1) (\1 (\\2 (2 1)) 1)`
BBλ(47)
BBλ(48)	$>10\uparrow \uparrow \uparrow 10\uparrow \uparrow \uparrow 10\uparrow \uparrow \uparrow 16$	`(\1 1 1 1 1) (\1 (\\2 (2 1)) 1)`
BBλ(49)	$>f_{\omega +1}({\frac {2\uparrow \uparrow 6}{2}})>{\text{Grahams Number}}$	`(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))`
BBλ(1850)	$>{\text{Loaders Number}}$	`Too large for this list`

Oracle Busy Beaver for Lambda Calculus (BBλ₁)

Note that $f(n)=6+5\times BB\lambda (n)$ .


	BBλ₁(n)	Champions
BBλ₁(1)	$0$
BBλ₁(2)	$1$	`1`
BBλ₁(3)	$0$
BBλ₁(4)	$4$	`\1`
BBλ₁(5)	$5$	`\2`
BBλ₁(6)	$6$	`\\1`
BBλ₁(7)	$7$	`\\2`
BBλ₁(8)	$26$	`1 (\1)`
BBλ₁(9)	$9$	`\\2`
BBλ₁(10)	$36$	`1 (\\1)`
BBλ₁(11)	$41$	`1 (\\2)`
BBλ₁(12)	$266$	`1 (1 (\1))`
BBλ₁(13)	$51$	`1 (\\2)`
BBλ₁(14)	$f(36)=25\times 2^{2^{2^{3}}}+36>2.85\times 10^{78}$	`1 (1 (\\1))`
BBλ₁(15)	$f(41)\geq 25\times 3^{3^{85}}+36>10^{10^{40}}$	`1 (1 (\\2))`
BBλ₁(16)	$f(266)$	`1 (1 (1 (\1)))`
BBλ₁(17)	$f(51)$	`1 (1 (\\\2))`
BBλ₁(18)	$f^{4}(4)=f(f(266))$	`1 (\1) 1 (\1)`
BBλ₁(19)	$f^{3}(7)=f(f(41))$	`1 (1 (1 (\\2)))`
BBλ₁(20)	$f^{6}(4)=f^{4}(266)$	`1 (\\1) 1 (\1)`
BBλ₁(21)	$f^{7}(4)=f^{5}(266)$	`1 (\\2) 1 (\1)`
BBλ₁(22)	$f^{52}(4)=f^{50}(266)$	`1 (1(\1)) 1(\1)`
BBλ₁(28)	$\geq f^{BB\lambda (f^{3}(4))}(4)$	`1 (\1) 1 (\1) 1 (\1)`

Doodle Function (doodle)

doodle(1,n) = 1 and doodle(2,n) = n

2 Symbols:


	Runtime	Champions
doodle(3,2)	$\geq 487$

Terminating Turmites (TT)

There are currently no known/available Champions for this function.

@@ Line 400: / Line 400: @@
 |{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
 |}
-==Maximum Score (Σi)==
+==Maximum Score ([[Σi]])==
 {| class="wikitable"
 |+
@@ Line 439: / Line 439: @@
 |{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
 |}
 ='''Reversible Turing Machines'''=
 ==Maximum Shifts Function ([[Reversible Turing Machine|BB<sub>rev</sub>]])==

User:Polygon/Collection of BB Champions: Difference between revisions

Revision as of 19:53, 14 August 2025

Contents

Original Busy Beaver Functions

Maximum Shifts Function (BB)

Maximum Score Function (Σ)

Beeping Busy Beavers

Beeping Busy Beaver (BBB)

Beeping Booping Busy Beaver (BBBB)

Maximum Consecutive Ones Function (Num)

Maximum Space Function (BB_space)

Instruction-Limited Busy Beaver

Maximum amount of steps (BBi)

Maximum Score (Σi)

Reversible Turing Machines

Maximum Shifts Function (BB_rev)

Maximum Score Function (Σ_rev)

Blanking Busy Beaver (BLB)

Lazy Beaver

Shifts Function (LB)

Period-oriented Busy Beavers

Busy Preperiodic Beaver (BBS)

Busy Periodic Beaver (BBP)

Busy Beaver for Lambda Calculus (BBλ)

Regular Busy Beaver for Lambda Calculus

Oracle Busy Beaver for Lambda Calculus (BBλ₁)

Doodle Function (doodle)

Terminating Turmites (TT)

Navigation menu

User:Polygon/Collection of BB Champions: Difference between revisions

Revision as of 19:53, 14 August 2025

Original Busy Beaver Functions

Maximum Shifts Function (BB)

Maximum Score Function (Σ)

Beeping Busy Beavers

Beeping Busy Beaver (BBB)

Beeping Booping Busy Beaver (BBBB)

Maximum Consecutive Ones Function (Num)

Maximum Space Function (BBspace)

Instruction-Limited Busy Beaver

Maximum amount of steps (BBi)

Maximum Score (Σi)

Reversible Turing Machines

Maximum Shifts Function (BBrev)

Maximum Score Function (Σrev)

Blanking Busy Beaver (BLB)

Lazy Beaver

Shifts Function (LB)

Period-oriented Busy Beavers

Busy Preperiodic Beaver (BBS)

Busy Periodic Beaver (BBP)

Busy Beaver for Lambda Calculus (BBλ)

Regular Busy Beaver for Lambda Calculus

Oracle Busy Beaver for Lambda Calculus (BBλ1)

Doodle Function (doodle)

Terminating Turmites (TT)

Navigation menu

Maximum Space Function (BB_space)

Maximum Shifts Function (BB_rev)

Maximum Score Function (Σ_rev)

Oracle Busy Beaver for Lambda Calculus (BBλ₁)