Latest revision as of 17:06, 12 November 2026

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))

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	Reference
BB(1)	1	`1RZ---` (bbch)
BB(2)	6	`1RB1LB_1LA1RZ` (bbch) `1RB0LB_1LA1RZ` (bbch) `1RB1RZ_1LB1LA` (bbch) `1RB1RZ_0LB1LA` (bbch) `0RB1RZ_1LA1RB` (bbch)	^[1]
BB(3)	21	`1RB1RZ_1LB0RC_1LC1LA` (bbch)	^[1]
BB(4)	107	`1RB1LB_1LA0LC_1RZ1LD_1RD0RA` (bbch)	^[1]
BB(5)	47,176,870	`1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA` (bbch)	^[1]
BB(6)	> 10 ↑↑ 10 ↑↑ 10 ↑↑ 8.10237	`1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE` (bbch)	^[2]
BB(7)	$> 3 ↑^{20} 3 > f_{ω} (20)$	`1RB0RG_1RC1RB_0RD1RZ_0LE0LD_0LF1LE_1RA1LF_1LC0RB` (bbch)	^[3]
BB(8)	$> f_{ω}^{3} (12)$	`1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LH0LC_1RG1LD_0RG0RF_1RG1RZ` (bbch)	^[4]
BB(9)	$> f_{ω}^{7} (12)$	`1RB0RA_1LC1LF_1RD0LB_1RA1LE_0LH0LC_1RG1LD_0RG0RF_0LI1RZ_1RI1RC` (bbch)	^[5]
BB(10)	$> f_{ω + 1} (2 ↑^{36} 3) > g_{64}$	`1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LI0LC_0RG1LD_1RH1LG_1LC0RG_0LJ0LI_1RZ0LC` (bbch)	^[6]
BB(11)	$> f_{ω + 1} (f_{ω}^{2} (5))$	`1RG0LH_1RK1RC_0LD1LA_0RE1LD_0LI1RF_1RZ0RG_1RB1RG_1LA1LH_0LK0LJ_1LH1LI_1LJ1RE` (bbch)	^[7]
BB(12)	$> f_{ω + 2} (f_{ω + 1}^{2} (3))$	`1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LK1RF_1RG0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_1RZ0LL_0LH1LK` (bbch)	^[8]
BB(13)	$> f_{ω + 2}^{6} (f_{ω + 1}^{2} (3))$	`1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LK1RF_1RM0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_0LL0LL_0LH1LK_1RZ0LH` (bbch)	^[9]
BB(14)	$> f_{ω + 3} (30)$	`1RB1RA_1RI1RC_0LD1LG_0RE1LD_0RN1RF_1RM0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_0LL0LL_0LH1LK_0LK0LH_1LM1RZ` (bbch)	^[10]
BB(15)	$> f_{ω \times 2} (13)$	`1LH0LG_0LI1RC_0LD1LA_0RE1LD_0RH1RL_1RB1RF_1LA1LG_1RF0RO_1LJ1LI_0LK1LI_1RL1RE_0RL0RF_0LA0RN_0LA1RO_1RZ1RM` (bbch)	^[11]
BB(16)	$> f_{ω \times 2} (f_{ω} (14))$	`1LG1LA_0LI1RC_0LD1LG_0RE1LD_0RM1RL_1RB1RF_1LH0LA_1RF0RN_1LJ1LI_0LK1LI_1RL1RE_0RL0RF_0LG0RN_0RP1RO_1RZ1RM_1LB1LI` (bbch)	^[12]
BB(17)
BB(18)	$> f_{ω \times 2} (f_{ω}^{9} (35))$	`0LB1LF_0RC1LB_0RD1RM_1LE0RN_1LF1LE_1LG0LE_1RH1RP_1RI1RH_0LJ1RA_1LK1LJ_0LL1LJ_1RM1RC_0RM0RH_1LE1RO_1RZ1RD_0LF0RQ_0LH1RR_0LF1RP` (bbch)	^[13]
BB(19)
BB(20)	$> f_{ω \times 2} (f_{ω + 2}^{2} (f_{10} (2)))$	`1LH0LG_0LI1RC_0LD1LA_0RE1LD_1RQ1RL_1RB1RF_1LA1LG_1RF1RM_1LJ1LI_0LK1LI_1RL1RE_0RL0RF_0LN1RM_0RF0LO_0RF1LP_0RF1LN_1LR1RQ_1RZ1LS_0LF1LT_0LF0LR` (bbch)	^[14]
BB(21)	$> f_{ω^{2}}^{3} (4 ↑ ↑ 341)$	`0LI0LF_0RJ1RG_0RD1LC_1RH1RE_1LO0RH_1LA1LF_0LC1LA_1RB1RH_1RD0LF_1LP0RK_0LM1LL_0LL1LM_1LE1LN_0RQ0LM_0RP1LO_1LR0RH_1LF1RQ_1RZ0LS_1LH0LT_1LH1LU_0LE1LR` (bbch)	^[15]
BB(25)	$> f_{ω^{2}}^{5} (4 ↑ ↑ 341)$	`0LI0LF_0RJ1RG_0RD1LC_1RH1RE_1LO0RH_1LA1LF_0LC1LA_1RB1RH_1RD0LF_1LP0RK_0LM1LL_0LL1LM_1LE1LN_0RQ0LM_0RP1LO_1LR0RH_1LF1RQ_1RZ0LS_1LH0LT_1LH1LU_0LE1LV_0LH0LW_1LH0LX_1LH1LY_0LE1LR` (bbch)	^[16]
BB(29)	$> f_{ω^{ω}} (3)$	[1]	^[17]
BB(31)	$> f_{ω ↑ ↑ 4} (3)$	[2]	^[18]
BB(33)	$> f_{ω ↑ ↑ 6} (3)$	[3]	^[19]
BB(48)	$> f_{ε_{0}} (f_{ω + 1} (f_{ω^{ε_{0} + 1}} (f_{ω^{2}} (f_{ω^{ω^{ε_{0} + 1}}} (f_{ω^{3}} (f_{ε_{1}} (3))))))) > f_{ε_{1}} (3)$	[4]	^[20]
BB(150)	$> f_{l i m (B M S)} (10 ↑ ↑ 15)$	too large to show	^[21]
BB(1015)	$> D^{5} (99)$ = Loader's Number


3 Symbols:	Runtime	Champions	Reference
BB(1,3)	1	`1RZ------` (bbch)
BB(2,3)	38	`1RB2LB1RZ_2LA2RB1LB` (bbch)	^[1]
BB(3,3)	$\geq 119 112 334 170 342 541 > 1 0^{17}$	`0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC` (bbch)
BB(4,3)	$> 2 ↑ ↑ ↑ 2 ↑ ↑ ↑ 2 ↑ ↑ ↑ (7.92 \times 1 0^{28})$	`1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD` (bbch)	^[22]
BB(6,3)	$> f_{ω}^{2} (10 ↑ ↑ ↑ 6)$	`1LE2LD2LC_2RB1RB1RA_2RB1LC0LC_1RB1LD2LD_2RD2LE2LF_1RZ---1LE` (bbch)
BB(8,3)	$> f_{ω + 1}^{4} (f_{ω}^{3} (14)) > g_{64}$	`2LH1LE2RA_0LE2LE2RB_1RF1RC2RB_0RC0LD0RH_1RC2LE1LG_0LA0LD1RZ_1LD2RA2LG_2RD1LF1LG` (bbch)	^[23]
BB(9,3)	$> f_{ω + 2}^{3} (f_{ω + 1}^{2} (f_{ω} (14)))$	`2LI1LE2RA_0LE2LE2RB_1RF1RC2RB_0RC0LD2RA_1RC2LE1LG_0LA0LD1RH_1LD2RA2LG_1RZ2LA2LI_1RD1LI2LH` (bbch)
BB(10,3)	$> f_{ω \times 2} (11)$	`1LD2RG2LA_2LE2LH2RB_1LI1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RZ---_0RC1LH2RG_1RC2LH1LA_1RJ2LI2LB_0RF1LA0RI` (bbch)
BB(11,3)	$> f_{ω \times 2} (46)$	`1LD2RG2LA_2LE2LH2RB_1LI1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RZ---_0RC1LH2RG_1RC2LH1LA_1RJ2LI2LB_0RF2LK0RI_2RG2RK1LK` (bbch)
BB(12,3)	$> f_{ω \times 2} (f_{ω + 2}^{2} (4))$	`1LD2RG2LA_2LE2LH2RB_1RK1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RI0RL_0RC1LH2RG_1RC2LH1LA_0RI2RI2RJ_0LJ1RZ1LA_1LK0LL2LA_2LL0LE2LB` (bbch)
BB(13,3)	$> f_{ω \times 2}^{3} (f_{ω + 1} (f_{ω}^{2} (5)))$	`1LD2RG2LA_2LE2LH2RB_1RK1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RI0LM_0RC1LH2RG_1RC2LH1LA_0RI2RI2RJ_0LJ1RZ1LA_1LK0LL2LB_2RG2RE2RL_1LL0RL0LK` (bbch)
BB(36,3)	$> f_{ε_{0}} (7)$
BB(37,3)	$> f_{ε_{0}} (373 676 378)$


4 Symbols:	Runtime	Champions	Reference
BB(1,4)	1	`1RZ---------` (bbch)
BB(2,4)	3,932,964	`1RB2LA1RA1RA_1LB1LA3RB1RZ` (bbch)	^[1]
BB(3,4)	$> (2 ↑^{15} 5) + 14$	`1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC` (bbch)	^[24]
BB(4,4)	$> 2 ↑^{509} 3 > f_{ω} (508)$	`0LB3LB1RZ2RA_1LD3RB1LC2RA_3RB1LB3LC2RC_0RD2RC2LB0RA` (bbch)	^[25]
BB(5,4)	$> f_{ω}^{3} (2 ↑^{15} 5)$	`1RB3LB1RD2RA_2LC3RB1LC2RA_3RB1LB3LC2RC_1LE2LE3LE1RZ_3RC2LE1LE3LE` (bbch)	^[26]
BB(6,4)	$> f_{ω + 1} (10 ↑ ↑ 65 533) > g_{64}$	`1LF2RA3RB---_3RD2LB3RB1LC_2RA1LB3RC2LC_1LD2RE0LA1LD_1RZ0LE1LF---_2RA---2LF2LF` (bbch)
BB(7,4)	$> f_{ω + 3} (21)$	`3RD1LE2RA1LF_0LE2LE2RB2RD_1RF1RC2RB0LB_0RC0LD3RC2RA_1RC2LE1LG2RF_0LA0LD3LG1RZ_1LD2RA2LG0RB` (bbch)
BB(8,4)	$> f_{ω + 8}^{2} (4)$	`1LD2RG2LA---_2LE2LH2RB1LF_2LA1RC2RB---_0RC1RD0LG1RH_0LF1LE1LE3LB_1RG1RZ3RD3LF_0RC1LH2RG0LC_1RC2LH1LA2LB` (bbch)	^[27]
BB(9,4)	$> f_{ω^{2}} (18)$	`1LD2RG2LA3LG_1LI2LH2RB2RF_1LG1RC2RB3LG_0RC1RC1LD3RD_3RD1LE1LE3LE_1RZ3LD2LE3LE_1RG1LH2RG3LG_1RC2LH1LA1LE_0RC3LI1RI0RI` (bbch)	^[28]
BB(10,4)	$> f_{ω^{2}}^{3} (3)$	`1LD2RG2LA3LG_0LI2LH2RB2RF_1LB1RC2RB3LG_0RC1RC1LD3RD_3RD1LE1LE3LE_1RI3LD2LE3LE_1RG1LH2RG3LG_1RC2LH1LA1LE_0LJ1RZ0RE0LI_3RC3LJ3LJ3LJ` (bbch)
BB(31,4)	$> f_{ε_{0}} (373 676 378)$


5 Symbols:	Runtime	Champions	Reference
BB(1,5)	1	`1RZ------------` (bbch)
BB(2,5)	$> 1 0^{1 0^{1 0^{3 314 360}}}$	`1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ` (bbch)
BB(3,5)	$> f_{ω} (2 ↑^{15} 5) > f_{ω}^{2} (15)$	`1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC` (bbch)
BB(4,5)	$> f_{ω + 1} (27 041) > g_{64}$	`1RB3LB4LC2RA4LB_2LC3RB1LC2RA4RD_3RB1LB3LC2RC4LC_2RC2RD1RZ4RB1LC` (bbch)	^[29]
BB(5,5)	$> f_{ω + 1} (f_{ω}^{4} (6))$	`1LB3LB4LC2RA4LB_3LE3RB1LC2RA4RD_3RB1LB3LC2RC4LC_1RZ------4LD4LE_3RB------4LE3LC` (bbch)	^[30]
BB(6,5)	$> f_{ω + 3} (11)$	`2LE3LB4LC2RA4LB_4RA3RB1LC2RA4RD_3RB1LB3LC2RC4LC_1RZ3LF1LA4LD4LE_3RB---0RD4LE3LC_4LC------3LF3LF` (bbch)	^[31]
BB(7,5)	$> f_{ω + 3}^{2} (10)$	`2LE3LB4LC2RA4LB_4RA3RB1LC2RA4RD_3RB1LB3LC2RC4LC_1RZ3LF1LG4LD4LE_3RB---1LE4LE3LC_4LC------3LF3LF_3LF------1LG1LG` (bbch)	^[32]
BB(8,5)	$> f_{ω^{2}} (13)$	`1LD2RG2LA3LG---_4LC2LH2RB2RF0RB_4LE1RC2RB3LG4RB_0RC1RC1LD3RD3LA_3RD1LE1LE3LE4RE_1RZ3LD2LE3LE4RD_1RG1LH2RG3LG3LF_1RC2LH1LA1LE---` (bbch)	^[33]
BB(17,5)	$> f_{ε_{0}} (20)$	`4RF---0RF0RD1LQ_1RZ1RB2LC------_0LI4RC2RC3RC2RC_1LE1RD2RD3RD---_3LA1LE2LE3LE1LG_0LG1RF2RF3RF---_0RI0LH4RD3RI2LH_2LA1LH2LH3LH0RC_3RN---0LA0LI2RF_2RO---2LJ3LJ---_2LL---2RK3RK4RK_3RO2RO2LL3LL4LL_3LL---2RM3RM4RM_4LJ1LN2LN3LN1LO_1RB4RP1RK0RM4LN_4LJ1RP2RP3RP0RP_1RB1LQ2LQ3LQ1LQ` (bbch)	^[34]
BB(19,5)	$> f_{ε_{0}} (1.7 \cdot 1 0^{352})$	`1RB2LA1RA2LB2LA_0LA2RB3RB4RA1LQ_1RZ1RC2LD------_0LK3RD2RD2RD4RD_------0RH1LS0RF_1LG1RF2RF---4RF_4LE1LG2LG---4LG_0LI1RH2RH---4RH_0RK0LJ------4RK_2LE1LJ2LJ---4LJ_4RP---0LE---0LK_2RQ---2LL---4LL_2LN---2RM3RM4RM_4RQ2RQ2LN3LN4LN_4LN---2RO3RO4RO_3LL1LP2LP1LQ4LP_1RC3RR1RM3LP0RO_3LL1RR2RR0RR4RR_1RC1LS2LS1LS4LS` (bbch)
BB(94,5)	$> f_{ψ_{0} (Ω_{2})} (1.7 \cdot 1 0^{352})$


6 Symbols:	Runtime	Champions	Reference
BB(1,6)	1	`1RZ---------------` (bbch)
BB(2,6)	$> 10 ↑ ↑ 10 ↑ ↑ 1 0^{1 0^{115}}$	`1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA` (bbch)	^[35]
BB(3,6)	$> f_{ω}^{3} (15)$	`1RB3LB4LC2RA4LB5LB_2LC3RB1LC2RA5LC1RZ_3RB1LB3LC2RC4LC5LC` (bbch)
BB(4,6)	$> f_{ω + 2} (29)$	`5LD3LB5LC2RA4LB4LC_1RZ3RB1LC2RA3RD0LA_3RB1LB3LC2RC4LC1RA_3RB0LA2LC4LD4LC4LD` (bbch)	^[36]
BB(5,6)	$> f_{ω + 3} (31)$	`1LE3LB2LA2RA4LB5RA_1RZ3RB1LC2RA3RD4RA_3RB1LB3LC2RC4LC2LB_3RB0LA5LE4LD4LC4LD_4LC5LC0RE5LE1RE2RE` (bbch)	^[37]
BB(6,6)	$> f_{ω \times 2} (89)$	`4LF3LB1RZ2RA4LB---_2LC3RB1LD2RA3LF---_3RB1LB3LC2RC4LC0LE_3RB---3LC2RC---3RE_5RC3RE---4LC------_5RB0LF1RB1LF---5RB` (bbch)	^[38]
BB(7,6)	$> f_{ω^{2}} (110)$	`5LG3LB---2RA4LB0LC_4LG3RB1LD2RA3LF5RB_3RB1LB3LC2RC4LC1LE_3RB3RE3LC2RC---3RD_5RB3RE0LE4LC3LG5LE_5RB5RB2LB1LF---5LF_5LF---0LG1LG---5LG` (bbch)	^[39]
BB(8,6)	$> f_{ω^{2} + 1} (f_{ω} (31))$	`5LH3LB2LH2RA4LB1RZ_2RC3RB1LD2RA3LF5RB_3RB1LB3LC2RC4LC1LE_3RB3RE3LC2RC---3RD_5RB3RE1LE4LC3LG5LE_5RB5RB---1LF5LF5LF_5LF4RG1LB1LG2LG5LG_0LC---1LH1LH---5RG` (bbch)	^[40]
BB(61,6)	$> f_{ψ_{0} (Ω_{2})} (1.7 \cdot 1 0^{352})$


7 Symbols:	Runtime	Champions	Reference
BB(2,7)	$> 10 ↑ ↑ 10 ↑ ↑ 10 ↑ ↑ 155$	`1RB3RB4LA1LB5LA6LB5RA_2LA6RA4RB1RZ3LB6LA4RB` (bbch)	^[41]
BB(3,7)	$> f_{ω + 1} (117) > g_{64}$	`6LC3LB6RA2RA4LB5LC5LA_1RZ3RB1LC2RA3RB4LC5RA_3RB1LB3LC2RC4LC1RA6LA` (bbch)	^[42]
BB(4,7)	$> f_{ω + 3} (34)$	`5LC3LB6LA2RA4LB2LD3RB_1RZ3RB1LC2RA3RD2LC4LC_3RB1LB3LC2RC4LC1RA5LD_3RB5LD6LC4LD4LC4LD6LC` (bbch)	^[43]
BB(18,7)	$> f_{ε_{0}} (1.7 \cdot 1 0^{352})$	`1RB2LA1RA2LB2LA------_0LA2RB3RB4RA1LP------_0RZ1RC2LD------------_0LJ3RD2RD2RD4RD------_4LE1LE0RG1LR0RF2LE4LE_1LE1RF5RF---6RF------_0LH1RG2RG---4RG------_0RJ0LI------4RJ------_2LE1LI2LI---4LI------_4RO---0LE---0LJ------_2RP---2LK---4LK------_2LM---2RL3RL4RL------_4RP2RP2LM3LM4LM------_4LM---2RN3RN4RN------_3LK1LO2LO1LP4LO------_1RC3RQ1RL3LO0RN------_3LK1RQ2RQ0RQ4RQ------_1RC1LR2LR1LR4LR------` (bbch)
BB(46,7)	$> f_{ψ_{0} (Ω_{2})} (1.7 \cdot 1 0^{352})$
BB(134,7)	$> f_{ψ_{0} (Ω_{ω})} (2050)$


8 Symbols or more:	Runtime	Champions	Reference
BB(2,8)	$> 10 ↑ ↑ 10 ↑ ↑ 10 ↑ ↑ 10 ↑ ↑ 155$	`1RB3RB4LA1LB5LA6LB5RA1RZ_2LA6RA4RB7RA3LB6LA4RB4RA` (bbch)	^[44]
BB(17,8)	$> f_{ε_{0}} (1.7 \cdot 1 0^{352})$	`1RB2LA1RA2LB2LA---------_0LA2RB3RB4RA1LO---------_0RZ1RC2LD---------------_0LJ3RD2RD2RD4RD---------_4LE1LE0RG1LQ0RF2LE4LE---_1LE1RF5RF---6RF---------_0LH1RG2RG---4RG---------_0RJ0LI------4RJ---------_2LE1LI2LI---4LI---------_4RN---0LE---0LJ---------_2RO2RO2LK3LK4LK2LK4LK4RO_2LK---5RL3RL6RL---------_4LK---5RM3RM6RM---------_3LK1LN2LN1LO4LN---------_1RC3RP1RL3LN7RM---------_3LK1RP2RP0RP4RP---------_1RC1LQ2LQ1LQ4LQ---------` (bbch)
BB(2,9)	$> 10 ↑ ↑ 10 ↑ ↑ 10 ↑ ↑ 10 ↑ ↑ 10 ↑ ↑ 155$	`1RB3RB4LA1LB5LA6LB5RA8RA1RZ_2LA6RA4RB7RA3LB6LA4RB4RA4RA` (bbch)	^[45]
BB(42,9)	$> f_{ψ_{0} (Ω_{2})} (1.7 \cdot 1 0^{352})$

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	Reference
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)	^[2]
Σ(7)	$> 3 ↑^{20} 3 > f_{ω} (20)$	`1RB0RG_1RC1RB_0RD1RZ_0LE0LD_0LF1LE_1RA1LF_1LC0RB` (bbch)	^[3]


3 Symbols:	Score	Champions	Reference
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)	$> 2 ↑ ↑ ↑ 2 ↑ ↑ ↑ 2 ↑ ↑ ↑ (7.92 \times 1 0^{28})$	`1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD` (bbch)	^[22]


4 Symbols:	Score	Champions	Reference
Σ(1,4)	1	`1RZ---------` (bbch)
Σ(2,4)	2050	`1RB2LA1RA1RA_1LB1LA3RB1RZ` (bbch)
Σ(3,4)	$\geq (2 ↑^{15} 5) + 14$	`1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC` (bbch)	^[24]


5 Symbols:	Score	Champions	Reference
Σ(1,5)	1	`1RZ------------` (bbch)
Σ(2,5)	$> 1 0^{1 0^{1 0^{3 314 360}}}$	`1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ` (bbch)


6 Symbols:	Score	Champions	Reference
Σ(1,6)	1	`1RZ---------------` (bbch)
Σ(2,6)	$> 10 ↑ ↑ 10 ↑ ↑ 1 0^{1 0^{115}}$	`1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA` (bbch)	^[35]

Beeping Busy Beavers

Beeping Busy Beaver (BBB(n,m))

Considering beep states


2 Symbols:	Steps taken	Champions	Reference
BBB(1)	1		^[46]
BBB(2)	6	`1RB1LB_1LB1LA` (bbch)	^[46]
BBB(3)	55	`1RB0LB_1LA0RC_1LC1LA` (bbch)
BBB(4)	32,779,478	`1RB1LD_1RC1RB_1LC1LA_0RC0RD` (bbch)	^[47]
BBB(5)	$\geq 1 0^{14006}$	`1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA` (bbch)	^[48]


3 Symbols:	Steps taken	Champions	Reference
BBB(1,3)
BBB(2,3)	59	`1RB2LB1LA_2LB2RA0RA` (bbch)	^[49]
BBB(3,3)	≥ 10 ↑↑ 6.74	`1RB0LB2LA_1LA0RC0LB_2RC2RB0LC` (bbch)	^[50]


4 Symbols:	Steps taken	Champions	Reference
BBB(1,4)
BBB(2,4)	$\geq 205 770 076 433 044 242 247 859 > 2 \times 1 0^{23}$	`1RB2LA1RA1LB_0LB2RB3RB1LA` (bbch)	^[51]

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


Domain	Value	Champions	Reference
BBBB(1)	2	`0RA---` (bbch) (beep: any, boop: A0)	^[52]
BBBB(2)	18	`0RB1LB_1LA1RB` (bbch) (beep: A0, boop: A1)	^[53]
BBBB(3)	≥ 127	`1RB1LB_0RC0LA_1LC0LA` (bbch) (beep: A0, boop: B1)	^[54]
BBBB(2,3)	≥ 187	`2RB2LA1RA_1LB1LA0RB` (bbch) (beep: B0, boop: B2)	^[54]

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


Domain	Number of Ones	Champions	Reference
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)	^[55]
num(6)	≥ ≈ 10↑↑15.60465	`1RB0LD_1RC0RF_1LC1LA_0LE0RZ_1LF0RB_0RC0RE` (bbch)
num(2,3)	9	`2RB1RZ1LB_1LA2LB1RB` (bbch)
num(3,3)	≥ 95524079	`1RB2RC1LA_2LA1RB1RZ_2RB2RA1LC` (bbch)	^[56]
num(4,3)	$> 10 ↑^{4} 4$	`1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD` (bbch)	^[22]
num(4,4)	$> 2 ↑^{15} 5$	`1RB3LB1LD2RA_2LC3RB1LC2RA_3RB1LB3LC2RC_1LZ---1LD1LD` (bbch)
num(17,5)	$> f_{ε_{0}} (20)$	Machine code (Not TNF) `4RF---0RF0RD1LQ_1RZ1RB2LC------_0LI4RC2RC3RC2RC_1LE1RD2RD3RD---_3LA1LE2LE3LE1LG_0LG1RF2RF3RF---_0RI0LH4RD3RI2LH_2LA1LH2LH3LH0RC_3RN---0LA0LI2RF_2RO---2LJ3LJ---_2LL---2RK3RK4RK_3RO2RO2LL3LL4LL_3LL---2RM3RM4RM_4LJ1LN2LN3LN1LO_1RB4RP1RK0RM4LN_4LJ1RP2RP3RP0RP_1RB1LQ2LQ3LQ1LQ` (bbch)

Maximum Space Function (BB_space(n,m))


Domain	Cells visited	Champions	Reference
BB_space(1,2)	1	`1RZ---` (bbch)
BB_space(2,2)	4	`1RB1LB_1LA1RZ` (bbch) and `1RB0LB_1LA1RZ` (bbch)
BB_space(3,2)	7	`1RB1RC_1LC1RZ_1RA0LB` (bbch) and `1RB0RC_1LC1RZ_1RA0LB` (bbch)
BB_space(4,2)	16	`1RB0RA_1LC0RD_0LD0LB_1RA1RZ` (bbch)
BB_space(5,2)	12289	`1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA` (bbch)	^[55]
BB_space(1,3)	1	`1RZ------` (bbch)
BB_space(2,3)	9	`1RB2LB1RZ_2LA2RB1LB` (bbch)
BB_space(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))


Domain	Steps	Champions	Reference
BB_rev(1)
BB_rev(2)	6	`0RB1RZ_1LA1RB` (bbch)	^[57]
BB_rev(3)	17	`0RB1RZ_0LC1RA_1RB1LC` (bbch)	^[58]
BB_rev(4)	48	`1RB0LD_0LC0RB_1LA1LD_1LC1RZ` (bbch)	^[59]^[60]
BB_rev(5)	388	`1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA` (bbch)	^[61]^[62]
BB_rev(6)	≥ 537,556	`1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA` (bbch)	^[63]
BB_rev(7)	$> 1 0^{19}$	`1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB1RZ` (bbch)

Maximum Score Function (Σ_rev(n,m))


Domain	Score	Champions	Reference
Σ_rev(1)
Σ_rev(2)	≥ 2	`0RB1RZ_1LA1RB` (bbch)
Σ_rev(3)	≥ 4	`0RB1RZ_0LC1RA_1RB1LC` (bbch)
Σ_rev(4)	≥ 8	`1RB0LA_1LC1RD_1LA0LC_0RB1RZ` (bbch)	^[59]
Σ_rev(5)	≥ 16	`1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA` (bbch)	^[62]
Σ_rev(6)	≥ 1161	`1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA` (bbch)	^[63]

Blanking Busy Beaver (BLB(n,m))


2 Symbols:	Steps	Champions	Reference
BLB(1)	nonexistent	nonexistent
BLB(2)	8	`1RB0RA_1LB1LA` (bbch)	^[51]
BLB(3)	34	`1RB1LB_1LA1LC_1RC0LC` (bbch)	^[64]
BLB(4)	≥ 32,779,477	`1RB1LD_1RC1RB_1LC1LA_0RC0RD` (bbch)	^[65]
BLB(5)	> 10^14,006	`1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA` (bbch)	^[66]
BLB(6)	> 10 ↑↑ 16	`1RB0LD_1RC0RF_1LC1LA_0LE1RE_1LF0RB_0RC0RE` (bbch)
BLB(7)	> 10^183.76332	`1RB0RF_1RC1RA_1LD0LA_0RG0RE_0RA1LF_0LE0LC_1RZ0RD` (bbch)	^[67]


3 Symbols:	Steps	Champions	Reference
BLB(1,3)	nonexistent	nonexistent
BLB(2,3)	77	`1RB2LA0RB_1LA0LB1RA` (bbch)	^[64]
BLB(3,3)	> 10 ↑↑ 6.54	`1RB1RC2RA_0RC2RA0RB_2LC2LB0LA` (bbch)	^[50]


4 Symbols:	Steps	Champions	Reference
BLB(1,4)	nonexistent	nonexistent
BLB(2,4)	≥ 1,367,361,263,049	`1RB2RA1RA2RB_2LB3LA0RB0RA` (bbch)	^[64]


5 Symbols:	Steps	Champions	Reference
BLB(2,5)	> 10³²	`1RB2RB4RA0RB2RA_2LB3LA0RB0RA1RB` (bbch)

Busy Cleaner (BB_clean(n,m))


Domain	Steps	Champions	Reference
BB_clean(2)	≥ 4
BB_clean(3)	≥ 12
BB_clean(4)	≥ 34
BB_clean(5)	≥ 187

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))


Domain	Preperiod	Champions	Reference
BBS(1,2)	0	`1RA---` (bbch)
BBS(2,2)	≥ 9	`1RB0LB_1LA0RB` (bbch) proven winner?	^[68]
BBS(3,2)	101	`1RB1LB_0RC0LA_1LC0LA` (bbch)	^[68]
BBS(4,2)	≥ 119,120,230,102	`1RB1LC_0LA1RD_0RB0LC_1LA0RD` (bbch)
BBS(5,2)	> 10^14,006	`1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA` (bbch)
BBS(1,3)	0	`1RA------` (bbch)
BBS(2,3)	≥ 165	`1RB0LA---_1LB2LA0RB` (bbch)	^[68]
BBS(3,3)	> 10 ↑↑ 6	`1RB0LB2LA_1LA0RC0LB_2RC2RB0LC` (bbch)
BBS(4,3)	> $10 ↑^{4} 4$	`1RB1RD1LC_2LB1RB1LC_1LB1LA1LD_0RB2RA2RD` (bbch)
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))


Domain	Period	Champions	Reference
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)
BBP(5,2)	≥ 8,468,569,863	`1LC1LE_---1LD_1RD0LD_1LA1RE_0LB0RC` (bbch)	^[69]
BBP(6,2)	$> 2.4154 \times 1 0^{2695}$	`1RB1LC_1LA1RB_1LD0LA_1RE1RD_---1RF_1LB0RF` (bbch), `1RB1LA_1LA1RC_1RD0RB_1LE1LD_---1LF_1RA0LF` (bbch)	^[70]
BBP(1,3)	1	`1RA------` (bbch)
BBP(2,3)
BBP(3,3)	≥ 1,195	`1RB2RC1LC_0RC0RB1LA_2LA2RC1LB` (bbch)
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	Classical Domain	Reference
BBi(1)	1	`0RH` (bbch) `1RH---` (bbch)	BB(1)
BBi(2)	3	`0RB---_1LA---` (bbch)	BB(2)	^[71]
BBi(3)	5	`1RB1LB_1LA---` (bbch)	BB(2)	^[72]
BBi(4)	16	`1RB---_0RC---_1LC0LA` (bbch)	BB(3)	^[73]
BBi(5)	37	`1RB2LB---_2LA2RB1LB` (bbch)	BB(2,3)	^[74]
BBi(6)	123	`1RB3LA1RA0LA_2LA------3RA` (bbch)	BB(2,4)	^[75]
BBi(7)	3,932,963	`1RB2LA1RA1RA_1LB1LA3RB---` (bbch) `1RB2LA1RA_1LC1LA2RB_---1LA---` (bbch)	BB(2,4)	^[76]
BBi(8)	$> 6.889 \times 1 0^{1565}$	`1RB1LA------_1RC3LB1RB---_2LA2LC---0LC` (bbch)	BB(3,4)	^[77]
BBi(9)	$> 1 0^{1 0^{1 0^{3 314 360}}}$	`1RB3LA4RB0RB2LA_1LB2LA3LA1RA---` (bbch)	BB(2,5)
BBi(10)
BBi(11)	$> 2 ↑^{15} 5$	`1RB3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC` (bbch)	BB(3,4)
BBi(12)	$> 2 ↑^{18} 3$	`0RD3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC_1RB---------` (bbch)	BB(4,4)	^[78]
BBi(13)	$> 2 ↑^{98} 4 > f_{ω} (97)$	`1LD3LB---2RA_2LC3RB1LC2RA_3RB1LB3LC2RC_0LB---2LB---` (bbch)	BB(4,4)	^[78]
BBi(14)	$> f_{ω} (2 ↑^{15} 5)$	`1RB3LB4LC2RA4LB_2LC3RB1LC2RA---_3RB1LB3LC2RC4LC` (bbch)	BB(3,5)
BBi(15)	$> f_{ω}^{3} (12)$	`1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LH0LC_1RG1LD_0RG0RF_1RG---` (bbch)	BB(8)	^[4]
BBi(16)
BBi(17)	$> f_{ω}^{11} (15)$	`1RB3LB4LC2RA4LB_2LC3RB1LC2RA4RD_3RB1LB3LC2RC4LC_3RC------3RA---` (bbch)	BB(4,5)	^[79]
BBi(18)	$> f_{ω + 1} (2611) > g_{64}$	`1RB3LB4LC2RA4LB_2LC3RB1LC2RA4RD_3RB1LB3LC2RC4LC_1LC3LC---1RD---` (bbch)	BB(4,5)	^[79]
BBi(19)	$> f_{ω + 1}^{2} (2 ↑^{6} 4)$	`5LD3LB---2RA4LB---_---3RB1LC2RA3RD2LC_3RB1LB3LC2RC4LC---_3RB---5LD4LD4LC4LD` (bbch)	BB(4,6)
BBi(20)	$> f_{ω + 1}^{4} (f_{ω}^{2} (2 ↑^{6} 4))$	`5LD3LB---2RA4LB2RB_---3RB1LC2RA3RD3LB_3RB1LB3LC2RC4LC---_3RB---5LD4LD4LC4LD` (bbch)	BB(4,6)
BBi(21)	$> f_{ω + 2} (f_{ω + 1}^{2} (2 ↑^{6} 4))$	`2LF3LB---2RA4LB---_---3RB1LC2RA3RD---_3RB1LB3LC2RC4LC---_3RB---5LE4LD4LC4LD_4LC------5LE------_2LE---------------` (bbch)	BB(6,6)
BBi(22)	$> f_{ω + 2}^{2} (f_{ω + 1}^{2} (2 ↑^{6} 4))$	`1RF3LB---2RA4LB---_---3RB1LC2RA3RD---_3RB1LB3LC2RC4LC---_3RB---5LE4LD4LC4LD_4LC2LF---5LE------_2LE---------------` (bbch)	BB(6,6)
BBi(23)	$> f_{ω + 3} (8)$	`1LF3LB---2RA4LB---_---3RB1LC2RA3RD---_3RB1LB3LC2RC4LC---_3RB2LF5LE4LD4LC4LD_4LC------5LE------_4LE------2LF------` (bbch)	BB(6,6)
BBi(24)
BBi(25)	$> f_{ω + 5} (28)$	`3LF3LB---2RA4LB---_2LF3RB1LD2RA3LF---_3RB1LB3LC2RC4LC0LE_3RB---3LC2RC---3RE_5RC3RE---4LC------_5RB---2LC1LF------` (bbch)	BB(6,6)
BBi(26)	$> f_{ω \times 2} (36)$	`5RB3LB---2RA4LB---_2LC3RB1LD2RA3LF---_3RB1LB3LC2RC4LC0LE_3RB3RD3LC2RC---3RE_5RC3RE1RB4LC------_5RB------1LF------` (bbch)	BB(6,6)	^[80]
BBi(27)	$> f_{ω \times 2} (89)$	`4LF3LB---2RA4LB---_2LC3RB1LD2RA3LF---_3RB1LB3LC2RC4LC0LE_3RB---3LC2RC---3RE_5RC3RE---4LC------_5RB0LF1RB1LF---5RB` (bbch)	BB(6,6)	^[38]
BBi(28)
BBi(29)	$> f_{ω \times 2} (f_{ω + 1} (f_{ω} (30)))$	`5LF3LB1LG2RA4LB---_---3RB1LD2RA3LF3LG_3RB1LB3LC2RC4LC0LE_3RB---3LC2RC---3RE_5RC3RE---4LC------_5RB---2LB1LF------_------1LG3LG---0LE` (bbch)	BB(7,6)
BBi(30)	$> f_{ω \times 2}^{3} (f_{ω + 2} (28))$	`5LF3LB1LG2RA4LB---_---3RB1LD2RA3LF3LF_3RB1LB3LC2RC4LC0LE_3RB---3LC2RC---3RE_5RC3RE---4LC------_5RB---2LF1LF---5RA_------1LG3LG---0LE` (bbch)	BB(7,6)
BBi(31)	$> f_{ω \times 2 + 1} (19)$	`5LF3LB1LG2RA4LB---_2LC3RB1LD2RA3LF2LB_3RB1LB3LC2RC4LC0LE_3RB---3LC2RC---3RE_5RC3RE2LE4LC------_5RB---2LE1LF------_------1LG3LG---0LE` (bbch)	BB(7,6)
BBi(32)	$> f_{ω^{2}} (4)$	`1LD2RG2LA3LG_---2LH2RB2RF_3LI1RC2RB3LG_0RC1RC1LD3RD_3RD1LE1LE3LE_1RZ3LD2LE3LE_1RG1LH2RG3LG_1RC2LH1LA1LE_3LA3LE------` (bbch)	BB(9,4)
BBi(33)	$> f_{ω^{2}} (7)$	`1LD2RG2LA3LG---_---2LH2RB2RF---_4LE1RC2RB3LG4RE_0RC1RC1LD3RD3LA_3RD1LE1LE3LE---_1RZ3LD2LE3LE---_1RG1LH2RG3LG3LF_1RC2LH1LA1LE---` (bbch)	BB(8,5)
BBi(34)	$> f_{ω^{2}} (16)$	`1LD2RG2LA3LG_3LI2LH2RB2RF_1LG1RC2RB3LG_0RC1RC1LD3RD_3RD1LE1LE3LE_1RZ3LD2LE3LE_1RG1LH2RG3LG_1RC2LH1LA1LE_3LD3LI3LI---` (bbch)	BB(9,4)
BBi(35)	$> f_{ω^{2}} (28)$	`2LF3LB---2RA4LB3LD_3LA3RB1LD2RA3LF5RB_3RB1LB3LC2RC4LC1LE_3RB3RE3LC2RC---3RD_5RB3RE---4LC3LG5LE_5RB5RB0LF1LF---5LF_5LF------1LG---5LG` (bbch)	BB(7,6)	^[81]
BBi(36)	$> f_{ω^{2}} (63)$	`2LE3LB---2RA4LB2LD_0LA3RB1LD2RA3LF5RB_3RB1LB3LC2RC4LC1LE_3RB3RE3LC2RC---3RD_5RB3RE0LG4LC3LG5LE_5RB5RB---1LF---5LF_5LF---3RC1LG---5LG` (bbch)	BB(7,6)	^[82]
BBi(37)	$> f_{ω^{2}} (110)$	`5LG3LB---2RA4LB0LC_4LG3RB1LD2RA3LF5RB_3RB1LB3LC2RC4LC1LE_3RB3RE3LC2RC---3RD_5RB3RE0LE4LC3LG5LE_5RB5RB2LB1LF---5LF_5LF---0LG1LG---5LG` (bbch)	BB(7,6)	^[39]
...
BBi(41)	$> f_{ω^{2}}^{3} (4 ↑ ↑ 341)$	`0LI0LF_0RJ1RG_0RD1LC_1RH1RE_1LO0RH_1LA1LF_0LC1LA_1RB1RH_1RD0LF_1LP0RK_0LM1LL_0LL1LM_1LE1LN_0RQ0LM_0RP1LO_1LR0RH_1LF1RQ_---0LS_1LH0LT_1LH1LU_0LE1LR` (bbch)	BB(21)	^[83]
BBi(42)	$f_{ω^{2} + 1} (29)$	`5LH3LB2LH2RA4LB---_---3RB1LD2RA3LF5RB_3RB1LB3LC2RC4LC1LE_3RB3RE3LC2RC---3RD_5RB3RE1LE4LC3LG5LE_5RB5RB---1LF5LF5LF_5LF4RG1LB1LG2LG5LG_3RC---1LH1LH---5RG` (bbch)	BB(8,6)
BBi(43)	$> f_{ω^{2} + 1} (f_{ω} (31))$	`5LH3LB2LH2RA4LB1RZ_2RC3RB1LD2RA3LF5RB_3RB1LB3LC2RC4LC1LE_3RB3RE3LC2RC---3RD_5RB3RE1LE4LC3LG5LE_5RB5RB---1LF5LF5LF_5LF4RG1LB1LG2LG5LG_0LC---1LH1LH---5RG` (bbch)	BB(8,6)	^[40]
...
BBi(57)	$> f_{ω^{ω}} (3)$	[5]	BB(29)	^[17]
BBi(61)	$> f_{ω ↑ ↑ 4} (3)$	[6]	BB(31)	^[18]
BBi(65)	$> f_{ω ↑ ↑ 6} (3)$	[7]	BB(33)	^[19]
BBi(78)	$> f_{ε_{0}} (1.7 \cdot 1 0^{352})$	BB(19,5) machine code `1RB2LA1RA2LB2LA_0LA2RB3RB4RA1LQ_---1RC2LD------_0LK3RD2RD2RD4RD_------0RH1LS0RF_1LG1RF2RF---4RF_4LE1LG2LG---4LG_0LI1RH2RH---4RH_0RK0LJ------4RK_2LE1LJ2LJ---4LJ_4RP---0LE---0LK_2RQ---2LL---4LL_2LN---2RM3RM4RM_4RQ2RQ2LN3LN4LN_4LN---2RO3RO4RO_3LL1LP2LP1LQ4LP_1RC3RR1RM3LP0RO_3LL1RR2RR0RR4RR_1RC1LS2LS1LS4LS` (bbch) BB(18,7) machine code `1RB2LA1RA2LB2LA------_0LA2RB3RB4RA1LP------_---1RC2LD------------_0LJ3RD2RD2RD4RD------_4LE1LE0RG1LR0RF2LE4LE_1LE1RF5RF---6RF------_0LH1RG2RG---4RG------_0RJ0LI------4RJ------_2LE1LI2LI---4LI------_4RO---0LE---0LJ------_2RP---2LK---4LK------_2LM---2RL3RL4RL------_4RP2RP2LM3LM4LM------_4LM---2RN3RN4RN------_3LK1LO2LO1LP4LO------_1RC3RQ1RL3LO0RN------_3LK1RQ2RQ0RQ4RQ------_1RC1LR2LR1LR4LR------` (bbch) BB(17,8) machine code `1RB2LA1RA2LB2LA---------_0LA2RB3RB4RA1LO---------_---1RC2LD---------------_0LJ3RD2RD2RD4RD---------_4LE1LE0RG1LQ0RF2LE4LE---_1LE1RF5RF---6RF---------_0LH1RG2RG---4RG---------_0RJ0LI------4RJ---------_2LE1LI2LI---4LI---------_4RN---0LE---0LJ---------_2RO2RO2LK3LK4LK2LK4LK4RO_2LK---5RL3RL6RL---------_4LK---5RM3RM6RM---------_3LK1LN2LN1LO4LN---------_1RC3RP1RL3LN7RM---------_3LK1RP2RP0RP4RP---------_1RC1LQ2LQ1LQ4LQ---------` (bbch)	BB(19,5), BB(18,7), BB(17,8)
BBi(94)	$> f_{ε_{1}} (3)$	[8]	BB(48)	^[84]
BBi(237)	$> f_{ψ_{0} (Ω_{2})} (1.7 \cdot 1 0^{352})$		BB(42,9)
BBi(299)	$> f_{l i m (B M S)} (10 ↑ ↑ 15)$	too large to show	BB(150)	^[21]
BBi(2029)	> D⁵(99) = Loader's Number		BB(1015)

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


	Score	Champions	Classical Domain	Reference
Σi(1)	1	`1RH---` (bbch)	BB(1)
Σi(2)	2	`1RB---_1LA---` (bbch)	BB(2)	^[71]
Σi(3)	4	`1RB1LB_1LA---` (bbch)	BB(2)	^[72]
Σi(4)	5	`1RB0LB---_1LA2RA---` (bbch)	BB(2,3)	^[73]
Σi(5)	9	`1RB2LB---_2LA2RB1LB` (bbch)	BB(2,3)	^[74]
Σi(6)	14	`1RB3LA1RA0LA_2LA------3RA` (bbch)	BB(2,4)	^[75]
Σi(7)	2050	`1RB2LA1RA1RA_1LB1LA3RB---` (bbch) `1RB2LA1RA_1LC1LA2RB_---1LA---` (bbch)	BB(2,4)	^[76]
Σi(8)	$> 1.355 \times 1 0^{783}$	`1RB1LA------_1RC3LB1RB---_2LA2LC---0LC` (bbch)	BB(3,4)	^[85]
Σi(9)	$> 1 0^{1 0^{1 0^{3 314 360}}}$	`1RB3LA4RB0RB2LA_1LB2LA3LA1RA---` (bbch)	BB(2,5)

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)	> 10^42,745	`1RB2RB1LA_2LC0LB2LB_2RC2RA0LC` (bbch)

Instruction-Limited Maximum Consecutive Ones Function (num_i(n))

Domain	num_i(n)	Champion
num_i(0)	0	`---` (bbch)
num_i(1)	1	`1RZ` (bbch)
num_i(2)	3	`1RB---_1LA---` (bbch)
num_i(3)	4	`1RB1LB_1LA---` (bbch)
num_i(4)
num_i(5)	9	`2RB---1LB_1LA2LB1RB` (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 $20 \geq n$ BBλ(n) = n.


	BBλ(n)	Champions
BBλ(4)	4	$λ 1$
BBλ(6)	6	$λ λ 1$
BBλ(7)	7	$λ λ 2$
BBλ(8)	8	$λ λ λ 1$
BBλ(9)	9	$λ λ λ 2$
BBλ(10)	10	$λ λ λ λ 1$
BBλ(11)	11	$λ λ λ λ 2$
BBλ(12)	12	$λ λ λ λ λ 1$
BBλ(13)	13	$λ λ λ λ λ 2$
BBλ(14)	14	$λ λ λ λ λ λ 1$
BBλ(15)	15	$λ λ λ λ λ λ 2$
BBλ(16)	16	$λ λ λ λ λ λ λ 1$
BBλ(17)	17	$λ λ λ λ λ λ λ 2$
BBλ(18)	18	$λ λ λ λ λ λ λ λ 1$
BBλ(19)	19	$λ λ λ λ λ λ λ λ 2$
BBλ(20)	20	$λ λ λ λ λ λ λ λ λ 1$
BBλ(21)	22	$λ (1 (λ 2)) (1 (λ 2))$
BBλ(22)	24	$λ (11) (11) (11)$
BBλ(23)	26	$λ (1 (λ λ 2)) (1 (λ λ 2))$
BBλ(24)	30	$λ (1 (λ 1)) (1 (λ 1)) (1 (λ 1))$
BBλ(25)	42	$λ 1 (λ 1 (21)) (1 (1 (λ 1 (21))))$
BBλ(26)	52	$λ λ 2 (λ λ 2 (12)) (1 (2 (λ λ 2 (12))))$
BBλ(27)	44	$λ λ 1 (λ 1 (21)) (1 (1 (λ 1 (21))))$
BBλ(28)	58	$λ 1 (λ λ 1 (3 (λ 2))) (1 (λ 2 (λ λ 1 (4 (λ 2)))))$
BBλ(29)	223	$λ (λ 11) (λ 1 (1 (21)))$
BBλ(30)	160	$(λ 111) (λ λ 2 (12))$
BBλ(31)	267	$(λ 11) (λ λ 2 (2 (12)))$
BBλ(32)	298	$λ (λ 11) (λ 1 (1 (2 (λ 2))))$
BBλ(33)	1812	$λ (λ 11) (λ 1 (1 (1 (21))))$
BBλ(34)	327,686	$(λ 1111) (λ λ 2 (21))$
BBλ(35)	$5 \times 3^{3^{3}} + 6 = 38 127 987 424 941 > 3.8 \times 1 0^{13}$	$(λ 111) (λ λ 2 (2 (21)))$
BBλ(36)	$5 \times 2^{2^{2^{3}}} + 6 > 5.7 \times 1 0^{77}$	$(λ 11) (λ 1 (1 (λ λ 2 (21))))$
BBλ(37)	$B B λ (35) + 2 = 5 \times 3^{3^{3}} + 8 = 38 127 987 424 943 > 3.8 \times 1 0^{13}$	$λ (λ 111) (λ λ 2 (2 (21)))$
BBλ(38)	$5 \times 2^{2^{2^{2^{2}}}} + 6 > 1 0^{19729}$	$(λ 11111) (λ λ 2 (21))$
BBλ(39)	$5 \times 3^{3^{3^{3}}} + 6 > 1 0^{3 638 334 640 024}$	$(λ 1111) (λ λ 2 (2 (21)))$
BBλ(40)	$\approx (2 ↑ ↑)^{15} 33 > 10 ↑ ↑ ↑ 16$	$(λ 111) (λ 1 (λ λ 2 (21)) 1)$
BBλ(41)	$5 \times 3^{3^{85}} + 6 > 1 0^{1.7 \times 1 0^{40}}$	$(λ 1 (λ 11) 1) (λ λ 2 (2 (21)))$
BBλ(42)	$\geq B B λ (40) + 2 > (2 ↑ ↑)^{15} 33 > 10 ↑ ↑ ↑ 16$	$λ (λ 111) (λ 1 (λ λ 2 (21)) 1)$
BBλ(43)	$> 2 ↑ ↑ ↑ 2 ↑ ↑ ↑ (\frac{2 ↑ ↑ 8}{2})$	$(λ 11) (λ 1 (λ 1 (λ λ 2 (21)) 2))$
BBλ(44)	$> 10 ↑ ↑ ↑ 10 ↑ ↑ ↑ 16$	$(λ 1111) (λ 1 (λ λ 2 (21)) 1)$
BBλ(45)	$\geq B B λ (43) + 2 > 2 ↑ ↑ ↑ 2 ↑ ↑ ↑ (\frac{2 ↑ ↑ 8}{2})$	$λ (λ 11) (λ 1 (λ 1 (λ λ 2 (21)) 2))$
BBλ(46)	$\geq B B λ (44) + 2 > 10 ↑ ↑ ↑ 10 ↑ ↑ ↑ 16$	$λ (λ 1111) (λ 1 (λ λ 2 (21)) 1)$
BBλ(47)	$> f_{ω} (f_{5} (2))$	$(λ 111) (λ λ 1 (12) (λ λ 2 (21)))$
BBλ(48)	$> 10 ↑ ↑ ↑ 10 ↑ ↑ ↑ 10 ↑ ↑ ↑ 16$	$(λ 11111) (λ 1 (λ λ 2 (21)) 1)$
BBλ(49)	$> f_{ω + 1} (\frac{2 ↑ ↑ 6}{2}) > Graham's Number$	$(λ 11) (λ 1 (1 (λ λ 12 (λ λ 2 (21)))))$
BBλ(61)	$> f_{ω^{2 ↑ ↑ 18 - 1}} (2)$	$(λ 111) (λ 1 (1 (λ λ λ 132 (λ λ 2 (21)))))$
BBλ(70)	$> f_{ω^{ω} 2} (f_{ω^{ω}} (f_{ω^{ω}} (3)))$	$(λ 11 (λ 1 (λ λ λ 1325) 2) 1) (λ λ 2 (21))$
BBλ(74)	$> f_{ω^{ω} 2 + 1} (7)$	$(λ 111 (λ 1 (λ λ λ 1325) 2) 1) (λ λ 2 (21))$
BBλ(76)	$> f_{ω^{ω^{3}}} (2)$	$(λ 11 (λ λ λ λ 14321) 1111) (λ λ 2 (21))$
BBλ(77)	$> f_{ω^{ω 2}} (3)$	$(λ 1 (λ λ λ λ 14321) 1111) (λ λ 2 (2 (21)))$
BBλ(80)	$> f_{ω^{ω^{14}}} (2)$	$(λ 111 (λ λ λ λ 14321) 1111) (λ λ 2 (21))$
BBλ(82)	$> f_{ω^{ω^{ω}}} (2 ↑ ↑ ↑ ↑ 18)$	$(λ 1 (λ 1 (λ λ λ λ 14321) 1111) 1) (λ λ 2 (21))$
BBλ(90)	$> f_{ζ_{0}} (15)$	$(λ 11 (λ λ λ λ 1444321) 1111) (λ λ 2 (21))$
BBλ(94)	$> f_{ψ (Ω_{ω})} (12)$	$(λ 111 (λ λ λ λ 1444321) 1111) (λ λ 2 (21))$
BBλ(95)	$> f_{ψ (Ω_{ω})} (23)$	$(λ 11 (λ λ λ λ 1444321) 1111) (λ λ 2 (2 (21)))$
BBλ(96)	$> f_{ψ (Ω_{ω})} (f_{ω^{ω^{2}}} (2))$	$(λ 1 (λ 1 (λ λ λ λ 1444321) 1111) 1) (λ λ 2 (21))$
BBλ(100)	$> f_{ψ (Ω_{ω}) + 1} (4)$	$(λ 11 (λ 1 (λ λ λ λ 1444321) 1111) 1) (λ λ 2 (21))$
BBλ(201)	$> q (5)$ (q is only proven to exceed all functions provably recursive in PA; the true value is likely much larger based on informal analysis)	`too large to show`
BBλ(331)	$> f_{\lim (BMS)}^{3} (5)$	`too large to show`
BBλ(1797)	$> D^{5} (99)$ (Loader's number)	`too large to show`

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

Note that $f (n) = 6 + 5 \times B B λ (n)$ .


	BBλ₁(n)	Champions
BBλ₁(1)	0
BBλ₁(2)	2	$1$
BBλ₁(3)	0
BBλ₁(4)	4	$λ 1$
BBλ₁(5)	5	$λ 2$
BBλ₁(6)	6	$λ λ 1$
BBλ₁(7)	7	$λ λ 2$
BBλ₁(8)	26	$1 (λ 1)$
BBλ₁(9)	9	$λ λ λ 2$
BBλ₁(10)	36	$1 (λ λ 1)$
BBλ₁(11)	41	$1 (λ λ 2)$
BBλ₁(12)	266	$1 (1 (λ 1))$
BBλ₁(13)	51	$1 (λ λ 2)$
BBλ₁(14)	$f (36) = 25 \times 2^{2^{2^{3}}} + 36 > 2.85 \times 1 0^{78}$	$1 (1 (λ λ 1))$
BBλ₁(15)	$f (41) = 25 \times 3^{3^{85}} + 36 > 1 0^{1.7 \times 1 0^{40}}$	$1 (1 (λ λ 2))$
BBλ₁(16)	$f (266)$	$1 (1 (1 (λ 1)))$
BBλ₁(17)	$f (51)$	$1 (1 (λ λ λ 2))$
BBλ₁(18)	$f^{4} (4) = f (f (266))$	$1 (λ 1) 1 (λ 1)$
BBλ₁(19)	$f^{3} (7) = f (f (41))$	$1 (1 (1 (λ λ 2)))$
BBλ₁(20)	$f^{6} (4) = f^{4} (266)$	$1 (λ λ 1) 1 (λ 1)$
BBλ₁(21)	$f^{7} (4) = f^{5} (266)$	$1 (λ λ 2) 1 (λ 1)$
BBλ₁(22)	$f^{26} (4) = f^{24} (266)$	$1 (1 (λ 1)) 1 (λ 1)$
BBλ₁(28)	$\geq f^{B B λ (f^{3} (4))} (4)$	$1 (λ 1) 1 (λ 1) 1 (λ 1)$
BBλ₁(29)	$\geq f^{B B λ (f^{B B λ (f^{4} (4)) + 4} (4)) + B B λ (f^{4} (4)) + 5} (4)$	$1 (λ 1) (λ 121) (λ 1)$

Busy Beaver for De Bruijn Lambda Calculus (De Bruijn index)

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	$> 7.625 \times 1 0^{12}$	`(\1 1 1) (\\2 (2 (2 1)))`
11	$> 1 0^{7.625 \times 1 0^{12}}$	`(\1 1 1 1) (\\2 (2 (2 1)))`
12	$> 10 ↑^{3} 16$	`(\1 1 1) (\1 (\\2 (2 1)) 1)`
13	$> 10 ↑^{3} 10 ↑^{3} 10 ↑^{2} 6$	`(\1 1) (\1 (\1 (\\2 (2 1)) 2))`
14	$> f_{ω} (f_{5} (2))$	`(\1 1 1) (\\1 (1 2) (\\2 (2 1)))`
15	$> f_{ω + 1} (1 0^{1 0^{19, 727}})$	`(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))`
18	$> f_{ω^{ω}} (2 ↑ ↑ 18)$	`(\1 1 1) (\1 (1 (\\\1 3 2 (\\2 (2 1)))))`
19	$> f_{ω^{ω} 2} (f_{ω^{ω}}^{2} (3))$	`(\1 1 (\1 (\\1 3 2 5) 2) 1) (\2 (2 1)))`
20	$> f_{ω^{ω} 2 + 1} (7)$	`(\1 1 1 (\1 (\\1 3 2 5) 2) 1) (\2 (2 1)))`
21	$> f_{ω^{ω^{2} + ω + 2}} (2)$	`(\1 1 (\\\\1 4 3 2 1) 1 1 1 1) (\\2 (2 1))`
22	$> f_{ω^{ω^{14}}} (2)$	`(\1 1 1 (\\1 4 3 2 1) 1 1 1 1) (\2 (2 1))`
23	$> f_{ζ_{0}} (15)$	`(\1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))`
24	$> f_{ψ (Ω_{ω})} (12)$	`(\1 1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))`
25	$> f_{ψ (Ω_{ω})} (f_{ω^{ω + 2}} (2))$	`(\1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))`
26	$> f_{ψ (Ω_{ω} + 1)} (4)$	`(\1 1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))`

Busy Beaver for SKI calculus

	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)	≥ 1561	S(SI)(SS)(SS)SS
Ξ₀(10)	≥ 1562	S(S(SI)(SS)(SS)SS)
Ξ₀(11)	≥ 1563	SS(S(SI)(SS)(SS)SS)
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↑↑↑17	SSI(SSK(S(S(KS)K)I))(S(K(S(SSK)))K)
Ξ₀(20)	> f_ω+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)	> f_ω+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)	> f_ω+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)	> f_ω+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)	> f_ω+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)	> f_ω+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)	> f_ω+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))
Ξ₀(61)	> f_ε₀ (255)	SSK(S(S(KS)K)I)(S(S(S(SI(K(S(K(SS(K(SI(KI)))))(S(K(SS(K(S(K(S(K(S(K(SS(K(S(K(S(S(KS)K)(KK)))S))))K))S))K))))K))))(K(KI)))(K(SS(KI))))I)
Ξ₀(71)	> f_{ψ(Ω_ω)} (f_ω^ω65536 (2))	SSK(S(S(KS)K)I)(S(S(S(S(S(K(SS(KI))))(K(S(K(S(K(S(K(S(K(SS(KI)))))(S(S(K(S(KS)K))S)(KK))))))(S(K(S(S(K(S(KS)K))S)(KK)))(S(K(S(S(K(SS(KI)))(SS(KI)))))K)))))I)I)I)

Busy Beaver for SK calculus

	Ξ₀_SK(n)	Champion
Ξ₀_SK(n)(1)	1	S
Ξ₀_SK(n)(2)	2	SS
Ξ₀_SK(n)(3)	3	SSS
Ξ₀_SK(n)(4)	4	SSSS
Ξ₀_SK(n)(5)	6	SSS(SS)
Ξ₀_SK(n)(6)	10	SSS(SS)S
Ξ₀_SK(n)(7)	41	SSS(S(SS))S
Ξ₀_SK(n)(8)	≥ 80	SSK(S(SS)S)S
Ξ₀_SK(n)(9)	≥ 169	S(SS)(SS)(SS)SS
Ξ₀_SK(n)(10)	≥ 1561	SS(SS)(SSS)SSS
Ξ₀_SK(n)(11)	≥ 1562	S(SS(SS)(SSS)SSS)
Ξ₀_SK(n)(12)	≥ 1563	S(S(SS(SS)(SSS)SSS))
Ξ₀_SK(n)(13)	≥ 7811	S(SS)(SS)(SS(SS(SSS)))S
Ξ₀_SK(n)(14)	≥ 2↑2097156*13-2	SSS(SSK)(SS(SK)(S(KS)K))
Ξ₀_SK(n)(15)	≥ 2↑2↑2097156*3-2	SSS(SSK)(SS(SK)(S(KS)K))S
Ξ₀_SK(n)(16)	> 2↑2↑2↑8193	S(SS(SS))(SSK)(SS(SK)(S(KS)K))
Ξ₀_SK(n)(17)	> 2↑↑2↑↑2↑258	SSS(S(SK))(SSK)(SS(SK)(S(KS)K))
Ξ₀_SK(n)(22)	> 2↑↑↑19	S(S(S(S(SK))))(S(SK))(S(SSK)(K(SS(SK)(S(KS)K)))
Ξ₀_SK(n)(23)	> 2↑↑↑↑2↑↑↑17	SS(S(SK))(SSK(SS(SK)(S(KS)K)))(S(K(S(SSK)))K)
Ξ₀_SK(n)(24)	> 2↑↑↑↑2↑↑↑19	S(SS(S(SK))(SSK(SS(SK)(S(KS)K)))(S(K(S(SSK)))K))
Ξ₀_SK(n)(25)	> 3↑⁷3	SS(S(SK))(SSK(SS(SS(SK))(S(KS)K)))(S(K(S(SSK)))K)
Ξ₀_SK(n)(26)	> f_ω+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

	Ξ₀_BCKW(n)	Champion
Ξ₀_BCKW(1)	1	B
Ξ₀_BCKW(2)	2	BB
Ξ₀_BCKW(3)	3	BBB
Ξ₀_BCKW(4)	11	WW(WB)
Ξ₀_BCKW(5)	47	W(WW)(WB)
Ξ₀_BCKW(6)	≥ 2↑256*6-1	WW(WC(WB))
Ξ₀_BCKW(7)	> 2↑↑↑17	WW(C(WC)(WB))
Ξ₀_BCKW(8)	> 2↑↑↑2↑↑↑17	W(WW)(C(WC)(WB))
Ξ₀_BCKW(9)	> 2↑⁵2↑↑↑↑2↑↑↑17	WW(WB(C(WC))(WB))
Ξ₀_BCKW(10)	> f_ω+1(2↑↑6/2-1)	WW(C(WB)(C(CC(WB))))
Ξ₀_BCKW(15)	> f_ω^{2^^18-1}(2)	WW(C(WB)(B(BC(CC(WB)))(C(BC))))
Ξ₀_BCKW(22)	> f_{ψ(Ω_ω)} (f_ω^ω65536 (2))	WB(W(W(W(C(BW)(B(B(B(BW)C))(BC(C(BWW))))))))(WB)

Brainfuck

	BB_brainf(n)	Champion	Reference
BB_{brainf₍₀₎}	0
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)	≥ 176	`++[>+++++[-<]>>]`	^[86]
BB_brainf(17)	≥ 3,125	`++[>++++++[-<]>>]`	^[86]
BB_brainf(18)
BB_brainf(19)
BB_brainf(20)	≥ 4,212	`+[->++++++++++[-<]>]`	^[86]
BB_brainf(21)	≥ 6,234	`+++[>+++++++++[-<]>>]`	^[86]
BB_brainf(22)	≥ 90,963	`+++[>++++++++++[-<]>>]`	^[86]
BB_brainf(23)	≥ 7,467,842	`+++[>+++++++++++[-<]>>]`	^[86]
BB_brainf(24)	≥ 239,071,921	`+++[>++++++++++++[-<]>>]`	^[86]

Boolfuck

Boolfuck score (Σ_boolf(n))

	Σ_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	`>>>`
BB_boolf(8)	4	`>>>>`
BB_boolf(9)	= 5	`>>>>*`
BB_boolf(10)	5	`>>>>*>`
BB_boolf(11)	6	`>>>>>`
BB_boolf(12)	7	`[>>>>*<<<]`
BB_boolf(13)	≥ 8	`[>>>><<<]`
BB_boolf(14)	≥ 11	`[>>[<<<*<]>]`
BB_boolf(15)	≥ 12	`[>>[<<<<]>]`
BB_boolf(16)	≥ 15	`[>>[[<<<]*<]>]`
BB_boolf(17)	≥ 31	`[>[>>[<<]>]>]`

Boolfuck runtime (S_boolf(n))

	S_boolf(n)	Champion	Reference
S_boolf(1)	1	`*`	^[87]
S_boolf(2)	2	`*>`	^[87]
S_boolf(3)	3	`>`	^[87]
S_boolf(4)	4	`>>`	^[87]
S_boolf(5)	5	`>>*`	^[87]
S_boolf(6)	8	`>[<]`	^[87]
S_boolf(7)	10	`>[*<]`	^[87]
S_boolf(8)	14	`[>>*<]`	^[87]
S_boolf(9)	18	`[[>]<<]`	^[87]
S_boolf(10)	30	`[>[<]>>]`	^[87]
S_boolf(11)	35	`[>[<]*>>]`	^[87]
S_boolf(12)	72	`[>[<<*<]>]`	^[87]
S_boolf(13)	≥ 120	`[>>[<<]>]`	^[87]
S_boolf(14)	≥ 158	`[>>[<<<*<]>]`	^[87]
S_boolf(15)	≥ 276	`[[>[<<<]>]>>]`	^[87]
S_boolf(16)	≥ 352	`[>>[[<<<]*<]>]`	^[87]
S_boolf(17)	≥ 871	`[[>>[<]<]>>>]`	^[87]
S_boolf(18)	≥ 7,981	`[[>>[<]<]>>><]`	^[87]
S_boolf(19)	≥ 32,167	`[>[>>>]<[<<]>>]`	^[87]
S_boolf(20)	≥ 4,359,966	`[>[<<]>>>*>[>]<]`	^[87]
S_boolf(21)	≥ 505,033,483,078	`[>>[[<]<[>>]>*]>]`	^[87]

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]`	$[\begin{matrix} - 1 \end{matrix}]$
BBf(3)	1	`[3/2]`	$[\begin{matrix} - 1 & 1 \end{matrix}]$
BBf(4)	1	`[9/2]`	$[\begin{matrix} - 1 & 2 \end{matrix}]$
BBf(5)	2	`[3/2, 1/3]`	$[\begin{matrix} - 1 & 1 \\ 0 & - 1 \end{matrix}]$
BBf(6)	3	`[9/2, 1/3]`	$[\begin{matrix} - 1 & 2 \\ 0 & - 1 \end{matrix}]$
BBf(7)	4	`[27/2, 1/3]`	$[\begin{matrix} - 1 & 3 \\ 0 & - 1 \end{matrix}]$
BBf(8)	5	`[81/2, 1/3]`	$[\begin{matrix} - 1 & 4 \\ 0 & - 1 \end{matrix}]$
BBf(9)	6	`[243/2, 1/3]`	$[\begin{matrix} - 1 & 5 \\ 0 & - 1 \end{matrix}]$
BBf(10)	7	`[729/2, 1/3]`	$[\begin{matrix} - 1 & 6 \\ 0 & - 1 \end{matrix}]$
BBf(11)	10	`[27/2, 25/3, 1/5]`	$[\begin{matrix} - 1 & 3 & 0 \\ 0 & - 1 & 2 \\ 0 & 0 & - 1 \end{matrix}]$
BBf(12)	13	`[81/2, 25/3, 1/5]`	$[\begin{matrix} - 1 & 4 & 0 \\ 0 & - 1 & 2 \\ 0 & 0 & - 1 \end{matrix}]$
BBf(13)	17	`[81/2, 125/3, 1/5]`	$[\begin{matrix} - 1 & 4 & 0 \\ 0 & - 1 & 3 \\ 0 & 0 & - 1 \end{matrix}]$
BBf(14)	21	`[243/2, 125/3, 1/5]`	$[\begin{matrix} - 1 & 5 & 0 \\ 0 & - 1 & 3 \\ 0 & 0 & - 1 \end{matrix}]$
BBf(15)	28	`[1/45, 4/5, 3/2, 25/3]`	$[\begin{matrix} 0 & - 2 & - 1 \\ 2 & 0 & - 1 \\ - 1 & 1 & 0 \\ 0 & - 1 & 2 \end{matrix}]$
BBf(16)	53	`[1/45, 4/5, 3/2, 125/3]`	$[\begin{matrix} 0 & - 2 & - 1 \\ 2 & 0 & - 1 \\ - 1 & 1 & 0 \\ 0 & - 1 & 3 \end{matrix}]$
BBf(17)	107	`[5/6, 49/2, 3/5, 40/7]`	$[\begin{matrix} - 1 & - 1 & 1 & 0 \\ - 1 & 0 & 0 & 2 \\ 0 & 1 & - 1 & 0 \\ 3 & 0 & 1 & - 1 \end{matrix}]$
BBf(18)	211	`[5/6, 49/2, 3/5, 80/7]`	$[\begin{matrix} - 1 & - 1 & 1 & 0 \\ - 1 & 0 & 0 & 2 \\ 0 & 1 & - 1 & 0 \\ 4 & 0 & 1 & - 1 \end{matrix}]$
BBf(19)	370	`[5/6, 49/2, 3/5, 160/7]`	$[\begin{matrix} - 1 & - 1 & 1 & 0 \\ - 1 & 0 & 0 & 2 \\ 0 & 1 & - 1 & 0 \\ 5 & 0 & 1 & - 1 \end{matrix}]$
BBf(20)	746	`[7/15, 22/3, 6/77, 5/2, 9/5]`	$[\begin{matrix} - 1 & - 1 & + 1 \\ + 1 & - 1 & + 1 \\ + 1 & + 1 & - 1 & - 1 \\ - 1 & + 1 \\ + 2 & - 1 \end{matrix}]$
BBf(21)	≥ 31,957,632	`[7/15, 4/3, 27/14, 5/2, 9/5]`	$[\begin{matrix} 0 & - 1 & - 1 & 1 \\ 2 & - 1 & 0 & 0 \\ - 1 & 3 & 0 & - 1 \\ - 1 & 0 & 1 & 0 \\ 0 & 2 & - 1 & 0 \end{matrix}]$
BBf(22)	$> 1.146 \times 1 0^{62}$	`[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]`	$[\begin{matrix} - 2 & - 1 & 0 & 0 & 0 \\ - 1 & 2 & - 1 & 0 & 0 \\ 1 & - 1 & 0 & 1 & 0 \\ - 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & - 1 \end{matrix}]$
BBf(23)	$> 4.393 \times 1 0^{124}$	`[10/3, 9/14, 5/4, 121/2, 7/5, 3/11]`	$[\begin{matrix} 1 & - 1 & 1 & 0 & 0 \\ - 1 & 2 & 0 & - 1 & 0 \\ - 2 & 0 & 1 & 0 & 0 \\ - 1 & 0 & 0 & 0 & 2 \\ 0 & 0 & - 1 & 1 & 0 \\ 0 & 1 & 0 & 0 & - 1 \end{matrix}]$
BBf(24)	$> 9.000 \times 1 0^{14448}$ ^[88]	`[14/15, 1/6, 11/3, 441/2, 5/7, 15/11]`	$[\begin{matrix} 1 & - 1 & - 1 & 1 & 0 \\ - 1 & - 1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 & 1 \\ - 1 & 2 & 0 & 2 & 0 \\ 0 & 0 & 1 & - 1 & 0 \\ 0 & 1 & 1 & 0 & - 1 \end{matrix}]$

Cyclic Tree Rewriting System (CTBB(n))


	Runtime	Champions
CTBB(2)	5^[89]
CTBB(3)	> 38^[90]
CTBB(4)	≥ 672^[91]
CTBB(5)	≥ 2^2^2^2^182^[92]
CTBB(6)	> 2↑↑↑131^[93]
CTBB(7)	> 4↑↑↑↑(4↑↑↑3)^[94] ^[95]

Cyclic Tag (BBCT(n))


BBCT(n)	Value	Champion
BBCT(1)	1	`empty`
BBCT(2)	2	`0`
BBCT(3)	3	`00`
BBCT(4)	4	`000`
BBCT(5)	7	`010_`
BBCT(6)	8	`010_0`
BBCT(7)	= 13	`0110__`
BBCT(8)	≥ 28	`101_10_`
BBCT(9)	≥ 717	`1011_00_`
BBCT(10)	≥ 1,626,862	`1001_1_00`
BBCT(11)	≥ 1,859,166	`10_10010_0`
BBCT(12)	≥ 6,328	`0111__010_0`
BBCT(13)	≥ 5,217,735	`001110_1_0_0`
BBCT(14)	≥ 154,171,565	`01011__0_0100`

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))

MBB(n)

Domain	Halting Time	Champion
MBB(1)	1	`0+*`
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)	≥ 3,394	`0+B_0+C_1-GD_1+E_0-DF_2-HG_2+A_2-D*`
MBB(9)	> 4.493 × 10⁷⁴	`0+B_0+C_1-HD_0-EG_0-FI_1+D_2+H_0+A_2-B*`
MBB(10)	> 10↑↑4.03	`0+B_1-AC_2-ED_1+E_1+F_0-DG_2-JH_2+I_1-HE_2-A*`
MBB(11)	> 10↑↑17.74	`0-KB_1-IC_1+D_1+E_2-CF_0-AG_0+H_1-GI_2+J_2+B_0-B*`

Register Sigma function (MΣ(n,r))

Domain	Sigma	Champion
MΣ(0)	0
MΣ(1)	1	`0+*`
MΣ(2)	2	`0+B_0+*`
MΣ(3)	3	`0+B_0+C_0+*`
MΣ(4)	4	`0+B_0+C_0+D_0+*`
MΣ(5)	6	`0+B_0+C_0+D_1-*E_1+A`
MΣ(6)	9	`0+B_0+C_1+D_1+E_1+F_0-C*`
MΣ(7)	34	`0+B_0+C_0+D_1-GE_1+F_0-EC_1-A*`
MΣ(8)	≥ 520	`0+B_1+C_2-AD_2+E_2+F_0-DG_1-H*_1-EB`
MΣ(9)	> 2.64 × 10⁷³	`0+B_0+C_1-HD_0-EG_0-FI_1+D_2+H_0+A_2-B*`
MΣ(10)	> 10↑↑4.03	`0+B_1-AC_2-ED_1+E_1+F_0-DG_2-JH_2+I_1-HE_2-A*`
MΣ(11)	> 10↑↑17.74	`0-KB_1-IC_1+D_1+E_2-CF_0-AG_0+H_1-GI_2+J_2+B_0-B*`

Pebble Automaton (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^[96]	`00_11_03_21_04`
peBBle(6)	17^[97]	`00_11_03_21_40_15`
peBBle(7)	28^[97]	`00_11_03_21_10_33_61`
peBBle(8)	63^[98]	`00_11_03_21_50_30_24_70`
peBBle(9)	63577^[99]	`00_02_01_40_14_12_00_52_81`
peBBle(10)	5778	`00_02_01_40_14_12_00_17_63_18`
peBBle(11)	≥ 810,626,821	`00_00_30_40_12_51_07_20_41_12_19`

PLUSMINUS


Initial string size	"steps"	Champion
1	1	`+`
2	3	`-+`
3	4	`-++`
4	7	`-+++`
5	15	`++++-`
6	18	`+++-++`
7	20	`++++---`
8	32	`--++++++`
9	44	`++++-+--+`
10	196	`+++++-+-+-`
11	913	`+++++-+--+-`
12	5940	`+++-++++++-+`
13	728	`+-++++++--++-`
14	608	`++++-++++++---`
15	2582	`-+-++++++---+++`
16	31441	`+++++-++++-+-++-`

Wang Tiles (BB_WT(n))


n	BB_WT(n)	Champion	Reference
BB_WT(1)	1	`[1,0,0,0]`
BB_WT(2)	2	`[1,0,2,0] [2,0,3,0]`	^[100]
BB_WT(3)	3	`[1,0,2,0] [2,0,3,0] [3,0,4,0]`	^[100]
BB_WT(4)	4	`[1,0,2,0] [2,0,3,0] [3,0,4,0] [4,0,5,0]`	^[101]
BB_WT(5)	6	`[1,0,2,0] [2,0,3,0] [3,0,4,0] [4,6,1,5] [4,5,2,6]`	^[102]
BB_WT(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]`	^[102]
BB_WT(7)	16	`[4,0,3,0] [3,0,4,1] [3,1,3,2] [5,2,4,0] [4,1,4,2] [4,1,3,1] [3,2,5,0]`	^[102]^[103]
BB_WT(8)	≥ 17	`[4,0,3,0] [3,0,4,1] [3,1,3,2] [5,2,4,0] [4,1,4,2] [4,1,3,1] [3,2,5,0] [0,0,3,1]`	^[102]^[103]
BB_WT(9)	≥ 17	`[1,0,2,0] [2,0,3,0] [3,0,4,0] [4,0,5,0] [5,0,6,0] [6,0,7,0] [7,9,1,8] [7,8,2,9] [6,8,3,9]`
BB_WT(10)	≥ 212		^[104]

Branching Beaver (BrB(n))

Maximum Steps Function (BrB(n))


Domain	Steps	Champions
BrB(1)	3	`1RA1RZ`
BrB(2)	10	`1RA0GB_1GA1RZ`
BrB(3)	≥ 52	`1RB0GB_0GC1RZ_1GC0RA`
BrB(4)	≥ 4636	`1RA0GB_1RC1RD_1GA0GC_1RZ0GA`
BrB(2,3)	≥ 68	`1RA2GA0BB_1BA2BA1RZ1RA2GA0BB_2BA1RZ1BA`
BrB(2,4)	≥ 332,372	`1RB2RB1GB1RZ_2GB3GA3RA1RA`

Maximum Space Function (BrB_space(n))


Domain	Max Space	Champions
BrB_space(1)	2	`1RA1RZ`
BrB_space(2)	4	`1RA0GB_1GA1RZ` (and 10 others)
BrB_space(3)	≥ 22	`1RB1RZ_0GC0BB_1GA0RA`
BrB_space(4)	≥ 384	`1RB0BB_0RC1RZ_0GA1GD_1RC0RA`
BrB_space(2,3)	≥ 20	`1RA2GA0BB_1BA2BA1RZ1RA2GA0BB_2BA1RZ1BA`
BrB_space(2,4)	≥ 5996	`1RB2GA3BA0RA_1GB2BB1RA1RZ`

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(1)	3	`1TA1PZ`
TT(2)	≥ 13	`1TB---_1PA0PB`
TT(3)	≥ 82	`1PB0PA_1TA0PC_1PA---`
TT(4)	≥ 48,186	`1TB1PA_1PC0PA_1TA0PD_---1TA`
TT(1,3)	≥ 5	`1TA2PA1PZ`
TT(2,3)	≥ 223	`1TB0PA2PA_2PA---1PA`
TT(3,3)	> $2.27 \times 1 0^{248}$ ^[105]	`1PB0TC1PA_2PC1PZ2PA_2TC0PA2PC`
TT(1,4)	≥ 9	`1TA2PA3PA1PZ`
TT(2,4)	> 3.467*10¹⁵	`1TA2PB3TB---_3TA1PB1TA1PA`

Maximum score for TT(n,k)

Domain	Score	Champion
TT(3,3)	≥ 32,778^[106]	`1PB2PA0PC_2TB1PC2PB_2PB1PZ0PA`

2D Turmites (turNing machines)

Domain	Runtime	Champion
TT_2D(1)	5	`1RA1PZ`
TT_2D(2)	≥ 121	`1LA0LB_1PZ1RA`
TT_2D(3)	≥ 21,790	`1RB1PA_1RB1RC_1PZ1PA`
TT_2D(4)	≥ 287,251	`1PB0TD_0RA0PC_1PZ0TB_1PA0PA`
TT_2D(1,3)	≥ 9	`1RA2RA1PZ`
TT_2D(2,3)	≥ 59,348,218	`1RB2TB2PZ_1RA1PB1TB`

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 Game)

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#
##########

References

↑ ^1.0 ^1.1 ^1.2 ^1.3 ^1.4 ^1.5 Determination of the fifth Busy Beaver value. The bbchallenge Collaboration et al. https://arxiv.org/abs/2509.12337
↑ ^2.0 ^2.1 https://discord.com/channels/960643023006490684/1387426381041893417/1429556125539369040
↑ ^3.0 ^3.1 Jeff Ketchersid, "BB(7): 1RB0RG_1RC1RB_0RD---_0LE0LD_0LF1LE_1RA1LF_0LD0RB (champ_231)". Github file, October 2026. Accessed 27 October 2026.
↑ ^4.0 ^4.1 Jeff Ketchersid, "M8: an 8-state machine with f_ω(f_ω(f_ω(12))) < σ < f_ω(f_ω(f_ω(13)))". Github file, October 2026. Accessed 27 October 2026.
↑ Jeff Ketchersid, "h003: a 9-state machine above f_ω⁷(12), four levels above M8". Github file, October 2026. Accessed 27 October 2026.
↑ Jeff Ketchersid, "n1: a 10-state machine at level f_(ω+1)". Github file, October 2026. Accessed 27 October 2026.
↑ Jeff Ketchersid, "BB11/: 11 states, 2 symbols". Github file, October 2026. Accessed 27 October 2026.
↑ Jeff Ketchersid, "M12: a 12-state machine at level f_(ω+2)". Github file, October 2026. Accessed 27 October 2026.
↑ Jeff Ketchersid, "M13b: a 13-state machine at level f_(ω+2), six rounds deep". Github file, October 2026. Accessed 30 October 2026.
↑ Jeff Ketchersid, "M14_31: a 14-state machine at level f_(ω+2), thirty-one rounds deep". Github file, November 2026. Accessed 9 November 2026.
↑ Jeff Ketchersid, "J15_fw13: a 15-state machine at level f_(ω+13), on Jacobzheng's two-list engine". Github file, November 2026. Accessed 9 November 2026.
↑ Jeff Ketchersid, "J16_w2: a 16-state machine at level f_(ω·2), on Jacobzheng's two-list engine". Github file, November 2026. Accessed 9 November 2026.
↑ Jeff Ketchersid, "J18_w2_C: an 18-state machine at level f_(ω·2), on Jacobzheng's two-list engine". Github file, November 2026. Accessed 9 November 2026.
↑ Jeff Ketchersid, "J20: Jacobzheng's BB(20) champion, at level f_(ω·2)". Github file, November 2026. Accessed 9 November 2026.
↑ Jeff Ketchersid, "B21: Racheline's BB(21) champion, at level f_(ω²)". Github file, November 2026. Accessed 9 November 2026.
↑ Jeff Ketchersid, "W2PR: a 25-state machine built on Racheline's B21, at level f_(ω²)^5". Github file, November 2026. Accessed 9 November 2026.
↑ ^17.0 ^17.1 Jeff Ketchersid, "BB29: a 29-state hydra machine at level f_{ω^ω}". Github file, October 2026. Accessed 30 October 2026.
↑ ^18.0 ^18.1 Jeff Ketchersid, "BB31: a 31-state hydra machine above f_{ω^ω^ω^ω}(3)". Github file, October 2026. Accessed 30 October 2026.
↑ ^19.0 ^19.1 Jeff Ketchersid, "BB33: a 33-state hydra machine above f_{ω^ω^ω^ω^ω^ω}(3)". Github file, October 2026. Accessed 30 October 2026.
↑ https://discord.com/channels/960643023006490684/1331570843829932063/1567585091172040726
↑ ^21.0 ^21.1 https://discord.com/channels/960643023006490684/1026577255754903572/1328863966688182345
↑ ^22.0 ^22.1 ^22.2 Jeff Ketchersid, "The BB(4,3) champion, machine-checked". Github file, October 2026. Accessed 2 November 2026.
↑ https://discord.com/channels/960643023006490684/1084047886494470185/1568460630002892892
↑ ^24.0 ^24.1 S. Ligocki. "BB(3,4) > Ack(14)." https://www.sligocki.com/2024/05/22/bb-3-4-a14.html Blog post, 2024. Accessed 15 August 2025.
↑ https://discord.com/channels/960643023006490684/1084047886494470185/1564816765031288876
↑ https://discord.com/channels/960643023006490684/1084047886494470185/1569814397801468005
↑ https://discord.com/channels/960643023006490684/1084047886494470185/1568659622204674234
↑ https://discord.com/channels/960643023006490684/1084047886494470185/1569676710771818526
↑ https://discord.com/channels/960643023006490684/1084047886494470185/1564866772941672489
↑ https://discord.com/channels/960643023006490684/1084047886494470185/1566039528819400716
↑ https://discord.com/channels/960643023006490684/1084047886494470185/1566103427316977789
↑ https://discord.com/channels/960643023006490684/1084047886494470185/1566111131926274121
↑ https://discord.com/channels/960643023006490684/1084047886494470185/1569678295098855628
↑ https://discord.com/channels/960643023006490684/1084047886494470185/1570308326602969159
↑ ^35.0 ^35.1 S. Ligocki, "BB(2,6) > 10↑↑10↑↑10↑↑3". Blog post, 2023. Accessed 15 August 2025.
↑ https://discord.com/channels/960643023006490684/1084047886494470185/1566700247881883699
↑ https://discord.com/channels/960643023006490684/1084047886494470185/1566703228169486477
↑ ^38.0 ^38.1 https://discord.com/channels/960643023006490684/1084047886494470185/1567192072165924935
↑ ^39.0 ^39.1 https://discord.com/channels/960643023006490684/1084047886494470185/1568599154039586929
↑ ^40.0 ^40.1 https://discord.com/channels/960643023006490684/1084047886494470185/1568093269123145840
↑ https://discord.com/channels/960643023006490684/960643023530762341/1563057558011252761
↑ https://discord.com/channels/960643023006490684/1084047886494470185/1567559567519522909
↑ https://discord.com/channels/960643023006490684/1084047886494470185/1567910427856470071
↑ https://discord.com/channels/960643023006490684/1084047886494470185/1563070360956698804
↑ https://discord.com/channels/960643023006490684/1084047886494470185/1563070360956698804
↑ ^46.0 ^46.1 Scott Aaronson. "The Busy Beaver Frontier". https://www.scottaaronson.com/papers/bb.pdf
↑ John Carrino, "Coq-BBB4"
↑ Nick Drozd. https://scottaaronson.blog/?p=8088#comment-1981333
↑ Nick Drozd. "BBB(3,3) > 10↑↑6". Accessed 15 August 2025.
↑ ^50.0 ^50.1 Nick Drozd. "A Tetrational Self-Cleaning Turing Machine". Blog post. Accessed 17 October 2026.
↑ ^51.0 ^51.1 Nick Drozd, "Blanking Beavers". Blog post, February 2021. Accessed 15 August 2025.
↑ racheline, September 2025. Discord message
↑ Charlie Reams. October 2026. Discord message
↑ ^54.0 ^54.1 Charlie Reams, GitHub https://github.com/calr3/bbbb
↑ ^55.0 ^55.1 J. Hertel, "Computing the Uncomputable Rado Sigma Function". The Mathematica Journal vol. 11 (2009).
↑ https://discord.com/channels/960643023006490684/1084047886494470185/1569172128165527636
↑ https://discord.com/channels/960643023006490684/1243312334907375676/1389807954013978756
↑ https://discord.com/channels/960643023006490684/1243312334907375676/138980795401397875
↑ ^59.0 ^59.1 https://discord.com/channels/960643023006490684/1243312334907375676/1389809599955210362
↑ https://discord.com/channels/960643023006490684/1243312334907375676/1389807954013978756
↑ https://discord.com/channels/960643023006490684/1243312334907375676/1389817569342652578
↑ ^62.0 ^62.1 https://discord.com/channels/960643023006490684/1243312334907375676/1389820614415618109
↑ ^63.0 ^63.1 https://discord.com/channels/960643023006490684/1239205785913790465/1400255249847025776
↑ ^64.0 ^64.1 ^64.2 Nick Drozd, "Latest Beeping Busy Beaver Results". Blog post, February 2022. Accessed 15 August 2025.
↑ Nick Drozd, "A New Record in Self-Cleaning Turing Machines". Blog post, July 2021. Accessed 3 November 2026.
↑ Nick Drozd, "Blanking Beavers: Results and Open Problems". Blog post, October 2026. Accessed 4 November 2026.
↑ https://discord.com/channels/960643023006490684/1369339127652159509/1374531970830307409
↑ ^68.0 ^68.1 ^68.2 Nick Drozd, "Blanking Beavers". Blog post, February 2021. Accessed 3 November 2026.
↑ Shawn Ligocki, "Skelet #1 is infinite ... we think". Blog post, March 2023. Accessed 3 November 2026.
↑ https://discord.com/channels/960643023006490684/1239205785913790465/1567517097360363621
↑ ^71.0 ^71.1 https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829
↑ ^72.0 ^72.1 https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353
↑ ^73.0 ^73.1 https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447
↑ ^74.0 ^74.1 https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341
↑ ^75.0 ^75.1 https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207
↑ ^76.0 ^76.1 Shawn Ligocki, List of long-running BB(3,3) TMs
↑ Nick Drozd, "The Shape of a Turing Machine". Blog post, September 2025. Accessed 4 November 2026.
↑ ^78.0 ^78.1 https://discord.com/channels/960643023006490684/1084047886494470185/1564808804024197152
↑ ^79.0 ^79.1 https://discord.com/channels/960643023006490684/1084047886494470185/1564866772941672489
↑ https://discord.com/channels/960643023006490684/1084047886494470185/1566710164482957434
↑ https://discord.com/channels/960643023006490684/1084047886494470185/1566818235800625315
↑ https://discord.com/channels/960643023006490684/1084047886494470185/1567350954573963377
↑ https://discord.com/channels/960643023006490684/1026577255754903572/1274471360206344213
↑ https://discord.com/channels/960643023006490684/1331570843829932063/1567585091172040726
↑ https://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252
↑ ^86.0 ^86.1 ^86.2 ^86.3 ^86.4 ^86.5 ^86.6 https://codegolf.stackexchange.com/questions/282018/maximize-the-number-of-steps-a-brainfuck-program-of-length-13-takes-to-terminate/
↑ ^87.00 ^87.01 ^87.02 ^87.03 ^87.04 ^87.05 ^87.06 ^87.07 ^87.08 ^87.09 ^87.10 ^87.11 ^87.12 ^87.13 ^87.14 ^87.15 ^87.16 ^87.17 ^87.18 ^87.19 ^87.20 https://discord.com/channels/960643023006490684/1243312334907375676/1346489746787926056
↑ https://discord.com/channels/960643023006490684/1438019511155691521/1547182811713380383
↑ https://discord.com/channels/960643023006490684/960643023530762341/1440163057463726233
↑ https://discord.com/channels/960643023006490684/1438694294042181742/1440193579006951517
↑ https://discord.com/channels/960643023006490684/1438694294042181742/1443246244142252043
↑ https://discord.com/channels/960643023006490684/1438694294042181742/1443298934217900063
↑ https://discord.com/channels/960643023006490684/1438694294042181742/1442847677883875479
↑ https://discord.com/channels/960643023006490684/1438694294042181742/1442950825545564281
↑ https://discord.com/channels/960643023006490684/1438694294042181742/1442819117735346217
↑ https://discord.com/channels/960643023006490684/1535695107448381591/1536023084975525999
↑ ^97.0 ^97.1 https://discord.com/channels/960643023006490684/1535695107448381591/1536608195156381747
↑ https://discord.com/channels/960643023006490684/1535695107448381591/1537528571394793563
↑ https://discord.com/channels/960643023006490684/1535695107448381591/1539048792400203796
↑ ^100.0 ^100.1 https://discord.com/channels/960643023006490684/1529586687926210691/1530032750579417198
↑ https://discord.com/channels/960643023006490684/1529586687926210691/1530697443820437645
↑ ^102.0 ^102.1 ^102.2 ^102.3 Yeyito, "BB_WT in Rocq: BB_WT(5) = 6, BB_WT(6) = 8, BB_WT(7) = 16, BB_WT(8) ≥ 17". Github document, November 2026. Accessed 7 November 2026.
↑ ^103.0 ^103.1 https://discord.com/channels/960643023006490684/1529586687926210691/1567754354671493152
↑ Emmanuel Jeandel, Michaël Rao, "An aperiodic set of 11 Wang tiles". January 2021. Accessed 7 November 2026.
↑ https://discord.com/channels/960643023006490684/1243312334907375676/1535305073259974728
↑ https://discord.com/channels/960643023006490684/1243312334907375676/1533503518906912778

[BB5-1] 1.0 ^1.1 ^1.2 ^1.3 ^1.4 ^1.5 Determination of the fifth Busy Beaver value. The bbchallenge Collaboration et al. https://arxiv.org/abs/2509.12337

[PeacemakerBB6champ-2] 2.0 ^2.1 https://discord.com/channels/960643023006490684/1387426381041893417/1429556125539369040

[KetchersidBB7A20-3] 3.0 ^3.1 Jeff Ketchersid, "BB(7): 1RB0RG_1RC1RB_0RD---_0LE0LD_0LF1LE_1RA1LF_0LD0RB (champ_231)". Github file, October 2026. Accessed 27 October 2026.

[KetchersidBB8f3w-4] 4.0 ^4.1 Jeff Ketchersid, "M8: an 8-state machine with f_ω(f_ω(f_ω(12))) < σ < f_ω(f_ω(f_ω(13)))". Github file, October 2026. Accessed 27 October 2026.

[5] Jeff Ketchersid, "h003: a 9-state machine above f_ω⁷(12), four levels above M8". Github file, October 2026. Accessed 27 October 2026.

[6] Jeff Ketchersid, "n1: a 10-state machine at level f_(ω+1)". Github file, October 2026. Accessed 27 October 2026.

[7] Jeff Ketchersid, "BB11/: 11 states, 2 symbols". Github file, October 2026. Accessed 27 October 2026.

[8] Jeff Ketchersid, "M12: a 12-state machine at level f_(ω+2)". Github file, October 2026. Accessed 27 October 2026.

[9] Jeff Ketchersid, "M13b: a 13-state machine at level f_(ω+2), six rounds deep". Github file, October 2026. Accessed 30 October 2026.

[10] Jeff Ketchersid, "M14_31: a 14-state machine at level f_(ω+2), thirty-one rounds deep". Github file, November 2026. Accessed 9 November 2026.

[11] Jeff Ketchersid, "J15_fw13: a 15-state machine at level f_(ω+13), on Jacobzheng's two-list engine". Github file, November 2026. Accessed 9 November 2026.

[12] Jeff Ketchersid, "J16_w2: a 16-state machine at level f_(ω·2), on Jacobzheng's two-list engine". Github file, November 2026. Accessed 9 November 2026.

[13] Jeff Ketchersid, "J18_w2_C: an 18-state machine at level f_(ω·2), on Jacobzheng's two-list engine". Github file, November 2026. Accessed 9 November 2026.

[14] Jeff Ketchersid, "J20: Jacobzheng's BB(20) champion, at level f_(ω·2)". Github file, November 2026. Accessed 9 November 2026.

[15] Jeff Ketchersid, "B21: Racheline's BB(21) champion, at level f_(ω²)". Github file, November 2026. Accessed 9 November 2026.

[16] Jeff Ketchersid, "W2PR: a 25-state machine built on Racheline's B21, at level f_(ω²)^5". Github file, November 2026. Accessed 9 November 2026.

[KetchersidBB29wtet2-17] 17.0 ^17.1 Jeff Ketchersid, "BB29: a 29-state hydra machine at level f_{ω^ω}". Github file, October 2026. Accessed 30 October 2026.

[KetchersidBB31wtet4-18] 18.0 ^18.1 Jeff Ketchersid, "BB31: a 31-state hydra machine above f_{ω^ω^ω^ω}(3)". Github file, October 2026. Accessed 30 October 2026.

[KetchersidBB33wtet6-19] 19.0 ^19.1 Jeff Ketchersid, "BB33: a 33-state hydra machine above f_{ω^ω^ω^ω^ω^ω}(3)". Github file, October 2026. Accessed 30 October 2026.

[20] ttps://discord.com/channels/960643023006490684/1331570843829932063/1567585091172040726

[PatcailBB150BMS-21] 21.0 ^21.1 https://discord.com/channels/960643023006490684/1026577255754903572/1328863966688182345

[KetchersidBB43hex1-22] 22.0 ^22.1 ^22.2 Jeff Ketchersid, "The BB(4,3) champion, machine-checked". Github file, October 2026. Accessed 2 November 2026.

[Loader3229BB7x4BB8x3BB9x3-23] ttps://discord.com/channels/960643023006490684/1084047886494470185/1568460630002892892

[:0-24] 24.0 ^24.1 S. Ligocki. "BB(3,4) > Ack(14)." https://www.sligocki.com/2024/05/22/bb-3-4-a14.html Blog post, 2024. Accessed 15 August 2025.

[25] ttps://discord.com/channels/960643023006490684/1084047886494470185/1564816765031288876

[26] ttps://discord.com/channels/960643023006490684/1084047886494470185/1569814397801468005

[27] ttps://discord.com/channels/960643023006490684/1084047886494470185/1568659622204674234

[28] ttps://discord.com/channels/960643023006490684/1084047886494470185/1569676710771818526

[29] ttps://discord.com/channels/960643023006490684/1084047886494470185/1564866772941672489

[30] ttps://discord.com/channels/960643023006490684/1084047886494470185/1566039528819400716

[31] ttps://discord.com/channels/960643023006490684/1084047886494470185/1566103427316977789

[32] ttps://discord.com/channels/960643023006490684/1084047886494470185/1566111131926274121

[33] ttps://discord.com/channels/960643023006490684/1084047886494470185/1569678295098855628

[Loader3329BB17x5Eps0of20-34] ttps://discord.com/channels/960643023006490684/1084047886494470185/1570308326602969159

[:1-35] 35.0 ^35.1 S. Ligocki, "BB(2,6) > 10↑↑10↑↑10↑↑3". Blog post, 2023. Accessed 15 August 2025.

[36] ttps://discord.com/channels/960643023006490684/1084047886494470185/1566700247881883699

[37] ttps://discord.com/channels/960643023006490684/1084047886494470185/1566703228169486477

[Loader3229BB6x6w2-38] 38.0 ^38.1 https://discord.com/channels/960643023006490684/1084047886494470185/1567192072165924935

[Loader3229BB7x6wsquaredof110-39] 39.0 ^39.1 https://discord.com/channels/960643023006490684/1084047886494470185/1568599154039586929

[Loader3221BB8x6fwsquarep1fw-40] 40.0 ^40.1 https://discord.com/channels/960643023006490684/1084047886494470185/1568093269123145840

[41] ttps://discord.com/channels/960643023006490684/960643023530762341/1563057558011252761

[42] ttps://discord.com/channels/960643023006490684/1084047886494470185/1567559567519522909

[43] ttps://discord.com/channels/960643023006490684/1084047886494470185/1567910427856470071

[44] ttps://discord.com/channels/960643023006490684/1084047886494470185/1563070360956698804

[45] ttps://discord.com/channels/960643023006490684/1084047886494470185/1563070360956698804

[:Aaronson2020-46] 46.0 ^46.1 Scott Aaronson. "The Busy Beaver Frontier". https://www.scottaaronson.com/papers/bb.pdf

[:Coq-BBB4-47] John Carrino, "Coq-BBB4"

[48] Nick Drozd. https://scottaaronson.blog/?p=8088#comment-1981333

[:2-49] Nick Drozd. "BBB(3,3) > 10↑↑6". Accessed 15 August 2025.

[:Nick_Drozd_Tetrational_Blank26-50] 50.0 ^50.1 Nick Drozd. "A Tetrational Self-Cleaning Turing Machine". Blog post. Accessed 17 October 2026.

[:3-51] 51.0 ^51.1 Nick Drozd, "Blanking Beavers". Blog post, February 2021. Accessed 15 August 2025.

[52] racheline, September 2025. Discord message

[53] Charlie Reams. October 2026. Discord message

[calr3-bbbb-54] 54.0 ^54.1 Charlie Reams, GitHub https://github.com/calr3/bbbb

[Hertel2009-55] 55.0 ^55.1 J. Hertel, "Computing the Uncomputable Rado Sigma Function". The Mathematica Journal vol. 11 (2009).

[56] ttps://discord.com/channels/960643023006490684/1084047886494470185/1569172128165527636

[57] ttps://discord.com/channels/960643023006490684/1243312334907375676/1389807954013978756

[58] ttps://discord.com/channels/960643023006490684/1243312334907375676/138980795401397875

[LegionBBrev4-59] 59.0 ^59.1 https://discord.com/channels/960643023006490684/1243312334907375676/1389809599955210362

[60] ttps://discord.com/channels/960643023006490684/1243312334907375676/1389807954013978756

[61] ttps://discord.com/channels/960643023006490684/1243312334907375676/1389817569342652578

[LegionBBrev5-62] 62.0 ^62.1 https://discord.com/channels/960643023006490684/1243312334907375676/1389820614415618109

[SligockiBBrev6-63] 63.0 ^63.1 https://discord.com/channels/960643023006490684/1239205785913790465/1400255249847025776

[:4-64] 64.0 ^64.1 ^64.2 Nick Drozd, "Latest Beeping Busy Beaver Results". Blog post, February 2022. Accessed 15 August 2025.

[NickdrozdJul2021-65] Nick Drozd, "A New Record in Self-Cleaning Turing Machines". Blog post, July 2021. Accessed 3 November 2026.

[66] Nick Drozd, "Blanking Beavers: Results and Open Problems". Blog post, October 2026. Accessed 4 November 2026.

[67] ttps://discord.com/channels/960643023006490684/1369339127652159509/1374531970830307409

[NickdrozdBlankingBeavers2021-68] 68.0 ^68.1 ^68.2 Nick Drozd, "Blanking Beavers". Blog post, February 2021. Accessed 3 November 2026.

[69] Shawn Ligocki, "Skelet #1 is infinite ... we think". Blog post, March 2023. Accessed 3 November 2026.

[70] ttps://discord.com/channels/960643023006490684/1239205785913790465/1567517097360363621

[BBiShawn1-71] 71.0 ^71.1 https://discord.com/channels/960643023006490684/960643023530762341/1393716114952818829

[BBiShawn2-72] 72.0 ^72.1 https://discord.com/channels/960643023006490684/960643023530762341/1393716181214564353

[BBiShawn3-73] 73.0 ^73.1 https://discord.com/channels/960643023006490684/960643023530762341/1393716260587700447

[BBiShawn5-74] 74.0 ^74.1 https://discord.com/channels/960643023006490684/960643023530762341/1393716514234040341

[BBiShawn6-75] 75.0 ^75.1 https://discord.com/channels/960643023006490684/960643023530762341/1393768821478785207

[BBiShawn7-76] 76.0 ^76.1 Shawn Ligocki, List of long-running BB(3,3) TMs

[77] Nick Drozd, "The Shape of a Turing Machine". Blog post, September 2025. Accessed 4 November 2026.

[Loaderi13i14-78] 78.0 ^78.1 https://discord.com/channels/960643023006490684/1084047886494470185/1564808804024197152

[LoaderBBi17y18-79] 79.0 ^79.1 https://discord.com/channels/960643023006490684/1084047886494470185/1564866772941672489

[80] ttps://discord.com/channels/960643023006490684/1084047886494470185/1566710164482957434

[81] ttps://discord.com/channels/960643023006490684/1084047886494470185/1566818235800625315

[Loader3229BB7x6wsquare-82] ttps://discord.com/channels/960643023006490684/1084047886494470185/1567350954573963377

[83] ttps://discord.com/channels/960643023006490684/1026577255754903572/1274471360206344213

[84] ttps://discord.com/channels/960643023006490684/1331570843829932063/1567585091172040726

[85] ttps://discord.com/channels/960643023006490684/1084047886494470185/1398753236835635252

[:CodegolfBBbrainf-86] 86.0 ^86.1 ^86.2 ^86.3 ^86.4 ^86.5 ^86.6 https://codegolf.stackexchange.com/questions/282018/maximize-the-number-of-steps-a-brainfuck-program-of-length-13-takes-to-terminate/

[truskboolfuckrecords-87] 87.00 ^87.01 ^87.02 ^87.03 ^87.04 ^87.05 ^87.06 ^87.07 ^87.08 ^87.09 ^87.10 ^87.11 ^87.12 ^87.13 ^87.14 ^87.15 ^87.16 ^87.17 ^87.18 ^87.19 ^87.20 https://discord.com/channels/960643023006490684/1243312334907375676/1346489746787926056

[88] ttps://discord.com/channels/960643023006490684/1438019511155691521/1547182811713380383

[89] ttps://discord.com/channels/960643023006490684/960643023530762341/1440163057463726233

[90] ttps://discord.com/channels/960643023006490684/1438694294042181742/1440193579006951517

[91] ttps://discord.com/channels/960643023006490684/1438694294042181742/1443246244142252043

[92] ttps://discord.com/channels/960643023006490684/1438694294042181742/1443298934217900063

[93] ttps://discord.com/channels/960643023006490684/1438694294042181742/1442847677883875479

[94] ttps://discord.com/channels/960643023006490684/1438694294042181742/1442950825545564281

[95] ttps://discord.com/channels/960643023006490684/1438694294042181742/1442819117735346217

[96] ttps://discord.com/channels/960643023006490684/1535695107448381591/1536023084975525999

[PA67-97] 97.0 ^97.1 https://discord.com/channels/960643023006490684/1535695107448381591/1536608195156381747

[98] ttps://discord.com/channels/960643023006490684/1535695107448381591/1537528571394793563

[99] ttps://discord.com/channels/960643023006490684/1535695107448381591/1539048792400203796

[WT23-100] 100.0 ^100.1 https://discord.com/channels/960643023006490684/1529586687926210691/1530032750579417198

[101] ttps://discord.com/channels/960643023006490684/1529586687926210691/1530697443820437645

[YeyitoWTRocq-102] 102.0 ^102.1 ^102.2 ^102.3 Yeyito, "BB_WT in Rocq: BB_WT(5) = 6, BB_WT(6) = 8, BB_WT(7) = 16, BB_WT(8) ≥ 17". Github document, November 2026. Accessed 7 November 2026.

[YeyitoWTdcannounce-103] 103.0 ^103.1 https://discord.com/channels/960643023006490684/1529586687926210691/1567754354671493152

[JeandelRao2021-104] Emmanuel Jeandel, Michaël Rao, "An aperiodic set of 11 Wang tiles". January 2021. Accessed 7 November 2026.

[105] ttps://discord.com/channels/960643023006490684/1243312334907375676/1535305073259974728

[TT(3,3)_715M-106] ttps://discord.com/channels/960643023006490684/1243312334907375676/1533503518906912778

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@@ Line 253: / Line 253: @@
 |-
 |BB(5,4)
-|<math>> f_\omega^2(10\uparrow^4 10\uparrow\uparrow 65533)</math>
+|<math>> f_\omega^3(2 \uparrow^{15} 5)</math>
-|<span style="word-break:break-all">{{TM|1LD2RA3RB1LC_1LD2LA3RB---_2RA1LA3RC2LC_2RA---2LD2LE_------1RZ1LD|halt}}</span>
+|<span style="word-break:break-all">{{TM|1RB3LB1RD2RA_2LC3RB1LC2RA_3RB1LB3LC2RC_1LE2LE3LE1RZ_3RC2LE1LE3LE|halt}}</span>
-|<ref>Jeff Ketchersid, "[https://github.com/jeff87654/busy-beaver/blob/main/BB54/1LD3LB---2RA_1LD3RB1LC2RA_3RB1LB3LC2RC_3RB---3LE3LD_------1LD1RZ.md 1LD3LB---2RA_1LD3RB1LC2RA_3RB1LB3LC2RC_3RB---3LE3LD_------1LD1RZ]". Github file, November 2026. Accessed 9 November 2026.</ref>
+|<ref>https://discord.com/channels/960643023006490684/1084047886494470185/1569814397801468005</ref>
 |-
 |BB(6,4)
@@ Line 273: / Line 273: @@
 |-
 |BB(9,4)
-|
+|<math>> f_{\omega^2}(18)</math>
-|
+|<span style="word-break:break-all">{{TM|1LD2RG2LA3LG_1LI2LH2RB2RF_1LG1RC2RB3LG_0RC1RC1LD3RD_3RD1LE1LE3LE_1RZ3LD2LE3LE_1RG1LH2RG3LG_1RC2LH1LA1LE_0RC3LI1RI0RI|halt}}</span>
-|
+|<ref>https://discord.com/channels/960643023006490684/1084047886494470185/1569676710771818526</ref>
-|-
-|BB(11,4)
-|<math>> f_{\omega \times 2}(2 \uparrow^{11} 6)</math>
-|<span style="word-break:break-all">{{TM|1RK3LB1RZ2RA_0LG1LE1LD2RA_3RB1LB3LC2RC_3RE0LI3LC2RC_1RJ3RE3LH2LF_3RF2RF2LE1LF_3LC3LH2LJ2RG_0LE3RH1LI2RG_3RH1LH3LI2RI_0LB0LJ3LJ3RK_2LC0LD2RA3RK|halt}}</span>
-|<ref>https://discord.com/channels/960643023006490684/1084047886494470185/1567782759185784953</ref>
 |-
-|BB(12,4)
+|BB(10,4)
-|<math>> f_{\omega \times 2}^2(2 \uparrow^{6} 3)</math>
+|<math>> f_{\omega^2}^3(3)</math>
-|<span style="word-break:break-all">{{TM|1LH---1LD2RB_2LL3LA1RZ2RB_3RA1LA3LC2RC_3RE---3LC2RC_2LI3RE3LH2LF_3RF2RF2LE1LF_1LL3LH2LJ2RG_2LC3RH1LI2RG_3RH1LH3LI2RI_------3LJ3RK_------2RB3RK_3RB---1LL3LL|halt}}</span>
+|<span style="word-break:break-all">{{TM|1LD2RG2LA3LG_0LI2LH2RB2RF_1LB1RC2RB3LG_0RC1RC1LD3RD_3RD1LE1LE3LE_1RI3LD2LE3LE_1RG1LH2RG3LG_1RC2LH1LA1LE_0LJ1RZ0RE0LI_3RC3LJ3LJ3LJ|halt}}</span>
 |
 |-
@@ Line 335: / Line 330: @@
 |-
 |BB(8,5)
-|<math>> f_{\omega \times 2+1}(93)</math>
+|<math>> f_{\omega^2}(13)</math>
-|<span style="word-break:break-all">{{TM|2RG3LB4LC2RA4LB_1RZ---1LD2RA4LD_3RB1LB3LC2RC4LC_3RE---3LC2RC3LC_---3RE4LE4LH4RB_4LE3LG3LB2RF4LG_2LH3RG1LH2RF2RA_3RG1LG3LH2RH4LH|halt}}</span>
+|<span style="word-break:break-all">{{TM|1LD2RG2LA3LG---_4LC2LH2RB2RF0RB_4LE1RC2RB3LG4RB_0RC1RC1LD3RD3LA_3RD1LE1LE3LE4RE_1RZ3LD2LE3LE4RD_1RG1LH2RG3LG3LF_1RC2LH1LA1LE---|halt}}</span>
-|<ref>https://discord.com/channels/960643023006490684/1084047886494470185/1566410612949385317</ref>
+|<ref>https://discord.com/channels/960643023006490684/1084047886494470185/1569678295098855628</ref>
+|-
+|BB(17,5)
+|<math>> f_{\varepsilon_0}(20)</math>
+|<span style="word-break:break-all">{{TM|4RF---0RF0RD1LQ_1RZ1RB2LC------_0LI4RC2RC3RC2RC_1LE1RD2RD3RD---_3LA1LE2LE3LE1LG_0LG1RF2RF3RF---_0RI0LH4RD3RI2LH_2LA1LH2LH3LH0RC_3RN---0LA0LI2RF_2RO---2LJ3LJ---_2LL---2RK3RK4RK_3RO2RO2LL3LL4LL_3LL---2RM3RM4RM_4LJ1LN2LN3LN1LO_1RB4RP1RK0RM4LN_4LJ1RP2RP3RP0RP_1RB1LQ2LQ3LQ1LQ|halt}}</span>
+|<ref name="Loader3329BB17x5Eps0of20">https://discord.com/channels/960643023006490684/1084047886494470185/1570308326602969159</ref>
 |-
 |BB(19,5)
@@ Line 737: / Line 737: @@
 |-
 |num(2,3)
-|6
+|9
-|{{TM|1RB1LA1LB_0LA2RA1RZ|halt}}
+|{{TM|2RB1RZ1LB_1LA2LB1RB|halt}}
 |
 |-
@@ Line 755: / Line 755: @@
 |{{TM|1RB3LB1LD2RA_2LC3RB1LC2RA_3RB1LB3LC2RC_1LZ---1LD1LD|halt}}
 |
+|-
+|num(17,5)
+|<math>> f_{\varepsilon_0}(20)</math>
+|{{WrapTM|Machine code (Not TNF)|4RF---0RF0RD1LQ_1RZ1RB2LC------_0LI4RC2RC3RC2RC_1LE1RD2RD3RD---_3LA1LE2LE3LE1LG_0LG1RF2RF3RF---_0RI0LH4RD3RI2LH_2LA1LH2LH3LH0RC_3RN---0LA0LI2RF_2RO---2LJ3LJ---_2LL---2RK3RK4RK_3RO2RO2LL3LL4LL_3LL---2RM3RM4RM_4LJ1LN2LN3LN1LO_1RB4RP1RK0RM4LN_4LJ1RP2RP3RP0RP_1RB1LQ2LQ3LQ1LQ|halt}}
 |}
@@ Line 1,395: / Line 1,399: @@
 |-
 |BBi(32)
-|<math>> f_{\omega \times 2+1}(130)</math>
+|<math>> f_{\omega^2}(4)</math>
-|<span style="word-break:break-all">{{TM|5LF3LB1LG2RA4LB---_2LC3RB1LD2RA3LF2LB_3RB1LB3LC2RC4LC0LE_3RB---3LC2RC---3RE_5RC3RE2LF4LC---0RB_5RB---2LE1LF------_------1LG3LG---0LE|halt}}</span>
+|<span style="word-break:break-all">{{TM|1LD2RG2LA3LG_---2LH2RB2RF_3LI1RC2RB3LG_0RC1RC1LD3RD_3RD1LE1LE3LE_1RZ3LD2LE3LE_1RG1LH2RG3LG_1RC2LH1LA1LE_3LA3LE------|halt}}</span>
-|BB(7,6)
+|BB(9,4)
 |
 |-
 |BBi(33)
-|
+|<math>> f_{\omega^2}(7)</math>
-|
+|<span style="word-break:break-all">{{TM|1LD2RG2LA3LG---_---2LH2RB2RF---_4LE1RC2RB3LG4RE_0RC1RC1LD3RD3LA_3RD1LE1LE3LE---_1RZ3LD2LE3LE---_1RG1LH2RG3LG3LF_1RC2LH1LA1LE---|halt}}</span>
+|BB(8,5)
 |
 |-
 |BBi(34)
-|<math>> f_{\omega^2}(5)</math>
+|<math>> f_{\omega^2}(16)</math>
-|<span style="word-break:break-all">{{TM|2LG3LB---2RA4LB---_4LG3RB1LD2RA3LF5RB_3RB1LB3LC2RC4LC1LE_3RB3RE3LC2RC---3RD_5RB3RE---4LC3LG5LE_5RB5RB---1LF---5LF_5LF---1LC1LG---5LG|halt}}</span>
+|<span style="word-break:break-all">{{TM|1LD2RG2LA3LG_3LI2LH2RB2RF_1LG1RC2RB3LG_0RC1RC1LD3RD_3RD1LE1LE3LE_1RZ3LD2LE3LE_1RG1LH2RG3LG_1RC2LH1LA1LE_3LD3LI3LI---|halt}}</span>
-|BB(7,6)
+|BB(9,4)
 |
 |-
@@ Line 1,614: / Line 1,619: @@
 |> 10<sup>42,745</sup>
 |{{TM|1RB2RB1LA_2LC0LB2LB_2RC2RA0LC}}
+|}
+== Instruction-Limited Maximum Consecutive Ones Function ([[Maximum Consecutive Ones Function#Instruction-Limited Maximum Consecutive Ones|num<sub>i</sub>(n)]]) ==
+{| class="wikitable"
+!Domain
+!num<sub>i</sub>(n)
+!Champion
+!Reference
+|-
+|num<sub>i</sub>(0)
+|0
+|{{TM|---|halt}}
+|
+|-
+|num<sub>i</sub>(1)
+|1
+|{{TM|1RZ|halt}}
+|
+|-
+|num<sub>i</sub>(2)
+|3
+|{{TM|1RB---_1LA---|halt}}
+|
+|-
+|num<sub>i</sub>(3)
+|4
+|{{TM|1RB1LB_1LA---|halt}}
+|
+|-
+|num<sub>i</sub>(4)
+|
+|
+|
+|-
+|num<sub>i</sub>(5)
+|9
+|{{TM|2RB---1LB_1LA2LB1RB|halt}}
+|
 |}

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

Latest revision as of 17:06, 12 November 2026

State-and-Symbol-Limited Busy Beaver functions

Original Busy Beaver Functions

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

Maximum Score Function (Σ(n,m))

Beeping Busy Beavers

Beeping Busy Beaver (BBB(n,m))

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

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

Maximum Space Function (BBspace(n,m))

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

Reversible Turing Machines

Maximum Shifts Function (BBrev(n,m))

Maximum Score Function (Σrev(n,m))

Blanking Busy Beaver (BLB(n,m))

Busy Cleaner (BB_clean(n,m))

Lazy Beaver

Shifts Function (LB(n,m))

Period-oriented Busy Beavers

Busy Preperiodic Beaver (BBS(n,m))

Busy Periodic Beaver (BBP(n,m))

Instruction-Limited Busy Beaver

Instruction-Limited Classical Busy Beaver Functions

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

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

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

Instruction-Limited Maximum Consecutive Ones Function (numi(n))

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

Program-Limited Busy Beaver

Busy Beaver for Lambda Calculus

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

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

Busy Beaver for De Bruijn Lambda Calculus (De Bruijn index)

Busy Beaver for SKI calculus

Busy Beaver for SKI calculus

Busy Beaver for SK calculus

Busy Beaver for BCKW calculus

Brainfuck

Boolfuck

Boolfuck score (Σboolf(n))

Boolfuck runtime (Sboolf(n))

CounterScript

Fractran (BBf(n))

Cyclic Tree Rewriting System (CTBB(n))

Cyclic Tag (BBCT(n))

Tiny Tag

Minsky Machines (MBB(n))

MBB(n)

Register Sigma function (MΣ(n,r))

Pebble Automaton (peBBle(n))

PLUSMINUS

Wang Tiles (BBWT(n))

Branching Beaver (BrB(n))

Maximum Steps Function (BrB(n))

Maximum Space Function (BrBspace(n))

Turmites

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

Maximum steps for TT(n,k)

Maximum score for TT(n,k)

2D Turmites (turNing machines)

Doodle Function (doodle(c,n))

Bug Function (Bug Game)

References

Navigation menu

Search

Maximum Space Function (BB_space(n,m))

Maximum Shifts Function (BB_rev(n,m))

Maximum Score Function (Σ_rev(n,m))

Instruction-Limited Maximum Consecutive Ones Function (num_i(n))

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

Boolfuck score (Σ_boolf(n))

Boolfuck runtime (S_boolf(n))

Wang Tiles (BB_WT(n))

Maximum Space Function (BrB_space(n))