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).

Note: highest ref name in use: 4

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))

Maximum amount of steps a Turing machine with n states and m symbols can run for before halting when started on the blank tape.

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(13)	$>f_{\omega +1}(2046)>g_{64}$	1RB1RA_1LC1RD_1LA1LC_1LG0RE_1LC1RB_0RL1LG_0LM0RH_1RI1RH_1LK0RI_---0LK_1LF1LK_1LJ1RL_1RZ1RH (bbch)
BB(14)	$>f_{\omega +1}(65536)$	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))

Maximum amount of non-zero symbols a Turing machine with n states and m symbols can leave on the tape before halting when started on the blank tape.

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))

Considering beep states

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))

Domain	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)
num(2,3)	6	1RB1LA1LB_0LA2RA1RZ (bbch)
num(3,3)	≥ 12	1RB1RA1RZ_1LC1LC2LA_2RA1LB1LA (bbch)
num(4,3)	$>10{\uparrow }^{4}4$	1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD (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)
BLB(5)	> 1014,006	1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA (bbch)
BLB(6)	> 10 ↑↑ 16	1RB0LD_1RC0RF_1LC1LA_0LE1RE_1LF0RB_0RC0RE (bbch)
BLB(7)	> 10^183.76332	1RB0RF_1RC1RA_1LD0LA_0RG0RE_0RA1LF_0LE0LC_1RZ0RD (bbch)

3 Symbols:	Steps	Champions
BLB(1,3)	nonexistent	nonexistent
BLB(2,3)	77	1RB2LA0RB_1LA0LB1RA (bbch)
BLB(3,3)	> 1042,745	1RB2RB1LA_2LC0LB2LB_2RC2RA0LC (bbch)

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

5 Symbols:	Steps	Champions
BLB(2,5)	> 1032	1RB2RB4RA0RB2RA_2LB3LA0RB0RA1RB (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)
BBS(5,2)	> 1014,006	1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA (bbch)

3 Symbols:	Preperiod	Champions
BBS(1,3)	0	1RA------ (bbch)
BBS(2,3)	≥ 165	1RB0LA---_1LB2LA0RB (bbch)
BBS(3,3)	> 10 ↑↑ 6	1RB0LB2LA_1LA0RC0LB_2RC2RB0LC (bbch)
BBS(4,3)	> $10{\uparrow }^{4}4$	1RB1RD1LC_2LB1RB1LC_1LB1LA1LD_0RB2RA2RD (bbch)

4 Symbols:	Preperiod	Champions
BBS(1,4)	0	1RA--------- (bbch)
BBS(2,4)	≥ 205,770,076,433,044,242,247,860	1RB2LA1RA1LB_0LB2RB3RB1LA (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)
BBP(2,3)
BBP(3,3)	≥ 1,195	1RB2RC1LC_0RC0RB1LA_2LA2RC1LB (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	1RB0RA_1LA--- (bbch)
BLBi(4)	12	1RB---_1RC---_1LC0RC (bbch)
BLBi(5)	30	1RB------_1RC------_2LC2RC0RC (bbch)
BLBi(6)	77	1RB2LA0RB_1LA0LB1RA (bbch)
BLBi(7)	808	1RB------_1RC------_0RD2LC---_1LD2RD0RC (bbch)
BLBi(8)	≥ 1,367,361,263,049	1RB2RA1RA2RB_2LB3LA0RB0RA (bbch)
BLBi(9)	> 1042,745	1RB2RB1LA_2LC0LB2LB_2RC2RA0LC (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(4)	4	$\lambda 1$
BBλ(6)	6	$\lambda \lambda 1$
BBλ(7)	7	$\lambda \lambda 2$
BBλ(8)	8	$\lambda \lambda \lambda 1$
BBλ(9)	9	$\lambda \lambda \lambda 2$
BBλ(10)	10	$\lambda \lambda \lambda \lambda 1$
BBλ(11)	11	$\lambda \lambda \lambda \lambda 2$
BBλ(12)	12	$\lambda \lambda \lambda \lambda \lambda 1$
BBλ(13)	13	$\lambda \lambda \lambda \lambda \lambda 2$
BBλ(14)	14	$\lambda \lambda \lambda \lambda \lambda \lambda 1$
BBλ(15)	15	$\lambda \lambda \lambda \lambda \lambda \lambda 2$
BBλ(16)	16	$\lambda \lambda \lambda \lambda \lambda \lambda \lambda 1$
BBλ(17)	17	$\lambda \lambda \lambda \lambda \lambda \lambda \lambda 2$
BBλ(18)	18	$\lambda \lambda \lambda \lambda \lambda \lambda \lambda \lambda 1$
BBλ(19)	19	$\lambda \lambda \lambda \lambda \lambda \lambda \lambda \lambda 2$
BBλ(20)	20	$\lambda \lambda \lambda \lambda \lambda \lambda \lambda \lambda \lambda 1$
BBλ(21)	22	$\lambda (1(\lambda 2))(1(\lambda 2))$
BBλ(22)	24	$\lambda (11)(11)(11)$
BBλ(23)	26	$\lambda (1(\lambda \lambda 2))(1(\lambda \lambda 2))$
BBλ(24)	30	$\lambda (1(\lambda 1))(1(\lambda 1))(1(\lambda 1))$
BBλ(25)	42	$\lambda 1(\lambda 1(21))(1(1(\lambda 1(21))))$
BBλ(26)	52	$\lambda \lambda 2(\lambda \lambda 2(12))(1(2(\lambda \lambda 2(12))))$
BBλ(27)	44	$\lambda \lambda 1(\lambda 1(21))(1(1(\lambda 1(21))))$
BBλ(28)	58	$\lambda 1(\lambda \lambda 1(3(\lambda 2)))(1(\lambda 2(\lambda \lambda 1(4(\lambda 2)))))$
BBλ(29)	223	$\lambda (\lambda 11)(\lambda 1(1(21)))$
BBλ(30)	160	$(\lambda 111)(\lambda \lambda 2(12))$
BBλ(31)	267	$(\lambda 11)(\lambda \lambda 2(2(12)))$
BBλ(32)	298	$\lambda (\lambda 11)(\lambda 1(1(2(\lambda 2))))$
BBλ(33)	1812	$\lambda (\lambda 11)(\lambda 1(1(1(21))))$
BBλ(34)	327,686	$(\lambda 1111)(\lambda \lambda 2(21))$
BBλ(35)	$5\times 3^{3^3}+6=38127987424941>3.8\times 10^{13}$	$(\lambda 111)(\lambda \lambda 2(2(21)))$
BBλ(36)	$5\times 2^{2^{2^3}}+6>5.7\times 10^{77}$	$(\lambda 11)(\lambda 1(1(\lambda \lambda 2(21))))$
BBλ(37)	$BB\lambda (35)+2=5\times 3^{3^3}+8=38127987424943>3.8\times 10^{13}$	$\lambda (\lambda 111)(\lambda \lambda 2(2(21)))$
BBλ(38)	$5\times 2^{2^{2^{2^2}}}+6>10^{19729}$	$(\lambda 11111)(\lambda \lambda 2(21))$
BBλ(39)	$5\times 3^{3^{3^3}}+6>10^{3638334640024}$	$(\lambda 1111)(\lambda \lambda 2(2(21)))$
BBλ(40)	$\approx (2\uparrow \uparrow {)}^{15}33>10\uparrow \uparrow \uparrow 16$	$(\lambda 111)(\lambda 1(\lambda \lambda 2(21))1)$
BBλ(41)	$5\times 3^{3^{85}}+6>10^{1.7\times 10^{40}}$	$(\lambda 1(\lambda 11)1)(\lambda \lambda 2(2(21)))$
BBλ(42)	$\ge BB\lambda (40)+2>(2\uparrow \uparrow {)}^{15}33>10\uparrow \uparrow \uparrow 16$	$\lambda (\lambda 111)(\lambda 1(\lambda \lambda 2(21))1)$
BBλ(43)	$>2\uparrow \uparrow \uparrow 2\uparrow \uparrow \uparrow 2\uparrow \uparrow 8$	$(\lambda 11)(\lambda 1(\lambda 1(\lambda \lambda 2(21))2))$
BBλ(44)	$>10\uparrow \uparrow \uparrow 10\uparrow \uparrow \uparrow 16$	$(\lambda 1111)(\lambda 1(\lambda \lambda 2(21))1)$
BBλ(45)	$\ge BB\lambda (43)+2>2\uparrow \uparrow \uparrow 2\uparrow \uparrow \uparrow 2\uparrow \uparrow 8$	$\lambda (\lambda 11)(\lambda 1(\lambda 1(\lambda \lambda 2(21))2))$
BBλ(46)	$\ge BB\lambda (44)+2>10\uparrow \uparrow \uparrow 10\uparrow \uparrow \uparrow 16$	$\lambda (\lambda 1111)(\lambda 1(\lambda \lambda 2(21))1)$
BBλ(47)	$>f_{\omega }\left(f_5\left(2\right)\right)$	$(\lambda 111)(\lambda \lambda 1(12)(\lambda \lambda 2(21)))$
BBλ(48)	$>10\uparrow \uparrow \uparrow 10\uparrow \uparrow \uparrow 10\uparrow \uparrow \uparrow 16$	$(\lambda 11111)(\lambda 1(\lambda \lambda 2(21))1)$
BBλ(49)	$>f_{\omega +1}(\frac{2\uparrow \uparrow 6}{2})>\text{Graham's Number}$	$(\lambda 11)(\lambda 1(1(\lambda \lambda 12(\lambda \lambda 2(21)))))$
BBλ(61)	$>f_{{\omega }^{2\uparrow \uparrow 18-1}}\left(2\right)$	$(\lambda 111)(\lambda 1(1(\lambda \lambda \lambda 132(\lambda \lambda 2(21)))))$
BBλ(86)	$>f_{{\omega }^{{\omega }^2}}\left(2\right)$	$(\lambda 1(\lambda \lambda \lambda \lambda 1444321)1111)(\lambda \lambda 2(21))$
BBλ(90)	$>f_{{\zeta }_0}\left(15\right)$	$(\lambda 11(\lambda \lambda \lambda \lambda 1444321)1111)(\lambda \lambda 2(21))$
BBλ(94)	$>f_{\psi ({\mathrm{\Omega }}_{\omega })}\left(12\right)$	$(\lambda 111(\lambda \lambda \lambda \lambda 1444321)1111)(\lambda \lambda 2(21))$
BBλ(95)	$>f_{\psi ({\mathrm{\Omega }}_{\omega })}\left(23\right)$	$(\lambda 11(\lambda \lambda \lambda \lambda 1444321)1111)(\lambda \lambda 2(2(21)))$
BBλ(96)	$>f_{\psi ({\mathrm{\Omega }}_{\omega })}\left(f_{{\omega }^{{\omega }^2}}\left(2\right)\right)$	$(\lambda 1(\lambda 1(\lambda \lambda \lambda \lambda 1444321)1111)1)(\lambda \lambda 2(21))$
BBλ(100)	$>f_{\psi ({\mathrm{\Omega }}_{\omega })+1}\left(4\right)$	$(\lambda 11(\lambda 1(\lambda \lambda \lambda \lambda 1444321)1111)1)(\lambda \lambda 2(21))$
BBλ(201)	> q(5)	too large to show
BBλ(331)	lim(BMS)	too large to show
BBλ(1850)	$>\text{Loader's Number}$	too large to show

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	$\lambda 1$
BBλ1(5)	5	$\lambda 2$
BBλ1(6)	6	$\lambda \lambda 1$
BBλ1(7)	7	$\lambda \lambda 2$
BBλ1(8)	26	$1(\lambda 1)$
BBλ1(9)	9	$\lambda \lambda 2$
BBλ1(10)	36	$1(\lambda \lambda 1)$
BBλ1(11)	41	$1(\lambda \lambda 2)$
BBλ1(12)	266	$1(1(\lambda 1))$
BBλ1(13)	51	$1(\lambda \lambda 2)$
BBλ1(14)	$f(36)=25\times 2^{2^{2^3}}+36>2.85\times 10^{78}$	$1(1(\lambda \lambda 1))$
BBλ1(15)	$f(41)\ge 25\times 3^{3^{85}}+36>10^{1.7\times 10^{40}}$	$1(1(\lambda \lambda 2))$
BBλ1(16)	$f(266)$	$1(1(1(\lambda 1)))$
BBλ1(17)	$f(51)$	$1(1(\lambda \lambda \lambda 2))$
BBλ1(18)	$f^4(4)=f(f(266))$	$1(\lambda 1)1(\lambda 1)$
BBλ1(19)	$f^3(7)=f(f(41))$	$1(1(1(\lambda \lambda 2)))$
BBλ1(20)	$f^6(4)=f^4(266)$	$1(\lambda \lambda 1)1(\lambda 1)$
BBλ1(21)	$f^7(4)=f^5(266)$	$1(\lambda \lambda 2)1(\lambda 1)$
BBλ1(22)	$f^{52}(4)=f^{50}(266)$	$1(1(\lambda 1))1(\lambda 1)$
BBλ1(28)	$\ge f^{BB\lambda (f^3(4))}(4)$	$1(\lambda 1)1(\lambda 1)1(\lambda 1)$
BBλ1(29)	$\ge f^{BB\lambda (f^{BB\lambda (f^4(4))+4}(4))+BB\lambda (f^4(4))+5}(4)$	$1(\lambda 1)(\lambda 121)(\lambda 1)$

Busy Beaver for De Bruijn Lambda Calculus

n	Value	Champion
7	≥ 7	\1 1 1 1 1 1
8	≥ 16	(\1 1) (\\2 (1 2))
9	≥ 68	(\1 1) (\\2 (2 (1 2)))
10	$>7.625\times 10^{12}$	(\1 1 1) (\\2 (2 (2 1)))
11	$>10^{7.625\times 10^{12}}$	(\1 1 1 1) (\\2 (2 (2 1)))
12	$>10{\uparrow }^{3}16$	(\1 1 1) (\1 (\\2 (2 1)) 1)
13	$>10{\uparrow }^{3}10{\uparrow }^{3}10{\uparrow }^{2}6$	(\1 1) (\1 (\1 (\\2 (2 1)) 2))
14	$>f_{\omega }\left(f_5\left(2\right)\right)$	(\1 1 1) (\\1 (1 2) (\\2 (2 1)))
15	$>f_{\omega +1}(10^{10^{19,727}})$	(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))
18	$>f_{{\omega }^{\omega }}(2\uparrow \uparrow 18)$	(\1 1 1) (\1 (1 (\\\1 3 2 (\\2 (2 1)))))
22	$>f_{{\omega }^{\omega +2}}(2)$	(\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))
23	$>f_{{\zeta }_0}(15)$	(\1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))
24	$>f_{\psi ({\mathrm{\Omega }}_{\omega })}(12)$	(\1 1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))
25	$>f_{\psi ({\mathrm{\Omega }}_{\omega })}(f_{{\omega }^{\omega +2}}(2))$	(\1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))
26	$>f_{\psi ({\mathrm{\Omega }}_{\omega }+1)}(4)$	(\1 1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))

Busy Beaver for SKI calculus

Busy Beaver for SKI calculus

	Ξ₀(n)	Champion
Ξ₀(1)	1	S
Ξ₀(2)	2	SS
Ξ₀(3)	3	SSS
Ξ₀(4)	4	SSSS
Ξ₀(5)	6	SSS(SS)
Ξ₀(6)	17	SSS(SI)S
Ξ₀(7)	≥ 41	SSS(S(SS))S
Ξ₀(8)	≥ 80	SSK(S(SS)S)S
Ξ₀(9)	≥ 169	S(SS)(SS)(SS)SS
Ξ₀(10)	≥ 376	S(S(SS(KK)))(SS)(SS)
Ξ₀(11)	≥ 912	S(SS)S(SS)(S(S(KS))S)
Ξ₀(12)	≥ 2↑2↑513*8-2	S(SI)(SSK)(S(S(KS)K)I)
Ξ₀(13)	> 2↑↑2↑↑2↑258	SI(SI)(SSK)(S(S(KS)K)I)
Ξ₀(14)	> 2↑↑2↑↑2↑258	SI(SI)(SSK)(S(S(KS)K)I)S
Ξ₀(15)	> 2↑↑2↑↑2↑↑34	S(S(SI)I(SSK))I(S(S(KS)K)I)
Ξ₀(16)	> 2↑↑↑19	S(S(SI))I(S(SSK)(K(S(S(KS)K)I)))
Ξ₀(17)	> 2↑↑↑↑2↑↑2↑↑258	S(SS(SSS)I)(SSK)(S(S(KS)K)I)
Ξ₀(18)	> 2↑↑↑↑2↑↑2↑↑258	S(SS(SSS)I)(SSK)(S(S(KS)K)I)S
Ξ₀(19)	> 2↑↑↑↑2↑↑2↑↑258	S(SS(SSS)I)(SSK)(S(S(KS)K)I)(SS)
Ξ₀(20)	> fω+1(2↑↑7/2-1) > Graham's Number	S(S(S(SI)))I(S(K(S(SS(K(K(S(S(KS)K)I))))))K)
Ξ₀(27)	> fω+2(2↑2↑21)	SI(SI(SI(S(S(K(SS(KI))))I)))(S(K(S(SSK)))K)(S(S(KS)K)I)
Ξ₀(28)	> fω+2(2↑2↑21)	S(SI(SI(SI(S(S(K(SS(KI))))I)))(S(K(S(SSK)))K)(S(S(KS)K)I))
Ξ₀(29)	> fω+3(2↑2↑21)	SI(SI(SI(SI(S(S(K(SS(KI))))I))))(S(K(S(SSK)))K)(S(S(KS)K)I)
Ξ₀(30)	> fω+3(2↑2↑21)	S(SI(SI(SI(SI(S(S(K(SS(KI))))I))))(S(K(S(SSK)))K)(S(S(KS)K)I))
Ξ₀(31)	> fω+4(2↑2↑21)	SI(SI(SI(SI(SI(S(S(K(SS(KI))))I)))))(S(K(S(SSK)))K)(S(S(KS)K)I)
Ξ₀(32)	> fω+4(2↑2↑21)	S(SI(SI(SI(SI(SI(S(S(K(SS(KI))))I)))))(S(K(S(SSK)))K)(S(S(KS)K)I))

Busy Beaver for SK calculus

	Ξ₀SK(n)	Champion
Ξ₀SK(n)(1)	1	S
Ξ₀SK(n)(2)	2	SS
Ξ₀SK(n)(3)	3	SSS
Ξ₀SK(n)(4)	4	SSSS
Ξ₀SK(n)(5)	6	SSS(SS)
Ξ₀SK(n)(6)	10	SSS(SS)S
Ξ₀SK(n)(7)	≥ 41	SSS(S(SS))S
Ξ₀SK(n)(8)	≥ 80	SSK(S(SS)S)S
Ξ₀SK(n)(9)	≥ 169	S(SS)(SS)(SS)SS
Ξ₀SK(n)(10)	≥ 376	S(S(SS(KK)))(SS)(SS)
Ξ₀SK(n)(11)	≥ 912	S(SS)S(SS)(S(S(KS))S)
Ξ₀SK(n)(12)	≥ 1530	S(SS)(SS)(SS(SS(SS)))S
Ξ₀SK(n)(13)	≥ 7811	S(SS)(SS)(SS(SS(SSS)))S
Ξ₀SK(n)(14)	≥ 2↑2097156*13-2	SSS(SSK)(SS(SK)(S(KS)K))
Ξ₀SK(n)(15)	≥ 2↑2↑2097156*3-2	SSS(SSK)(SS(SK)(S(KS)K))S
Ξ₀SK(n)(16)	> 2↑2↑2↑8193	S(SS(SS))(SSK)(SS(SK)(S(KS)K))
Ξ₀SK(n)(17)	> 2↑↑2↑↑2↑258	SSS(SKS)(SSK)(SS(SK)(S(KS)K))
Ξ₀SK(n)(18)	> 2↑↑33	S(SKS)(SSK)(SSK(SS(SK)(S(KS)K)))
Ξ₀SK(n)(22)	> 2↑↑↑19	S(S(S(S(SK))))(SKS)(S(SSK)(K(SS(SK)(S(KS)K)))
Ξ₀SK(n)(23)	> 2↑↑↑↑2↑↑2↑↑258	S(SS(SSS)(SKS))(SKS)(SSK)(SS(SK)(S(KS)K))
Ξ₀SK(n)(24)	> 2↑↑↑↑2↑↑2↑↑258	S(SS(SSS)(SKS))(SKS)(SSK)(SS(SK)(S(KS)K))S
Ξ₀SK(n)(25)	> 3↑↑↑↑3↑↑3↑↑3↑↑3↑↑3↑3↑3↑82 > 3↑↑↑↑3↑↑↑6	S(SS(SSS)(SKS))(SKS)(SSK)(SS(SS(SK))(S(KS)K))
Ξ₀SK(n)(26)	> fω+1(2↑↑7/2-1) > Graham's number	S(S(S(S(SKS))))(SKS)(S(K(S(SS(K(K(SS(SK)(S(KS)K)))))))K)

Busy Beaver for BCKW calculus

	Ξ₀BCKW(n)	Champion
Ξ₀BCKW(1)	1	B
Ξ₀BCKW(2)	2	BB
Ξ₀BCKW(3)	3	BBB
Ξ₀BCKW(4)	≥ 11	WW(WB)
Ξ₀BCKW(5)	≥ 47	W(WW)(WB)
Ξ₀BCKW(6)	≥ 2↑256*6-1	WW(WC(WB))
Ξ₀BCKW(7)	> 2↑↑↑17	WW(C(WC)(WB))
Ξ₀BCKW(8)	> 2↑↑↑2↑↑↑17	W(WW)(C(WC)(WB))
Ξ₀BCKW(9)	> 2↑↑↑2↑↑↑2↑↑↑17	W(W(WW))(C(WC)(WB))
Ξ₀BCKW(10)	> fω+1(2↑↑6/2-1)	WW(C(WB)(C(CC(WB))))

Brainfuck

	BB_brainf(n)	Champion
BB_brainf(1)	1	+
BB_brainf(2)	2	++
BB_brainf(3)	3	+++
BB_brainf(4)	4	++++
BB_brainf(5)	≥ 5	+++++
BB_brainf(6)	≥ 6	++++++
BB_brainf(7)	≥ 7	+++++++
BB_brainf(8)	≥ 8	++++++++
BB_brainf(9)	≥ 9	+++++++++
BB_brainf(10)	≥ 10	++++++++++
BB_brainf(11)	≥ 11	+++++++++++
BB_brainf(12)	≥ 12	++++++++++++
BB_brainf(13)	≥ 16	++++[->++++<]
BB_brainf(14)	≥ 20	+++++[->++++<]
BB_brainf(15)	≥ 25	+++++[->+++++<]
BB_brainf(16)	≥ 30	++++++[->+++++<]
BB_brainf(17)	≥ 36	++++++[->++++++<]
BB_brainf(18)	≥ 42	+++++++[->++++++<]
BB_brainf(19)	≥ 49	+++++++[->+++++++<]
BB_brainf(20)	≥ 56	++++++++[->+++++++<]
BB_brainf(21)	≥ 64	++++++++[->++++++++<]
BB_brainf(22)	≥ 72	+++++++++[->++++++++<]
BB_brainf(23)	≥ 81	+++++++++[->+++++++++<]
BB_brainf(24)	≥ 90	+++++++++[->++++++++++<]
BB_brainf(25)	≥ 340	++++[>+[->++<]>[-<++>]<<]
BB_brainf(26)	≥ 1364	+++++[>+[->++<]>[-<++>]<<]

Boolfuck

	BB_boolf(n)	Champion
BB_boolf(1)	1	*
BB_boolf(2)	1	*>
BB_boolf(3)	2	*>*
BB_boolf(4)	2	*>*>
BB_boolf(5)	3	*>*>*
BB_boolf(6)	3	*>*>*>
BB_boolf(7)	4	*>*>*>*

CounterScript

Macros for compression:

	Macro	Definition	Size	Function
Constant Addition	A+=n;	increment A by n	n	A++; repeated n times
Transfer	A>>B*n;	increment B by A*n then set A to 0	n+2	while A {A--; (B++; repeated n times)}

	BBCS(n)	Champion
BBCS(1)	1	A++;
BBCS(2)	2	A++; A++;
BBCS(3)	3	A++; A++; A++;
BBCS(4)	4	A++; A++; A++; A++;
BBCS(5)	5	A++; A++; A++; A++; A++;
BBCS(6)	6	A++; A++; A++; A++; A++; A++;
BBCS(7)	7	A++; A++; A++; A++; A++; A++; A++;
BBCS(8)	9	A++; A++; A++; while A {A--; B++; B++; B++;}
BBCS(9)	12	A++; A++; A++; A++; while A {A--; B++; B++; B++;}
BBCS(10)	16	A++; A++; A++; A++; while A {A--; B++; B++; B++; B++;}
BBCS(11)	≥ 20	A++; A++; A++; A++; A++; while A {A--; B++; B++; B++; B++;}
BBCS(12)	≥ 25	A++; A++; A++; A++; A++; while A {A--; B++; B++; B++; B++; B++;}
BBCS(13)	≥ 42	A++; while A {A++; while B {A--; B--; C++; C++;} C>>B*2; C++;}
BBCS(14)	≥ 129	A++; while A {A++; while B {A--; B--; C++; C++;} C>>B*3; C++;}
BBCS(15)	≥ 340	A++; A++; A++; A++; while A {A--; B++; while B {B--; C++; C++;} while C {C--; B++; B++;}}
BBCS(16)	≥ 1,554	A++; A++; A++; A++; while A {A--; B++; while B {B--; C++; C++; C++;} while C {C--; B++; B++;}}
BBCS(17)	≥ 9,330	A++; A++; A++; A++; A++; while A {A--; B++; while B {B--; C++; C++; C++;} while C {C--; B++; B++;}}
BBCS(18)	≥ 66,429	A++; A++; A++; A++; A++; while A {A--; B++; while B {B--; C++; C++; C++;} while C {C--; B++; B++; B++;}}
BBCS(19)	≥ (3^122 - 3)/2	A++; while A {A++; while B {A--; B--; C++;} while C {B++; C--; B>>D*3; D>>B;} C++;}
BBCS(20)	≥ 2^65536 - 2	A+=4; while A {A--; B++; while B {B--; C++; C>>D*2; D>>C;} C>>B;}
BBCS(21)	≥ 2^2^65536 - 2	A+=5; while A {A--; B++; while B {B--; C++; C>>D*2; D>>C;} C>>B;}
BBCS(22)	≥ 2^^7 - 2	A+=6; while A {A--; B++; while B {B--; C++; C>>D*2; D>>C;} C>>B;}
BBCS(23)	≥ 2^^8 - 2	A+=7; while A {A--; B++; while B {B--; C++; C>>D*2; D>>C;} C>>B;}
BBCS(24)	≥ 2^^65536 - 2	A++; while A {A++; while B {A--; B--; C++;} while C {B++; C--; while B {B--; D++; D>>E*2; E>>D;} D>>B;} C++;}
BBCS(25)	> 2^^2^^6	A++; while A {A++; while B {A--; B--; C+=2;} while C {B++; C--; while B {B--; D++; D>>E*2; E>>D;} D>>B;} C++;}
BBCS(26)	≥ 2^^2^^65536 - 2	A++; A++; A++; A++; while A {A--; B++; while B {B--; C++; while C {C--; D++; while D {D--; E++; E++;} while E {E--; D++;}} while D {D--; C++;}} while C {C--; B++;}}
BBCS(64)	> Graham's number	too large to show
BBCS(85)	> q(5)	too large to show, see Discord

Fractran (BBf(n))

n	BBf(n)	Example Champion	Vector Representation
BBf(2)	1	[1/2]	$\left[\begin{array}{c}-1\end{array}\right]$
BBf(3)	1	[3/2]	$\left[\begin{array}{cc}-1& 1\end{array}\right]$
BBf(4)	1	[9/2]	$\left[\begin{array}{cc}-1& 2\end{array}\right]$
BBf(5)	2	[3/2, 1/3]	$\left[\begin{array}{cc}-1& 1\\ 0& -1\end{array}\right]$
BBf(6)	3	[9/2, 1/3]	$\left[\begin{array}{cc}-1& 2\\ 0& -1\end{array}\right]$
BBf(7)	4	[27/2, 1/3]	$\left[\begin{array}{cc}-1& 3\\ 0& -1\end{array}\right]$
BBf(8)	5	[81/2, 1/3]	$\left[\begin{array}{cc}-1& 4\\ 0& -1\end{array}\right]$
BBf(9)	6	[243/2, 1/3]	$\left[\begin{array}{cc}-1& 5\\ 0& -1\end{array}\right]$
BBf(10)	7	[729/2, 1/3]	$\left[\begin{array}{cc}-1& 6\\ 0& -1\end{array}\right]$
BBf(11)	10	[27/2, 25/3, 1/5]	$\left[\begin{array}{ccc}-1& 3& 0\\ 0& -1& 2\\ 0& 0& -1\end{array}\right]$
BBf(12)	13	[81/2, 25/3, 1/5]	$\left[\begin{array}{ccc}-1& 4& 0\\ 0& -1& 2\\ 0& 0& -1\end{array}\right]$
BBf(13)	17	[81/2, 125/3, 1/5]	$\left[\begin{array}{ccc}-1& 4& 0\\ 0& -1& 3\\ 0& 0& -1\end{array}\right]$
BBf(14)	21	[243/2, 125/3, 1/5]	$\left[\begin{array}{ccc}-1& 5& 0\\ 0& -1& 3\\ 0& 0& -1\end{array}\right]$
BBf(15)	28	[1/45, 4/5, 3/2, 25/3]	$\left[\begin{array}{ccc}0& -2& -1\\ 2& 0& -1\\ -1& 1& 0\\ 0& -1& 2\end{array}\right]$
BBf(16)	53	[1/45, 4/5, 3/2, 125/3]	$\left[\begin{array}{ccc}0& -2& -1\\ 2& 0& -1\\ -1& 1& 0\\ 0& -1& 3\end{array}\right]$
BBf(17)	107	[5/6, 49/2, 3/5, 40/7]	$\left[\begin{array}{cccc}-1& -1& 1& 0\\ -1& 0& 0& 2\\ 0& 1& -1& 0\\ 3& 0& 1& -1\end{array}\right]$
BBf(18)	211	[5/6, 49/2, 3/5, 80/7]	$\left[\begin{array}{cccc}-1& -1& 1& 0\\ -1& 0& 0& 2\\ 0& 1& -1& 0\\ 4& 0& 1& -1\end{array}\right]$
BBf(19)	370	[5/6, 49/2, 3/5, 160/7]	$\left[\begin{array}{cccc}-1& -1& 1& 0\\ -1& 0& 0& 2\\ 0& 1& -1& 0\\ 5& 0& 1& -1\end{array}\right]$
BBf(20)	746	[7/15, 22/3, 6/77, 5/2, 9/5]	$\left[\begin{array}{ccccc}& -1& -1& +1& \\ +1& -1& & & +1\\ +1& +1& & -1& -1\\ -1& & +1& & \\ & +2& -1& & \end{array}\right]$
BBf(21)	≥ 31,957,632	[7/15, 4/3, 27/14, 5/2, 9/5]	$\left[\begin{array}{cccc}0& -1& -1& 1\\ 2& -1& 0& 0\\ -1& 3& 0& -1\\ -1& 0& 1& 0\\ 0& 2& -1& 0\end{array}\right]$
BBf(22)	$>1.146\times 10^{62}$	[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]	$\left[\begin{array}{ccccc}-2& -1& 0& 0& 0\\ -1& 2& -1& 0& 0\\ 1& -1& 0& 1& 0\\ -1& 0& 0& 0& 1\\ 0& 0& 1& -1& 0\\ 0& 1& 0& 0& -1\end{array}\right]$
BBf(23)	$>4.393\times 10^{124}$	[10/3, 9/14, 5/4, 121/2, 7/5, 3/11]	$\left[\begin{array}{ccccc}1& -1& 1& 0& 0\\ -1& 2& 0& -1& 0\\ -2& 0& 1& 0& 0\\ -1& 0& 0& 0& 2\\ 0& 0& -1& 1& 0\\ 0& 1& 0& 0& -1\end{array}\right]$
BBf(24)	$>9.263\times 10^{9595}$	[18/35, 1/10, 11/5, 75/2, 49/3, 5/11]	$\left[\begin{array}{ccccc}1& 2& -1& -1& 0\\ -1& 0& -1& 0& 0\\ 0& 0& -1& 0& 1\\ -1& 1& 2& 0& 0\\ 0& -1& 0& 2& 0\\ 0& 0& 1& 0& -1\end{array}\right]$

Cyclic Tag (CTBB(n))

	Runtime	Champions
CTBB(2)	5
CTBB(3)	> 38
CTBB(4)	≥ 672
CTBB(5)	≥ 2^2^2^2^182
CTBB(6)	> 2↑↑↑131
CTBB(7)	> 4↑↑↑↑(4↑↑↑3)

Tiny Tag

n	Runtime	Champions
1	1	0
2	5	10
3	9	101
4	13	1001
5	15	10101
6	334	010111
7	404	1010111
8	670	11100101
9	12584	001101110
10	2180995	0100011110

Minsky Machines (MBB(n))

Domain	Halting Time	Champion
MBB(1)	1	0+Z
MBB(2)	3	0+B_0-B*
MBB(3)	5	0+B_0+C_0-C*
MBB(4)	10	0+B_1+C_0-BD_1-C*
MBB(5)	24	0-DB_0+C_1-ED_1+A_1-B*
MBB(6)	49	0+B_1-FC_1+D_0-CE_0+A_1-A*
MBB(7)	≥ 231	0+B_0+C_0+D_1-GE_1+F_0-EC_1-A*
MBB(8)	≥ 3394	0+B_0+C_1-GD_1+E_0-DF_2-HG_2+A_2-D*
MBB(9)	≥ 9870	0+B_0+C_0+D_1-IE_1+F_0-GI_0-HC_0-E*_0+A

Pebble Automation (peBBle(n))

n	peBBle(n)	Champion
peBBle(1)	0	00
peBBle(2)	1	00_01
peBBle(3)	3	00_01_20
peBBle(4)	6	00_01_20_03
peBBle(5)	≥ 11	00_11_03_21_04
peBBle(6)	≥ 17	00_11_03_21_40_15
peBBle(7)	≥ 28	00_11_03_21_10_33_61
peBBle(8)	≥ 63	00_11_03_21_50_30_24_70
peBBle(9)	≥ 63577	00_02_01_40_14_12_00_52_81

Wang Tiles (BB_WT(n))

n	BBWT(n)	Champion	Image
BBWT(1)	1	[1,0,0,0]
BBWT(2)	2	[1,0,2,0] [2,0,3,0]
BBWT(3)	3	[1,0,2,0] [2,0,3,0] [3,0,4,0]
BBWT(4)	4	[1,0,2,0] [2,0,3,0] [3,0,4,0] [4,0,5,0]
BBWT(5)	≥ 6	[1,0,2,0] [2,0,3,0] [3,0,4,0] [4,6,1,5] [4,5,2,6]
BBWT(6)	≥ 8	[1,0,2,0] [2,0,3,0] [3,0,4,0] [4,0,5,0] [5,7,1,6] [5,6,2,7]
BBWT(7)	≥ 11
BBWT(8)	≥ 14
BBWT(9)	≥ 17
BBWT(10)	≥ 212

Turmites

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

Where n is the amount of states and k is the amount of symbols.

Maximum steps for TT(n,k)

Domain	Runtime	Champions
TT(2)	≥ 13	1TB---_1PA0PB
TT(3)	≥ 82	1PB0PA_1TA0PC_1PA---
TT(4)	≥ 48,186	1TB1PA_1PC0PA_1TA0PD_---1TA
TT(2,3)	≥ 223	1TB0PA2PA_2PA---1PA
TT(3,3)	> $2.27\times 10^{248}$	1PB0TC1PA_2PC1PZ2PA_2TC0PA2PC
TT(2,4)	> 3.467*1015	1TA2PB3TB---_3TA1PB1TA1PA

Maximum score for TT(n,k)

Domain	Score	Champion
TT(3,3)	≥ 32,778	1PB2PA0PC_2TB1PC2PB_2PB1PZ0PA

2D Turmites (turNing machines)

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

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

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#
##########