User:Polygon/Collection of BB Champions

A collection of Busy Beaver Champions including Champions for BB-Adjacent functions. Note that for all functions with the input format f(n,m), n denotes the number of states and m denotes the number of symbols of the relevant Busy Beaver domain. Note that for functions with this input format f(n) = f(n,2).

State-and-Symbol-Limited Busy Beaver functions

Busy Beaver functions where the programs size is limited by the amount of states and symbols.

Original Busy Beaver Functions

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

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)	> 10 ↑↑ 10 ↑↑ 10 ↑↑ 8.10237	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)	$>f_{\omega +1}(65536)>g_{64}$	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)	$>f_{{\omega }^{\omega }}(75500)$
BB(41)	$>f_{{\omega }^{\omega }}^4(32)$
BB(51)	$>f_{{\epsilon }_0+1}(8)$

3 Symbols:	Runtime	Champions
BB(1,3)	1	1RZ------ (bbch)
BB(2,3)	38	1RB2LB1RZ_2LA2RB1LB (bbch)
BB(3,3)	$\ge 119112334170342541>10^{17}$	0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC (bbch)
BB(4,3)	$>2\uparrow \uparrow \uparrow 2\uparrow \uparrow \uparrow 2\uparrow \uparrow \uparrow (7.92\times 10^{28})$	1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD (bbch)

4 Symbols:	Runtime	Champions
BB(1,4)	1	1RZ--------- (bbch)
BB(2,4)	3,932,964	1RB2LA1RA1RA_1LB1LA3RB1RZ (bbch)
BB(3,4)	$>(2{\uparrow }^{15}5)+14$	1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC (bbch)

5 Symbols:	Runtime	Champions
BB(1,5)	1	1RZ------------ (bbch)
BB(2,5)	$>10^{10^{10^{3314360}}}$	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(n,m))

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)	> 10 ↑↑ 10 ↑↑ 10 ↑↑ 8.10237	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)	≥ 374,676,383	0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC (bbch)
Σ(4,3)	$>2\uparrow \uparrow \uparrow 2\uparrow \uparrow \uparrow 2\uparrow \uparrow \uparrow (7.92\times 10^{28})$	1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD (bbch)

4 Symbols:	Score	Champions
Σ(1,4)	1	1RZ--------- (bbch)
Σ(2,4)	2050	1RB2LA1RA1RA_1LB1LA3RB1RZ (bbch)
Σ(3,4)	$\ge (2{\uparrow }^{15}5)+14$	1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC (bbch)

5 Symbols:	Score	Champions
Σ(1,5)	1	1RZ------------ (bbch)
Σ(2,5)	$>10^{10^{10^{3314360}}}$	1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ (bbch)

6 Symbols:	Score	Champions
Σ(1,6)	1	1RZ--------------- (bbch)
Σ(2,6)	$>10\uparrow \uparrow 10\uparrow \uparrow 10^{10^{115}}$	1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA (bbch)

Beeping Busy Beavers

Beeping Busy Beaver (BBB(n,m))

2 Symbols:	Steps taken	Champions
BBB(1)	1
BBB(2)	6	1RB1LB_1LB1LA (bbch)
BBB(3)	55	1RB0LB_1LA0RC_1LC1LA (bbch)
BBB(4)	≥ 32,779,478	1RB1LD_1RC1RB_1LC1LA_0RC0RD (bbch)
BBB(5)	$\ge 10^{14006}$	1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA (bbch)

3 Symbols:	Steps taken	Champions
BBB(1,3)
BBB(2,3)	59	1RB2LB1LA_2LB2RA0RA (bbch)
BBB(3,3)	≥ 10 ↑↑ 6	1RB0LB2LA_1LA0RC0LB_2RC2RB0LC (bbch)

4 Symbols:	Steps taken	Champions
BBB(1,4)
BBB(2,4)	$\ge 205770076433044242247859>2\times 10^{23}$	1RB2LA1RA1LB_0LB2RB3RB1LA (bbch)

Beeping Booping Busy Beaver (BBBB(n,m))

2 Symbols:	Steps taken	Champions
BBBB(1)	2
BBBB(2)	17

Maximum Consecutive Ones Function (Num(n,m))

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(n,m))

2 Symbols:	Cells visited	Champions
BBspace(1,2)	1	1RZ--- (bbch)
BBspace(2,2)	4	1RB1LB_1LA1RZ (bbch) and 1RB0LB_1LA1RZ (bbch)
BBspace(3,2)	7	1RB1RC_1LC1RZ_1RA0LB (bbch) and 1RB0RC_1LC1RZ_1RA0LB (bbch)
BBspace(4,2)	16	1RB0RA_1LC0RD_0LD0LB_1RA1RZ (bbch)
BBspace(5,2)	12289	1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA (bbch)

3 Symbols:	Cells visited	Champions
BBspace(1,3)
BBspace(2,3)	9	1RB2LB1RZ_2LA2RB1LB (bbch)
BBspace(2,4)	2050	1RB2LA1RA1RA_1LB1LA3RB1RZ (bbch)

Size of the runtime spectrum (R(n,m))

There currently doesn't seem to be any available information about values of this function.

Reversible Turing Machines

Maximum Shifts Function (BB_rev(n,m))

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

Maximum Score Function (Σ_rev(n,m))

2 Symbols:	Score	Champions
Σrev(1)
Σrev(2)	≥ 2	0RB1RZ_1LA1RB (bbch)
Σrev(3)	≥ 4	0RB1RZ_0LC1RA_1RB1LC (bbch)
Σrev(4)	≥ 6	1RB0LD_0LC0RB_1LA1LD_1LC1RZ (bbch)
Σrev(5)	≥ 16	1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA (bbch)
Σrev(6)	≥ 1161	1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA (bbch)

Blanking Busy Beaver (BLB(n,m))

2 Symbols:	Steps	Champions
BLB(1)	nonexistent	nonexistent
BLB(2)	≥ 8	1RB0RA_1LB1LA (bbch)
BLB(3)	≥ 34	1RB1LB_1LA1LC_1RC0LC (bbch)
BLB(4)	≥ 32,779,477	1RB1LD_1RC1RB_1LC1LA_0RC0RD (bbch)

3 Symbols:	Steps	Champions
BLB(1,3)	nonexistent	nonexistent
BLB(2,3)	≥ 77	1RB2LA0RB_1LA0LB1RA (bbch)

4 Symbols:	Steps	Champions
BLB(1,4)	nonexistent	nonexistent
BLB(2,4)	≥ 1,367,361,263,049	1RB2RA1RA2RB_2LB3LA0RB0RA (bbch)

Lazy Beaver

Shifts Function (LB(n,m))

	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(n,m))

2 Symbols:	Preperiod	Champions
BBS(1,2)	0	1RA--- (bbch)
BBS(2,2)	≥ 9	1RB0LB_1LA0RB (bbch) proven winner?
BBS(3,2)	101	1RB1LB_0RC0LA_1LC0LA (bbch)
BBS(4,2)	≥ 119,120,230,102	1RB1LC_0LA1RD_0RB0LC_1LA0RD (bbch)

3 Symbols:	Preperiod	Champions
BBS(1,3)	0	1RA------ (bbch)

4 Symbols:	Preperiod	Champions
BBS(1,4)	0	1RA--------- (bbch)
BBS(2,4)	≥ 293,225,660,896	1RB2LA0RA3LA_1LA1LB3RB1RA (bbch)

Busy Periodic Beaver (BBP(n,m))

2 Symbols:	Period	Champions
BBP(1,2)	1	1RA--- (bbch)
BBP(2,2)	≥ 9	1RB0RB_1LB1RA (bbch) proven winner?
BBP(3,2)	92	1RB0LA_0RC1LA_1LC0RB (bbch)
BBP(4,2)	≥ 212,081,736	1RB0LA_0RC1RD_1LD0RB_1LA1RB (bbch)

3 Symbols:	Period	Champions
BBP(1,3)	1	1RA------ (bbch)

4 Symbols:	Period	Champions
BBP(1,4)	1	1RA--------- (bbch)
BBP(2,4)	≥ 33,209,131	1RB0RA3LB1RB_2LA0LB1RA2RB (bbch)

Instruction-Limited Busy Beaver

Instruction-Limited Classical Busy Beaver Functions

Instruction-Limited Maximum Shifts Function (BBi(n))

	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)
BBi(9)	$>10^{10^{10^{3314360}}}$	1RB3LA4RB0RB2LA_1LB2LA3LA1RA--- (bbch)

Instruction-Limited Maximum Score Function (Σi(n))

	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)
Σi(9)	$>10^{10^{10^{3314360}}}$	1RB3LA4RB0RB2LA_1LB2LA3LA1RA--- (bbch)

Instruction-Limited Blanking Busy Beaver (BLBi(n))

	Steps	Champions
BLBi(1)	nonexistent	nonexistent
BLBi(2)	nonexistent	nonexistent
BLBi(3)	4
BLBi(4)	12
BLBi(5)	30
BLBi(6)	77	1RB2LA0RB_1LA0LB1RA (bbch)
BLBi(7)	808
BLBi(8)	≥ 1,367,361,263,049	1RB2RA1RA2RB_2LB3LA0RB0RA (bbch)

Instruction-Limited Greedy Busy Beaver (gBBi(n))

	Steps	Champions
gBBi(1)	1
gBBi(2)	3
gBBi(3)	5
gBBi(4)	13
gBBi(5)	19
gBBi(6)	25
gBBi(7)	41
gBBi(8)	55
gBBi(9)	238
gBBi(10)	941
gBBi(11)	1341
gBBi(12)	10465
gBBi(13)	10675
gBBi(14)	≥ 9,874,580	0RB6RB1LB---3LA1RB7RB2LB_1LA2RB3LA4RB5RB1LB5LA--- (bbch)

Program-Limited Busy Beaver

Busy Beaver for Lambda Calculus

Regular Busy Beaver for Lambda Calculus (BBλ(n))

For n = 0,1,2,3,5 BBλ(n) is undefined, while for the rest of $20\ge n$ 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)	$5\times 3^{3^3}+6=38127987424941>3.8\times 10^{13}$	(\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)	$BB\lambda (35)+2=5\times 3^{3^3}+8=38127987424943>3.8\times 10^{13}$	\(\1 1 1) (\\2 (2 (2 1)))
BBλ(38)	$\ge 5\times 2^{2^{2^{2^2}}}+6>10^{19729}$	(\1 1 1 1 1) (\\2 (2 1)) and (\1 (1 1) 1 1) (\\2 (2 1))
BBλ(39)	$\ge 5\times 3^{3^{3^3}}+6>10^{3638334640024}$	(\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)	$\ge 5\times 3^{3^{85}}+6>10^{1.7\times 10^{40}}$	(\1 (\1 1) 1) (\\2 (2 (2 1)))
BBλ(42)	$\ge BB\lambda (40)+2>(2\uparrow \uparrow {)}^{15}33>10\uparrow \uparrow \uparrow 16$	\(\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)	$\ge BB\lambda (43)+2>2\uparrow \uparrow \uparrow 2\uparrow \uparrow \uparrow 2\uparrow \uparrow 8$	\(\1 1) (\1 (\1 (\\2 (2 1)) 2))
BBλ(46)	$\ge BB\lambda (44)+2>10\uparrow \uparrow \uparrow 10\uparrow \uparrow \uparrow 16$	\(\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)	$>f_{\omega +1}(\frac{2\uparrow \uparrow 6}{2})>\text{Graham's Number}$	(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))
BBλ(1850)	$>\text{Loader's Number}$	Too large for this list

Oracle Busy Beaver for Lambda Calculus (BBλ₁(n))

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)	$f(41)\ge 25\times 3^{3^{85}}+36>10^{1.7\times 10^{40}}$	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)	$\ge f^{BB\lambda (f^3(4))}(4)$	1 (\1) 1 (\1) 1 (\1)
BBλ1(29)	$\ge f^{BB\lambda (f^{BB\lambda (f^4(4))+4}(4))+BB\lambda (f^4(4))+5}(4)$	1(\1)(\1 2 1)(\1)

Fractran (BBf(n))

n	BBf(n)	Example Champion	Vector Representation
2	1	[1/2]	$\left[\begin{array}{c}-1\end{array}\right]$
3	1	[3/2]	$\left[\begin{array}{cc}-1& 1\end{array}\right]$
4	1	[9/2]	$\left[\begin{array}{cc}-1& 2\end{array}\right]$
5	2	[3/2, 1/3]	$\left[\begin{array}{cc}-1& 1\\ 0& -1\end{array}\right]$
6	3	[9/2, 1/3]	$\left[\begin{array}{cc}-1& 2\\ 0& -1\end{array}\right]$
7	4	[27/2, 1/3]	$\left[\begin{array}{cc}-1& 3\\ 0& -1\end{array}\right]$
8	5	[81/2, 1/3]	$\left[\begin{array}{cc}-1& 4\\ 0& -1\end{array}\right]$
9	6	[243/2, 1/3]	$\left[\begin{array}{cc}-1& 5\\ 0& -1\end{array}\right]$
10	7	[729/2, 1/3]	$\left[\begin{array}{cc}-1& 6\\ 0& -1\end{array}\right]$
11	10	[27/2, 25/3, 1/5]	$\left[\begin{array}{ccc}-1& 3& 0\\ 0& -1& 2\\ 0& 0& -1\end{array}\right]$
12	13	[81/2, 25/3, 1/5]	$\left[\begin{array}{ccc}-1& 4& 0\\ 0& -1& 2\\ 0& 0& -1\end{array}\right]$
13	17	[81/2, 125/3, 1/5]	$\left[\begin{array}{ccc}-1& 4& 0\\ 0& -1& 3\\ 0& 0& -1\end{array}\right]$
14	21	[243/2, 125/3, 1/5]	$\left[\begin{array}{ccc}-1& 5& 0\\ 0& -1& 3\\ 0& 0& -1\end{array}\right]$
15	28	[1/45, 4/5, 3/2, 25/3]	$\left[\begin{array}{ccc}0& -2& -1\\ 2& 0& -1\\ -1& 1& 0\\ 0& -1& 2\end{array}\right]$
16	53	[1/45, 4/5, 3/2, 125/3]	$\left[\begin{array}{ccc}0& -2& -1\\ 2& 0& -1\\ -1& 1& 0\\ 0& -1& 3\end{array}\right]$
17	107	[5/6, 49/2, 3/5, 40/7]	$\left[\begin{array}{cccc}-1& -1& 1& 0\\ -1& 0& 0& 2\\ 0& 1& -1& 0\\ 3& 0& 1& -1\end{array}\right]$
18	211	[5/6, 49/2, 3/5, 80/7]	$\left[\begin{array}{cccc}-1& -1& 1& 0\\ -1& 0& 0& 2\\ 0& 1& -1& 0\\ 4& 0& 1& -1\end{array}\right]$
19	370	[5/6, 49/2, 3/5, 160/7]	$\left[\begin{array}{cccc}-1& -1& 1& 0\\ -1& 0& 0& 2\\ 0& 1& -1& 0\\ 5& 0& 1& -1\end{array}\right]$
20	≥ 746	[7/15, 22/3, 6/77, 5/2, 9/5]	$\left[\begin{array}{ccccc}& -1& -1& +1& \\ +1& -1& & & +1\\ +1& +1& & -1& -1\\ -1& & +1& & \\ & +2& -1& & \end{array}\right]$
21	≥ 55,213	[7/15, 22/3, 3/77, 175/2, 9/5]
22	≥ 1,362,233	[121/6, 35/3, 15/77, 4/5, 9/2]
23	≥ 3,441,433	[11/6, 35/3, 15/77, 16/5, 9/2]

Doodle Function (doodle(c,n))

doodle(1,n) = 1 and doodle(2,n) = n. Also note that doodle(c) = doodle(c,2).

2 Symbols:	Runtime	Champions
doodle(3,2)	≥ 487

Turmites

Terminating Turmites (TT(n,k), 1D Turmites)

Where n is the amount of states and k is the amount of symbols. There are currently no known/available Champions for this function.

2D Turmites (turNing machines)

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

Bug Function

Bug(2,2) = 2

####
#S-#
#.F#
####

Bug(3,3) = 8

#####
#S..#
#.#.#
#.#F#
#####

Bug(4,4) = 20

######
#S...#
#..###
#....#
#..#F#
######

Bug(5,5) = 42

#######
#S...##
#.....#
#.....#
#.#.###
#.#..F#
#######

Bug(6,6) = 96

########
#S.....#
#.###.##
#.#...##
#.#....#
#..#.###
#..#..F#
########

Bug(7,7) = 218

#########
#S.###..#
#......##
#.#.##..#
#..#...##
#..#....#
#...#.###
#..#-..F#
#########

Bug(8,8) = 506

##########
#S.#.....#
#.#..##.##
#.#.##..##
#....##..#
#.#.#...##
#..##....#
#....#.###
#....#..F#
##########