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

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A collection of Busy Beaver [[Champions]] including Champions for BB-Adjacent [[:Category:Functions|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).
A collection of Busy Beaver [[Champions]] including Champions for BB-Adjacent [[:Category:Functions|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'''=
 
= '''State-and-Symbol-Limited Busy Beaver functions''' =
Busy Beaver functions where the programs size is limited by the amount of states and symbols.
Busy Beaver functions where the programs size is limited by the amount of states and symbols.
=='''Original Busy Beaver Functions'''==
== Original Busy Beaver Functions ==
===Maximum Shifts Function ([[Busy Beaver Functions|S(n,m)]], also commonly called BB(n,m))===
=== Maximum Shifts Function ([[Busy Beaver Functions|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.
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 10: Line 12:
!Champions
!Champions
|-
|-
|BB(1)
|[[BB(1)]]
|1
|1
|{{TM|1RZ---|halt}}
|{{TM|1RZ---|halt}}
Line 39: Line 41:
|-
|-
|[[BB(8)]]
|[[BB(8)]]
|
|<math>> 2 \uparrow^{37} 3</math><ref>https://github.com/jeff87654/busy-beaver/tree/main/BB8</ref>
|
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_1LC0RH_1RG1LF|halt}}
|-
|-
|BB(9)
|BB(9)
|<math>> f_\omega(f_9(2))</math>
|<math>> f_\omega(2 \uparrow^{13} 3) > f_\omega^2(12)</math><ref>[https://github.com/jeff87654/busy-beaver/tree/main/BB9]</ref>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LI0LC_1RG1LD_0RH0RF_0RG0RE_0LH1RZ|halt}}
|-
|-
|BB(10)
|BB(10)
|<math>> f_\omega^2(25)</math>
|<math>> f_{\omega+1}(2 \uparrow^{36} 3) > g_{64}</math><ref>https://github.com/jeff87654/busy-beaver/tree/main/BB10</ref>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|<span style="word-break:break-all">{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LI0LC_0RG1LD_1RH1LG_1LC0RG_0LJ0LI_1RZ0LC|halt}}</span>
|-
|-
|BB(11)
|BB(11)
|<math>> f_\omega^2(2 \uparrow\uparrow 12) > f_\omega^2(f_3(9))</math>
|
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RZ0LI_0LD1LE|halt}}
|
|-
|-
|BB(12)
|BB(12)
|<math>> f_\omega^4(2 \uparrow\uparrow\uparrow 4-3) > f_\omega^4(f_4(2))</math>
|<math>> f_{\omega+1}^2(2 \uparrow^{36} 3)</math><ref>https://github.com/jeff87654/busy-beaver/tree/main/BB12</ref>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_1RZ1LA_1RF1LL|halt}}
|<span style="word-break:break-all">{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LI0LC_0RG1LD_1RH1LG_1LC0RG_0LJ0LI_1RK0LC_0RL---_0RF1RZ|halt}}</span>
|-
|BB(13)
|<math>> f_{\omega+2}(f_\omega(20))</math><ref>https://github.com/jeff87654/busy-beaver/tree/main/BB13</ref>
|<span style="word-break:break-all">{{TM|1LB1LA_1RC0LA_1RD1RC_1RE1RK_1RF0LJ_0LG1RI_1RE0RH_1LF0RC_1RM0LE_1LA1LE_0LL1LB_0RM1LL_1RZ1RH|halt}}</span>
|-
|-
|BB(14)
|BB(14)
|<math>> f_{\omega + 1}(65\,536) > g_{64}</math>
|<math>> f_{\omega+2}^3(15)</math><ref>https://github.com/jeff87654/busy-beaver/tree/main/BB14</ref>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ---|halt}}
|<span style="word-break:break-all">{{TM|1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LN1RF_1LJ0RA_1RA0LH_1LG1LH_1RK0LM_1RZ0LE_0LE0RJ_1LH1LM_0LH0LL_0LG1LN|halt}}</span>
|-
|BB(15)
|<math>> f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_1RZ0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|BB(16)
|<math>> f_{\omega + 1}^2(10^{10^{57}})</math>
|
|-
|BB(18)
|<math>> f_{\omega + 2}(f_{\omega + 1}^3(f_{\omega}^2(60)))</math>
|
|-
|-
|BB(20)
|BB(20)
|<math>> f_{\omega + 2}^2(21)</math>
|<math>> f_{\omega + 6}^2(4)</math>
|
|<span style="word-break:break-all">{{TM|1RR---_0LJ0RC_1RB1LD_0LD1LE_1LF1RG_1LI0LE_0RE1LD_1RB1RH_1RH1RN_1LK1LJ_0LL1LJ_1RM1RA_0RM0RH_0LO1RN_1RH0LP_1RH1LQ_1RH1LO_1LS1RR_1RZ1LT_0LH0LS|halt}}</span>
|-
|-
|BB(21)
|BB(21)
|<math>> f_{\omega^2}^2(4 \uparrow\uparrow 341)</math>
|<math>> f_{\omega^2}^2(4 \uparrow\uparrow 341)</math><ref>https://discord.com/channels/960643023006490684/1026577255754903572/1274471360206344213</ref>
|
|<span style="word-break:break-all">{{TM|0LI0LF_0RJ1RG_0RD1LC_1RH1RE_1LO0RH_1LA1LF_0LC1LA_1RB1RH_1RD0LF_1LP0RK_0LM1LL_0LL1LM_1LE1LN_0RQ0LM_0RP1LO_1LR0RH_1LF1RQ_1RZ0LS_1LH0LT_1LH1LU_0LE1LR|halt}}</span>
|-
|-
|BB(40)
|BB(40)
Line 92: Line 86:
|BB(51)
|BB(51)
|<math>> f_{\varepsilon_0 + 1}(8)</math>
|<math>> f_{\varepsilon_0 + 1}(8)</math>
|
|-
|BB(150)
|<math>> f_{lim(BMS)}(10\uparrow\uparrow 15)</math><ref>https://discord.com/channels/960643023006490684/1026577255754903572/1328863966688182345</ref>
|[https://morphett.info/turing/turing.html?c95a199c8e8a3dd56452f8b7e28fabbf too large to show]
|-
|BB(1015)
|<math>> D^{5}(99)</math> = Loader's Number
|
|
|}
|}
Line 115: Line 117:
|<math>> 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow (7.92 \times 10^{28})</math>
|<math>> 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow (7.92 \times 10^{28})</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|-
|BB(6,3)
|<math>> f_\omega^2(10\uparrow\uparrow\uparrow 6)</math>
|{{TM|1LE2LD2LC_2RB1RB1RA_2RB1LC0LC_1RB1LD2LD_2RD2LE2LF_1RZ---1LE|halt}}
|-
|BB(8,3)
|<math>> f_{\omega+1}(f_\omega(f_7(2)))</math>
|<span style="word-break:break-all">{{TM|1LD1LE2RA_1RE2LE2RB_1RD2RA2LC_1RB1RD2RB_1LH2RF1LC_1RZ0LF1LG_1RD2LG2LG_0LD1LH1LH|halt}}</span>
|-
|BB(11,3)
|<math>> f_{\omega+3}^2(4)</math>
|<span style="word-break:break-all">{{TM|1LD2RG2LA_2LE2LH2RB_2LE1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RI---_0RC1LH2RG_1RC2LH1LA_1RI1RI1LJ_0LA2LK---_1RZ2LJ---|halt}}</span>
|-
|BB(36,3)
|<math>> f_{\varepsilon_0}(7)</math>
|
|-
|BB(37,3)
|<math>> f_{\varepsilon_0}(373\,676\,378)</math>
|
|}
|}
{| class="wikitable"
{| class="wikitable"
Line 130: Line 152:
|{{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
|{{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
|-
|-
|BB(3,4)
|[[BB(3,4)]]
|<math>> (2 \uparrow^{15} 5) + 14</math><ref name=":0">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.</ref>
|<math>> (2 \uparrow^{15} 5) + 14</math><ref name=":0">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.</ref>
|{{TM|1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|{{TM|1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|-
|BB(5,4)
|<math>> f_\omega^2(10\uparrow^4 10\uparrow\uparrow 65533)</math>
|<span style="word-break:break-all">{{TM|1LD2RA3RB1LC_1LD2LA3RB---_2RA1LA3RC2LC_2RA---2LD2LE_------1RZ1LD|halt}}</span>
|-
|BB(31,4)
|<math>> f_{\varepsilon_0}(373\,676\,378)</math>
|
|}
|}
{| class="wikitable"
{| class="wikitable"
Line 148: Line 178:
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}
|-
|-
|BB(3,5)
|[[BB(3,5)]]
|<math>> f_\omega(2 \uparrow^{15} 5) > f_\omega^2(15)</math>
|<math>> f_\omega(2 \uparrow^{15} 5) > f_\omega^2(15)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC|halt}}
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC|halt}}
|-
|BB(19,5)
|<math>> f_{\varepsilon_0}(1.7\cdot10^{352})</math>
|<span style="word-break:break-all">{{TM|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|halt}}</span>
|-
|BB(94,5)
|<math>> f_{\psi_0(\Omega_2)}(1.7\cdot10^{352})</math>
|
|}
|}
{| class="wikitable"
{| class="wikitable"
Line 162: Line 200:
|{{TM|1RZ---------------|halt}}
|{{TM|1RZ---------------|halt}}
|-
|-
|BB(2,6)
|[[BB(2,6)]]
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10^{10^{115}}</math><ref name=":1">S. Ligocki, "[https://www.sligocki.com/2023/05/20/bb-2-6-p3.html BB(2,6) > 10↑↑10↑↑10↑↑3]". Blog post, 2023. Accessed 15 August 2025.</ref>
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10^{10^{115}}</math><ref name=":1">S. Ligocki, "[https://www.sligocki.com/2023/05/20/bb-2-6-p3.html BB(2,6) > 10↑↑10↑↑10↑↑3]". Blog post, 2023. Accessed 15 August 2025.</ref>
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
|-
|BB(61,6)
|<math>> f_{\psi_0(\Omega_2)}(1.7\cdot10^{352})</math>
|
|}
{| class="wikitable"
|+
!'''7 Symbols:'''
!Runtime
!Champions
|-
|BB(18,7)
|<math>> f_{\varepsilon_0}(1.7\cdot10^{352})</math>
|<span style="word-break:break-all">{{TM|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------|halt}}</span>
|-
|BB(46,7)
|<math>> f_{\psi_0(\Omega_2)}(1.7\cdot10^{352})</math>
|
|-
|BB(134,7)
|<math>> f_{\psi_0(\Omega_\omega)}(2050)</math>
|
|}
|}
===Maximum Score Function ([[Busy Beaver Functions|Σ(n,m)]])===
{| class="wikitable"
|+
!'''8 Symbols or more:'''
!Runtime
!Champions
|-
|BB(17,8)
|<math>> f_{\varepsilon_0}(1.7\cdot10^{352})</math>
|<span style="word-break:break-all">{{TM|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---------|halt}}</span>
|-
|BB(42,9)
|<math>> f_{\psi_0(\Omega_2)}(1.7\cdot10^{352})</math>
|
|}
 
=== Maximum Score Function ([[Busy Beaver Functions|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.
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 270: Line 346:
|}
|}


=='''Beeping Busy Beavers'''==
== Beeping Busy Beavers ==
===Beeping Busy Beaver ([[BBB]](n,m))===
=== Beeping Busy Beaver ([[BBB]](n,m)) ===
Considering beep states
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 291: Line 368:
|-
|-
|BBB(4)
|BBB(4)
|≥ 32,779,478
|32,779,478<ref name=":Coq-BBB4">[https://github.com/carrino/Coq-BBB4 Coq-BBB4]</ref>
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|-
|-
Line 313: Line 390:
|-
|-
|BBB(3,3)
|BBB(3,3)
|≥ 10 ↑↑ 6
|≥ 10 ↑↑ 6.74<ref name=":Nick Drozd Tetrational Blank26" />
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}
|}
|}
Line 330: Line 407:
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA}}
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA}}
|}
|}
===Beeping Booping Busy Beaver ([[Beeping Busy Beaver#Beeping Booping Busy Beavers|BBBB]](n,m))===
 
=== Beeping Booping Busy Beaver ([[Beeping Busy Beaver#Beeping Booping Busy Beavers|BBBB]](n,m)) ===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 346: Line 424:
|}
|}


=='''Maximum Consecutive Ones Function ([[Num]](n,m))'''==
== Maximum Consecutive Ones Function ([[Num]](n,m)) ==
{| class="wikitable"
{| class="wikitable"
|+
|+
!'''2 Symbols:'''
!Domain
!Number of Ones
!Number of Ones
!Champions
!Champions
Line 372: Line 450:
|165
|165
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_1RZ0LB|halt}} {{TM|0RB1LD_1LC1RB_1LD1RE_1LA1LE_1LZ0RC|halt}}
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_1RZ0LB|halt}} {{TM|0RB1LD_1LC1RB_1LD1RE_1LA1LE_1LZ0RC|halt}}
|-
|num(6)
|≥ ≈ 10↑↑15.60465
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE0RZ_1LF0RB_0RC0RE|halt}}
|-
|num(2,3)
|6
|{{TM|1RB1LA1LB_0LA2RA1RZ|halt}}
|-
|num(3,3)
|≥ 12
|{{TM|1RB1RA1RZ_1LC1LC2LA_2RA1LB1LA|halt}}
|-
|num(4,3)
|<math>> 10 {\uparrow}^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|}
|}
=='''Maximum Space Function ([[Maximum Space Function|BB<sub>space</sub>]](n,m))'''==
 
== Maximum Space Function ([[Maximum Space Function|BB<sub>space</sub>]](n,m)) ==
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 407: Line 502:
|-
|-
|BB<sub>space</sub>(1,3)
|BB<sub>space</sub>(1,3)
|
|1
|
|{{TM|1RZ------|halt}}
|-
|-
|BB<sub>space</sub>(2,3)
|BB<sub>space</sub>(2,3)
Line 419: Line 514:
|}
|}


=='''Size of the runtime spectrum (R(n,m))'''==
== Size of the runtime spectrum (R(n,m)) ==
There currently doesn't seem to be any available information about values of this function.
There currently doesn't seem to be any available information about values of this function.


=='''Reversible Turing Machines'''==
== Reversible Turing Machines ==
===Maximum Shifts Function ([[Reversible Turing Machine|BB<sub>rev</sub>]](n,m))===
=== Maximum Shifts Function ([[Reversible Turing Machine|BB<sub>rev</sub>]](n,m)) ===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 458: Line 553:
|{{TM|1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB1RZ|halt}}
|{{TM|1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB1RZ|halt}}
|}
|}
===Maximum Score Function (Σ<sub>rev</sub>(n,m))===
 
=== Maximum Score Function (Σ<sub>rev</sub>(n,m)) ===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 489: Line 585:
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA|halt}}
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA|halt}}
|}
|}
=='''Blanking Busy Beaver ([[Busy Beaver Functions#Other Busy Beaver functions|BLB(n,m)]])'''==
 
== Blanking Busy Beaver ([[Blanking Busy Beaver|BLB(n,m)]]) ==
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 501: Line 598:
|-
|-
|BLB(2)
|BLB(2)
|≥ 8<ref name=":3">Nick Drozd. "[https://nickdrozd.github.io/2021/02/14/blanking-beavers.html Blanking Beavers]". Accessed 15 August 2025.</ref>
|8<ref name=":3">Nick Drozd, "[https://nickdrozd.github.io/2021/02/14/blanking-beavers.html Blanking Beavers]". Accessed 15 August 2025.</ref>
|{{TM|1RB0RA_1LB1LA}}
|{{TM|1RB0RA_1LB1LA}}
|-
|-
|BLB(3)
|BLB(3)
|≥ 34<ref name=":4">Nick Drozd. "[https://nickdrozd.github.io/2022/02/11/latest-beeping-busy-beaver-results.html Latest Beeping Busy Beaver Results]". Accessed 15 August 2025.</ref>
|34<ref name=":4">Nick Drozd, "[https://nickdrozd.github.io/2022/02/11/latest-beeping-busy-beaver-results.html Latest Beeping Busy Beaver Results]". Accessed 15 August 2025.</ref>
|{{TM|1RB1LB_1LA1LC_1RC0LC}}
|{{TM|1RB1LB_1LA1LC_1RC0LC}}
|-
|-
Line 511: Line 608:
|≥ 32,779,477
|≥ 32,779,477
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|-
|BLB(5)
|> 10<sup>14,006</sup>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA}}
|-
|BLB(6)
|> 10 ↑↑ 16
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RE_1LF0RB_0RC0RE}}
|-
|BLB(7)
|> 10^183.76332<ref>https://discord.com/channels/960643023006490684/1369339127652159509/1374531970830307409</ref>
|{{TM|1RB0RF_1RC1RA_1LD0LA_0RG0RE_0RA1LF_0LE0LC_1RZ0RD|halt}}
|}
|}
{| class="wikitable"
{| class="wikitable"
Line 523: Line 632:
|-
|-
|BLB(2,3)
|BLB(2,3)
|≥ 77<ref name=":4" />
|77<ref name=":4" />
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLB(3,3)
|> 10 ↑↑ 6.54<ref name=":Nick Drozd Tetrational Blank26">Nick Drozd. "[https://nickdrozd.github.io/2026/09/01/tetrational-self-cleaning-turing-machine.html A Tetrational Self-Cleaning Turing Machine]". Blog post. Accessed 17 October 2026.</ref>
|{{TM|1RB1RC2RA_0RC2RA0RB_2LC2LB0LA}}
|}
|}
{| class="wikitable"
{| class="wikitable"
Line 539: Line 652:
|≥ 1,367,361,263,049<ref name=":4" />
|≥ 1,367,361,263,049<ref name=":4" />
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}
{| class="wikitable"
|+
!'''5 Symbols:'''
!Steps
!Champions
|-
|BLB(2,5)
|> 10<sup>32</sup>
|{{TM|1RB2RB4RA0RB2RA_2LB3LA0RB0RA1RB}}
|}
|}


=='''Lazy Beaver'''==
== Lazy Beaver ==
===Shifts Function ([[Lazy Beaver|LB]](n,m))===
=== Shifts Function ([[Lazy Beaver|LB]](n,m)) ===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 594: Line 717:
|}
|}


=='''Period-oriented Busy Beavers'''==
== Period-oriented Busy Beavers ==
===Busy Preperiodic Beaver ([[BBS]](n,m))===
=== Busy Preperiodic Beaver ([[BBS]](n,m)) ===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 617: Line 740:
|≥ 119,120,230,102
|≥ 119,120,230,102
|{{TM|1RB1LC_0LA1RD_0RB0LC_1LA0RD}}
|{{TM|1RB1LC_0LA1RD_0RB0LC_1LA0RD}}
|-
|BBS(5,2)
|> 10<sup>14,006</sup>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA}}
|}
|}
{| class="wikitable"
{| class="wikitable"
Line 627: Line 754:
|0
|0
|{{TM|1RA------}}
|{{TM|1RA------}}
|-
|BBS(2,3)
|≥ 165<ref>Nick Drozd. [https://nickdrozd.github.io/2021/02/14/blanking-beavers.html Blanking Beavers]</ref>
|{{TM|1RB0LA---_1LB2LA0RB}}
|-
|BBS(3,3)
|> 10 ↑↑ 6
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}
|-
|BBS(4,3)
|> <math>10 \uparrow^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_1LB1LA1LD_0RB2RA2RD}}
|}
|}
{| class="wikitable"
{| class="wikitable"
Line 639: Line 778:
|-
|-
|BBS(2,4)
|BBS(2,4)
|≥ 293,225,660,896
|≥ 205,770,076,433,044,242,247,860
|{{TM|1RB2LA0RA3LA_1LA1LB3RB1RA}}
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA}}
|}
|}
===Busy Periodic Beaver ([[BBP]](n,m))===
 
=== Busy Periodic Beaver ([[BBP]](n,m)) ===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 674: Line 814:
|1
|1
|{{TM|1RA------}}
|{{TM|1RA------}}
|-
|BBP(2,3)
|
|
|-
|BBP(3,3)
|≥ 1,195
|{{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB}}
|}
|}
{| class="wikitable"
{| class="wikitable"
Line 690: Line 838:
|}
|}


='''Instruction-Limited Busy Beaver'''=
= '''Instruction-Limited Busy Beaver''' =
=='''Instruction-Limited Classical Busy Beaver Functions'''==
== Instruction-Limited Classical Busy Beaver Functions ==
===Instruction-Limited Maximum Shifts Function ([[BBi]](n))===
=== Instruction-Limited Maximum Shifts Function ([[BBi]](n)) ===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 735: Line 883:
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|}
|}
===Instruction-Limited Maximum Score Function ([[Σi]](n))===
 
=== Instruction-Limited Maximum Score Function ([[Σi]](n)) ===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 779: Line 928:
|}
|}


=='''Instruction-Limited Blanking Busy Beaver ([[Instruction-Limited Busy Beaver#Instruction-Limited Busy Beaver Variants|BLBi(n)]])'''==
== Instruction-Limited Blanking Busy Beaver ([[Blanking Busy Beaver#Instruction-limited Blanking Busy Beaver (BLBi(n))|BLBi(n)]]) ==
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 796: Line 945:
|BLBi(3)
|BLBi(3)
|4
|4
|
|{{TM|1RB0RA_1LA---}}
|-
|-
|BLBi(4)
|BLBi(4)
|12
|12
|
|{{TM|1RB---_1RC---_1LC0RC}}
|-
|-
|BLBi(5)
|BLBi(5)
|30
|30
|
|{{TM|1RB------_1RC------_2LC2RC0RC}}
|-
|-
|BLBi(6)
|BLBi(6)
Line 812: Line 961:
|BLBi(7)
|BLBi(7)
|808
|808
|
||{{TM|1RB------_1RC------_0RD2LC---_1LD2RD0RC}}
|-
|-
|BLBi(8)
|BLBi(8)
|≥ 1,367,361,263,049
|≥ 1,367,361,263,049
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|-
|BLBi(9)
|> 10<sup>42,745</sup>
|{{TM|1RB2RB1LA_2LC0LB2LB_2RC2RA0LC}}
|}
|}


=='''Instruction-Limited Greedy Busy Beaver ([[Instruction-Limited Busy Beaver#Instruction-Limited Busy Beaver Variants|gBBi(n)]])'''==
== Instruction-Limited Greedy Busy Beaver ([[Instruction-Limited Busy Beaver#Instruction-Limited Busy Beaver Variants|gBBi(n)]]) ==
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 882: Line 1,035:
|{{TM|0RB6RB1LB---3LA1RB7RB2LB_1LA2RB3LA4RB5RB1LB5LA---|halt}}
|{{TM|0RB6RB1LB---3LA1RB7RB2LB_1LA2RB3LA4RB5RB1LB5LA---|halt}}
|}
|}
='''Program-Limited Busy Beaver'''=
 
=='''Busy Beaver for Lambda Calculus'''==
= '''Program-Limited Busy Beaver''' =
===Regular Busy Beaver for Lambda Calculus ([[BBλ]](n))===
== 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 <math> 20 \geq n </math> BBλ(n) = n.
For n = 0,1,2,3,5 BBλ(n) is undefined, while for the rest of <math> 20 \geq n </math> BBλ(n) = n.
{| class="wikitable"
{| class="wikitable"
Line 892: Line 1,046:
!Champions
!Champions
|-
|-
|BBλ(21)
|BBλ(4)
|22
|4
|<code>\(\1 1) (1 (\2))</code>
|<math>\lambda 1</math>
|-
|-
|BBλ(22)
|BBλ(6)
|24
|6
|<code>\(\1 1) (1 (\\1))\(\1 1 1) (1 1)</code>
|<math>\lambda\lambda 1</math>
|-
|-
|BBλ(23)
|BBλ(7)
|26
|7
|<code>\(\1 1) (1 (\\2))</code>
|<math>\lambda\lambda 2</math>
|-
|-
|BBλ(24)
|BBλ(8)
|30
|8
|<code>\(\1 1 1) (1 (\1))</code>
|<math>\lambda\lambda\lambda 1</math>
|-
|-
|BBλ(25)
|BBλ(9)
|42
|9
|<code>\(\1 1) (\1 (2 1))</code>
|<math>\lambda\lambda\lambda 2</math>
|-
|-
|BBλ(26)
|BBλ(10)
|52
|10
|<code>(\1 1) (\\2 (1 2))</code>
|<math>\lambda\lambda\lambda\lambda 1</math>
|-
|-
|BBλ(27)
|BBλ(11)
|44
|11
|<code>\\(\1 1) (\1 (2 1))</code>
|<math>\lambda\lambda\lambda\lambda 2</math>
|-
|-
|BBλ(28)
|BBλ(12)
|58
|12
|<code>\(\1 1) (\1 (2 (\2))))</code>
|<math>\lambda\lambda\lambda\lambda\lambda 1</math>
|-
|-
|BBλ(29)
|BBλ(13)
|223
|13
|<code>\(\1 1) (\1 (1 (2 1)))</code>
|<math>\lambda\lambda\lambda\lambda\lambda 2</math>
|-
|-
|BBλ(30)
|BBλ(14)
|160
|14
|<code>(\1 1 1) (\\2 (1 2))</code> and <code>(\1 (1 1)) (\\2 (1 2))</code>
|<math>\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
|-
|-
|BBλ(31)
|BBλ(15)
|267
|15
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|<math>\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
|-
|-
|BBλ(32)
|BBλ(16)
|298
|16
|<code>\(\1 1) (\1 (1 (2 (\2))))</code>
|<math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
|-
|-
|BBλ(33)
|BBλ(17)
|1812
|17
|<code>\(\1 1) (\1 (1 (1 (2 1))))</code>
|<math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
|-
|-
|BBλ(34)
|BBλ(18)
|327,686
|18
|<code>(\1 1 1 1) (\\2 (2 1))</code> and <code>(\1 (1 1) 1) (\\2 (2 1))</code>
|<math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
|-
|-
|BBλ(35)
|BBλ(19)
|<math>5 \times 3^{3^{3}} +6 = 38\,127\,987\,424\,941 > 3.8 \times 10^{13}</math>
|19
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|<math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
|-
|-
|BBλ(36)
|BBλ(20)
|<math>5 \times 2^{2^{2^{3}}} +6 > 5.7 \times 10^{77}</math>
|20
|<code>(\1 1) (\1 (1 (\\2 (2 1))))</code>
|<math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
|-
|-
|BBλ(37)
|BBλ(21)
|<math>BB \lambda(35) +2 =5 \times 3^{3^{3}} +8 = 38\,127\,987\,424\,943 > 3.8 \times 10^{13}</math>
|22
|<code>\(\1 1 1) (\\2 (2 (2 1)))</code>
|<math>\lambda(1(\lambda 2))(1(\lambda 2))</math>
|-
|-
|BBλ(38)
|BBλ(22)
|<math>\geq 5 \times 2^{2^{2^{2^{2}}}} +6 > 10^{19729}</math>
|24
|<code>(\1 1 1 1 1) (\\2 (2 1))</code> and <code>(\1 (1 1) 1 1) (\\2 (2 1))</code>
|<math>\lambda(1 1) (1 1) (1 1)</math>
|-
|-
|BBλ(39)
|BBλ(23)
|<math>\geq 5 \times 3^{3^{3^{3}}} +6 > 10^{3\,638\,334\,640\,024}</math>
|26
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|<math>\lambda(1 (\lambda\lambda 2)) (1 (\lambda\lambda 2))</math>
|-
|-
|BBλ(40)
|BBλ(24)
|<math>> (2 \uparrow\uparrow)^{15}33 > 10 \uparrow\uparrow\uparrow 16</math>
|30
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|<math>\lambda(1 (\lambda 1)) (1 (\lambda 1)) (1 (\lambda 1))</math>
|-
|-
|BBλ(41)
|BBλ(25)
|<math>\geq 5 \times 3^{3^{85}} +6 > 10^{1.7 \times 10^{40}}</math>
|42
|<code>(\1 (\1 1) 1) (\\2 (2 (2 1)))</code>
|<math>\lambda 1 (\lambda 1 (2 1)) (1 (1 (\lambda 1 (2 1))))</math>
|-
|-
|BBλ(42)
|BBλ(26)
|<math>\geq BB \lambda(40) + 2 > (2 \uparrow\uparrow)^{15}33 > 10 \uparrow\uparrow\uparrow 16</math>
|52
|<code>\(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|<math>\lambda\lambda 2 (\lambda\lambda 2 (1 2)) (1 (2 (\lambda\lambda 2 (1 2))))</math>
|-
|-
|BBλ(43)
|BBλ(27)
|<math>> 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
|44
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|<math>\lambda\lambda 1 (\lambda 1 (2 1)) (1 (1 (\lambda 1 (2 1))))</math>
|-
|-
|BBλ(44)
|BBλ(28)
|<math>> 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
|58
|<code>(\1 1 1 1) (\1 (\\2 (2 1)) 1)</code>
|<math>\lambda 1 (\lambda\lambda 1 (3 (\lambda 2))) (1 (\lambda 2 (\lambda\lambda 1 (4 (\lambda 2)))))</math>
|-
|-
|BBλ(45)
|BBλ(29)
|<math>\geq BB \lambda(43) + 2 > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
|223
|<code>\(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|<math>\lambda(\lambda 1 1) (\lambda 1 (1 (2 1)))</math>
|-
|-
|BBλ(46)
|BBλ(30)
|<math>\geq BB \lambda(44) + 2 > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
|160
|<code>\(\1 1 1 1) (\1 (\\2 (2 1)) 1)</code>
|<math>(\lambda 1 1 1) (\lambda\lambda 2 (1 2))</math>
|-
|-
|BBλ(47)
|BBλ(31)
|
|267
|
|<math>(\lambda 1 1) (\lambda\lambda 2 (2 (1 2)))</math>
|-
|-
|BBλ(48)
|BBλ(32)
|<math>> 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
|298
|<code>(\1 1 1 1 1) (\1 (\\2 (2 1)) 1)</code>
|<math>\lambda(\lambda 1 1) (\lambda 1 (1 (2 (\lambda 2))))</math>
|-
|-
|BBλ(49)
|BBλ(33)
|<math>> f_{\omega+1}(\frac{2 \uparrow\uparrow 6}{2}) > \text{Graham's Number}</math>
|1812
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|<math>\lambda(\lambda 1 1) (\lambda 1 (1 (1 (2 1))))</math>
|-
|-
|BBλ(1850)
|BBλ(34)
|<math>> \text{Loader's Number}</math>
|327,686
|<code>Too large for this list</code>
|<math>(\lambda 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|}
===Oracle Busy Beaver for Lambda Calculus ([[Busy Beaver for lambda calculus#Oracle Busy Beaver|BBλ<sub>1</sub>(n)]])===
Note that <math>f(n) = 6 + 5 \times BB \lambda(n)</math>.
{| class="wikitable"
|+
!
!BBλ<sub>1</sub>(n)
!Champions
|-
|-
|BBλ<sub>1</sub>(1)
|BBλ(35)
|0
|<math>5 \times 3^{3^{3}} +6 = 38\,127\,987\,424\,941 > 3.8 \times 10^{13}</math>
|
|<math>(\lambda 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
|-
|-
|BBλ<sub>1</sub>(2)
|BBλ(36)
|1
|<math>5 \times 2^{2^{2^{3}}} +6 > 5.7 \times 10^{77}</math>
|<code>1</code>
|<math>(\lambda 1 1) (\lambda 1 (1 (\lambda\lambda 2 (2 1))))</math>
|-
|-
|BBλ<sub>1</sub>(3)
|BBλ(37)
|0
|<math>BB \lambda(35) +2 =5 \times 3^{3^{3}} +8 = 38\,127\,987\,424\,943 > 3.8 \times 10^{13}</math>
|
|<math>\lambda(\lambda 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
|-
|-
|BBλ<sub>1</sub>(4)
|BBλ(38)
|4
|<math> 5 \times 2^{2^{2^{2^{2}}}} +6 > 10^{19729}</math>
|<code>\1</code>
|<math>(\lambda 1 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|-
|-
|BBλ<sub>1</sub>(5)
|BBλ(39)
|5
|<math> 5 \times 3^{3^{3^{3}}} +6 > 10^{3\,638\,334\,640\,024}</math>
|<code>\2</code>
|<math>(\lambda 1 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
|-
|-
|BBλ<sub>1</sub>(6)
|BBλ(40)
|6
|<math>\approx (2 \uparrow\uparrow)^{15}33 > 10 \uparrow\uparrow\uparrow 16</math>
|<code>\\1</code>
|<math>(\lambda 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
|-
|-
|BBλ<sub>1</sub>(7)
|BBλ(41)
|7
|<math>5 \times 3^{3^{85}} +6 > 10^{1.7 \times 10^{40}}</math>
|<code>\\2</code>
|<math>(\lambda 1 (\lambda 1 1) 1) (\lambda\lambda 2 (2 (2 1)))</math>
|-
|-
|BBλ<sub>1</sub>(8)
|BBλ(42)
|26
|<math>\geq BB \lambda(40) + 2 > (2 \uparrow\uparrow)^{15}33 > 10 \uparrow\uparrow\uparrow 16</math>
|<code>1 (\1)</code>
|<math>\lambda(\lambda 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
|-
|-
|BBλ<sub>1</sub>(9)
|BBλ(43)
|9
|<math>> 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
|<code>\\2</code>
|<math>(\lambda 1 1) (\lambda 1 (\lambda 1 (\lambda\lambda 2 (2 1)) 2))</math>
|-
|-
|BBλ<sub>1</sub>(10)
|BBλ(44)
|36
|<math>> 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
|<code>1 (\\1)</code>
|<math>(\lambda 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
|-
|-
|BBλ<sub>1</sub>(11)
|BBλ(45)
|41
|<math>\geq BB \lambda(43) + 2 > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
|<code>1 (\\2)</code>
|<math>\lambda(\lambda 1 1) (\lambda 1 (\lambda 1 (\lambda\lambda 2 (2 1)) 2))</math>
|-
|-
|BBλ<sub>1</sub>(12)
|BBλ(46)
|266
|<math>\geq BB \lambda(44) + 2 > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
|<code>1 (1 (\1))</code>
|<math>\lambda(\lambda 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
|-
|-
|BBλ<sub>1</sub>(13)
|BBλ(47)
|51
|<math>> f_{\omega}\left(f_{5}\left(2\right)\right)</math>
|<code>1 (\\2)</code>
|<math>(\lambda 1 1 1)(\lambda\lambda 1 (1 2) (\lambda\lambda 2 (2 1)))</math>
|-
|-
|BBλ<sub>1</sub>(14)
|BBλ(48)
|<math>f(36) = 25 \times 2^{2^{2^{3}}}+36 > 2.85 \times 10^{78}</math>
|<math>> 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
|<code>1 (1 (\\1))</code>
|<math>(\lambda 1 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
|-
|-
|BBλ<sub>1</sub>(15)
|BBλ(49)
|<math>f(41) \geq 25 \times 3^{3^{85}}+36 > 10^{1.7 \times 10^{40}}</math>
|<math>> f_{\omega+1}(\frac{2 \uparrow\uparrow 6}{2}) > \text{Graham's Number}</math>
|<code>1 (1 (\\2))</code>
|<math>(\lambda 1 1) (\lambda 1 (1 (\lambda\lambda 1 2 (\lambda\lambda 2 (2 1)))))</math>
|-
|-
|BBλ<sub>1</sub>(16)
|BBλ(61)
|<math>f(266)</math>
|<math>> f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right)</math>
|<code>1 (1 (1 (\1)))</code>
|<math>(\lambda 1 1 1) (\lambda 1 (1 (\lambda\lambda\lambda 1 3 2 (\lambda\lambda 2 (2 1)))))</math>
|-
|BBλ(70)
|<math>> f_{\omega^{\omega}2}\left(f_{\omega^{\omega}}\left(f_{\omega^{\omega}}\left(3\right)\right)\right)</math>
|<math>(\lambda 1 1 (\lambda 1 (\lambda\lambda\lambda 1 3 2 5) 2) 1) (\lambda\lambda 2 (2 1))</math>
|-
|BBλ(74)
|<math>> f_{\omega^{\omega}2 + 1}\left(7\right)</math>
|<math>(\lambda 1 1 1 (\lambda 1 (\lambda\lambda\lambda 1 3 2 5) 2) 1) (\lambda\lambda 2 (2 1))</math>
|-
|BBλ(76)
|<math>> f_{\omega^{\omega^{3}}}(2)</math>
|<math>(\lambda11(\lambda\lambda\lambda\lambda14321)1111)(\lambda\lambda2(21))</math>
|-
|BBλ(77)
|<math>> f_{\omega^{\omega2}}(3)</math>
|<math>(\lambda1(\lambda\lambda\lambda\lambda14321)1111)(\lambda\lambda2(2(21)))</math>
|-
|BBλ(80)
|<math>>f_{\omega^{\omega^{14}}}(2)</math>
|<math>(\lambda111(\lambda\lambda\lambda\lambda14321)1111)(\lambda\lambda2(21))</math>
|-
|BBλ(82)
|<math>> f_{\omega^{\omega^{\omega}}}(2\uparrow\uparrow\uparrow\uparrow18)</math>
|<math>(\lambda1(\lambda1(\lambda\lambda\lambda\lambda14321)1111)1)(\lambda\lambda2(21))</math>
|-
|BBλ(86)
|<math>> f_{\omega^{\omega65536}}\left(2\right)</math>
|<math>(\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|-
|-
|BBλ<sub>1</sub>(17)
|BBλ(90)
|<math>f(51)</math>
|<math>> f_{\zeta_0}\left(15\right)</math>
|<code>1 (1 (\\\2))</code>
|<math>(\lambda 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|-
|-
|BBλ<sub>1</sub>(18)
|BBλ(94)
|<math>f^{4}(4) = f(f(266))</math>
|<math>> f_{\psi(\Omega_\omega)}\left(12\right)</math>
|<code>1 (\1) 1 (\1)</code>
|<math>(\lambda 1 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|-
|-
|BBλ<sub>1</sub>(19)
|BBλ(95)
|<math>f^{3}(7) = f(f(41))</math>
|<math>> f_{\psi(\Omega_\omega)}\left(23\right)</math>
|<code>1 (1 (1 (\\2)))</code>
|<math>(\lambda 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
|-
|-
|BBλ<sub>1</sub>(20)
|BBλ(96)
|<math>f^{6}(4) = f^{4}(266)</math>
|<math>> f_{\psi(\Omega_\omega)}\left(f_{\omega^{\omega^{2}}}\left(2\right)\right)</math>
|<code>1 (\\1) 1 (\1)</code>
|<math>(\lambda 1 (\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) 1) (\lambda\lambda 2 (2 1))</math>
|-
|-
|BBλ<sub>1</sub>(21)
|BBλ(100)
|<math>f^{7}(4) = f^{5}(266)</math>
|<math>> f_{\psi(\Omega_\omega)+1}\left(4\right)</math>
|<code>1 (\\2) 1 (\1)</code>
|<math>(\lambda 1 1 (\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) 1) (\lambda\lambda 2 (2 1))</math>
|-
|-
|BBλ<sub>1</sub>(22)
|BBλ(201)
|<math>f^{52}(4) = f^{50}(266)</math>
|<math>> q(5)</math> (q is only proven to exceed all functions provably recursive in PA; the true value is likely much larger based on informal analysis)
|<code>1 (1(\1)) 1(\1)</code>
|<code>too large to show</code>
|-
|-
|BBλ<sub>1</sub>(28)
|BBλ(331)
|<math>\geq f^{BB \lambda(f^{3}(4))}(4)</math>
|<math>> f_{\operatorname{lim}(\operatorname{BMS})}^3(5)</math>
|<code>1 (\1) 1 (\1) 1 (\1)</code>
|<code>too large to show</code>
|-
|-
|BBλ<sub>1</sub>(29)
|BBλ(1797)
|<math>\geq f^{BB \lambda(f^{BB \lambda(f^{4}(4))+4}(4))+BB \lambda(f^{4}(4))+5}(4)</math>
|<math>> D^{5}(99)</math> (Loader's number)
|<code>1(\1)(\1 2 1)(\1)</code>
|<code>too large to show</code>
|}
|}


='''Fractran ([[Fractran|BBf]](n))'''=
=== Oracle Busy Beaver for Lambda Calculus ([[Busy Beaver for lambda calculus#Oracle Busy Beaver|BBλ<sub>1</sub>(n)]]) ===
Note that <math>f(n) = 6 + 5 \times BB \lambda(n)</math>.
{| class="wikitable"
{| class="wikitable"
|+
|+
!n
!
!BBf(n)
!BBλ<sub>1</sub>(n)
!Example Champion
!Champions
!Vector Representation
|-
|-
| 2 || 1 || <code>[1/2]</code>
|BBλ<sub>1</sub>(1)
|<math>\begin{bmatrix}
|0
  -1
|
\end{bmatrix}</math>
|-
|-
| 3 || 1 || <code>[3/2]</code>
|BBλ<sub>1</sub>(2)
|<math>\begin{bmatrix}
|1
  -1 & 1
|<math>1</math>
\end{bmatrix}</math>
|-
|-
| 4 || 1 || <code>[9/2]</code>
|BBλ<sub>1</sub>(3)
|<math>\begin{bmatrix}
|0
  -1 & 2
|
\end{bmatrix}</math>
|-
|-
| 5 || 2 || <code>[3/2, 1/3]</code>
|BBλ<sub>1</sub>(4)
|<math>\begin{bmatrix}
|4
  -1 &  1 \\
|<math>\lambda 1</math>
  0 & -1
|-
\end{bmatrix}</math>
|BBλ<sub>1</sub>(5)
|5
|<math>\lambda 2</math>
|-
|-
| 6 || 3 || <code>[9/2, 1/3]</code>
|BBλ<sub>1</sub>(6)
|<math>\begin{bmatrix}
|6
  -1 &  2 \\
|<math>\lambda \lambda 1</math>
  0 & -1
\end{bmatrix}</math>
|-
|-
| 7 || 4 || <code>[27/2, 1/3]</code>
|BBλ<sub>1</sub>(7)
|<math>\begin{bmatrix}
|7
  -1 &  3 \\
|<math>\lambda \lambda 2</math>
  0 & -1
\end{bmatrix}</math>
|-
|-
| 8 || 5 || <code>[81/2, 1/3]</code>
|BBλ<sub>1</sub>(8)
|<math>\begin{bmatrix}
|26
  -1 &  4 \\
|<math>1 (\lambda 1)</math>
  0 & -1
\end{bmatrix}</math>
|-
|-
| 9 || 6 || <code>[243/2, 1/3]</code>
|BBλ<sub>1</sub>(9)
|<math>\begin{bmatrix}
|9
  -1 &  5 \\
|<math>\lambda \lambda \lambda 2</math>
  0 & -1
\end{bmatrix}</math>
|-
|-
| 10 || 7 || <code>[729/2, 1/3]</code>
|BBλ<sub>1</sub>(10)
|<math>\begin{bmatrix}
|36
  -1 &  6 \\
|<math>1 (\lambda \lambda 1)</math>
  0 & -1
\end{bmatrix}</math>
|-
|-
| 11 || 10 || <code>[27/2, 25/3, 1/5]</code>
|BBλ<sub>1</sub>(11)
|<math>\begin{bmatrix}
|41
  -1 &  3 &  0 \\
|<math>1 (\lambda \lambda 2)</math>
  0 & -1 &  2 \\
  0 &  0 & -1
\end{bmatrix}</math>
|-
|-
| 12 || 13 || <code>[81/2, 25/3, 1/5]</code>
|BBλ<sub>1</sub>(12)
|<math>\begin{bmatrix}
|266
  -1 &  4 &  0 \\
|<math>1 (1 (\lambda 1))</math>
  0 & -1 &  2 \\
  0 &  0 & -1
\end{bmatrix}</math>
|-
|-
| 13 || 17 || <code>[81/2, 125/3, 1/5]</code>
|BBλ<sub>1</sub>(13)
|<math>\begin{bmatrix}
|51
  -1 &  4 &  0 \\
|<math>1 (\lambda \lambda 2)</math>
  0 & -1 &  3 \\
|-
  0 &  0 & -1
|BBλ<sub>1</sub>(14)
\end{bmatrix}</math>
|<math>f(36) = 25 \times 2^{2^{2^{3}}}+36 > 2.85 \times 10^{78}</math>
|<math>1 (1 (\lambda \lambda 1))</math>
|-
|-
| 14 || 21 || <code>[243/2, 125/3, 1/5]</code>
|BBλ<sub>1</sub>(15)
|<math>\begin{bmatrix}
|<math>f(41) = 25 \times 3^{3^{85}}+36 > 10^{1.7 \times 10^{40}}</math>
  -1 &  5 &  0 \\
|<math>1 (1 (\lambda \lambda 2))</math>
  0 & -1 &  3 \\
  0 &  0 & -1
\end{bmatrix}</math>
|-
|-
| 15 || 28 || <code>[1/45, 4/5, 3/2, 25/3]</code>
|BBλ<sub>1</sub>(16)
|<math>\begin{bmatrix}
|<math>f(266)</math>
  0 & -2 & -1 \\
|<math>1 (1 (1 (\lambda 1)))</math>
  2 &  0 & -1 \\
  -1 &  1 &  0 \\
  0 & -1 &  2
\end{bmatrix}</math>
|-
|-
| 16 || 53 || <code>[1/45, 4/5, 3/2, 125/3]</code>
|BBλ<sub>1</sub>(17)
|<math>\begin{bmatrix}
|<math>f(51)</math>
  0 & -2 & -1 \\
|<math>1 (1 (\lambda \lambda \lambda 2))</math>
  2 &  0 & -1 \\
|-
  -1 &  1 &  0 \\
|BBλ<sub>1</sub>(18)
  0 & -1 &  3
|<math>f^{4}(4) = f(f(266))</math>
\end{bmatrix}</math>
|<math>1 (\lambda 1) 1 (\lambda 1)</math>
|-
|-
| 17 || 107 || <code>[5/6, 49/2, 3/5, 40/7]</code>
|BBλ<sub>1</sub>(19)
|<math>\begin{bmatrix}
|<math>f^{3}(7) = f(f(41))</math>
  -1 & -1 &  1 &  0 \\
|<math>1 (1 (1 (\lambda \lambda 2)))</math>
  -1 &  0 &  0 &  2 \\
    0 &  1 & -1 &  0 \\
    3 &  0 &  1 & -1
\end{bmatrix}</math>
|-
|-
| 18 || 211 || <code>[5/6, 49/2, 3/5, 80/7]</code>
|BBλ<sub>1</sub>(20)
|<math>\begin{bmatrix}
|<math>f^{6}(4) = f^{4}(266)</math>
  -1 & -1 &  1 &  0 \\
|<math>1 (\lambda \lambda 1) 1 (\lambda 1)</math>
  -1 &  0 &  0 &  2 \\
    0 &  1 & -1 &  0 \\
    4 &  0 &  1 & -1
\end{bmatrix}</math>
|-
|-
| 19 || 370 || <code>[5/6, 49/2, 3/5, 160/7]</code>
|BBλ<sub>1</sub>(21)
|<math>\begin{bmatrix}
|<math>f^{7}(4) = f^{5}(266)</math>
|<math>1 (\lambda \lambda 2) 1 (\lambda 1)</math>
|-
|BBλ<sub>1</sub>(22)
|<math>f^{26}(4) = f^{24}(266)</math>
|<math>1 (1 (\lambda 1)) 1 (\lambda 1)</math>
|-
|BBλ<sub>1</sub>(28)
|<math>\geq f^{BB \lambda(f^{3}(4))}(4)</math>
|<math>1 (\lambda 1) 1 (\lambda 1) 1 (\lambda 1)</math>
|-
|BBλ<sub>1</sub>(29)
|<math>\geq f^{BB \lambda(f^{BB \lambda(f^{4}(4))+4}(4))+BB \lambda(f^{4}(4))+5}(4)</math>
|<math>1(\lambda 1)(\lambda 1 2 1)(\lambda 1)</math>
|}
 
=== Busy Beaver for De Bruijn Lambda Calculus ([[De Bruijn index]]) ===
{| class="wikitable"
!n
!Value
!Champion
|-
|7
|≥ 7
|<code>\1 1 1 1 1 1</code>
|-
|8
|≥ 16
|<code>(\1 1) (\\2 (1 2))</code>
|-
|9
|≥ 68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|-
|10
|<math>> 7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|-
|11
|<math>> 10^{7.625 \times 10^{12}}</math>
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|12
|<math>> 10 {\uparrow}^{3} 16</math>
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|13
|<math>> 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|-
|14
|<math>> f_{\omega}\left(f_{5}\left(2\right)\right)</math>
|<code>(\1 1 1) (\\1 (1 2) (\\2 (2 1)))</code>
|-
|15
|<math>> f_{\omega+1}(10^{10^{19,727}})</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|-
|18
|<math>> f_{\omega^\omega}(2 \uparrow\uparrow 18)</math>
|<code>(\1 1 1) (\1 (1 (\\\1 3 2 (\\2 (2 1)))))</code>
|-
|19
|<math>> f_{\omega^\omega2}(f^2_{\omega^\omega}(3))</math>
|<code>(\1 1 (\1 (\\1 3 2 5) 2) 1) (\2 (2 1)))</code>
|-
|20
|<math>> f_{\omega^\omega2+1}(7)</math>
|<code>(\1 1 1 (\1 (\\1 3 2 5) 2) 1) (\2 (2 1)))</code>
|-
|21
|<math>> f_{\omega^{\omega^{2}+\omega+2}}(2)</math>
|<code>(\1 1 (\\\\1 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|-
|22
|<math>> f_{\omega^{\omega^{14}}}(2)</math>
|<code>(\1 1 1 (\\1 4 3 2 1) 1 1 1 1) (\2 (2 1))</code>
|-
|23
|<math>> f_{\zeta_0}(15)</math>
|<code>(\1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|-
|24
|<math>> f_{\psi(\Omega_\omega)}(12)</math>
|<code>(\1 1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|-
|25
|<math>> f_{\psi(\Omega_\omega)}(f_{\omega^{\omega+2}}(2))</math>
|<code>(\1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|-
|26
|<math>> f_{\psi(\Omega_\omega+1)}(4)</math>
|<code>(\1 1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|}
 
== [[Busy Beaver for SKI calculus]] ==
=== Busy Beaver for SKI calculus ===
{| class="wikitable"
! !! 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↑↑↑17
|SSI(SSK(S(S(KS)K)I))(S(K(S(SSK)))K)
|-
|Ξ₀(20)
|> f<sub>ω+1</sub>(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<sub>ω+2</sub>(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<sub>ω+2</sub>(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<sub>ω+3</sub>(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<sub>ω+3</sub>(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<sub>ω+4</sub>(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<sub>ω+4</sub>(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 ===
{| class="wikitable"
! !! Ξ₀<sub>SK(n)</sub> !! Champion
|-
| Ξ₀<sub>SK(n)</sub>(1) || 1 || S
|-
| Ξ₀<sub>SK(n)</sub>(2) || 2 || SS
|-
| Ξ₀<sub>SK(n)</sub>(3) || 3 || SSS
|-
| Ξ₀<sub>SK(n)</sub>(4) || 4 || SSSS
|-
| Ξ₀<sub>SK(n)</sub>(5) || 6 || SSS(SS)
|-
| Ξ₀<sub>SK(n)</sub>(6) || 10 || SSS(SS)S
|-
|Ξ₀<sub>SK(n)</sub>(7)
| 41
|SSS(S(SS))S
|-
|Ξ₀<sub>SK(n)</sub>(8)
|≥ 80
|SSK(S(SS)S)S
|-
|Ξ₀<sub>SK(n)</sub>(9)
|≥ 169
|S(SS)(SS)(SS)SS
|-
|Ξ₀<sub>SK(n)</sub>(10)
|≥ 376
|S(S(SS(KK)))(SS)(SS)
|-
|Ξ₀<sub>SK(n)</sub>(11)
|≥ 912
|S(SS)S(SS)(S(S(KS))S)
|-
|Ξ₀<sub>SK(n)</sub>(12)
|≥ 1530
|S(SS)(SS)(SS(SS(SS)))S
|-
|Ξ₀<sub>SK(n)</sub>(13)
|≥ 7811
|S(SS)(SS)(SS(SS(SSS)))S
|-
|Ξ₀<sub>SK(n)</sub>(14)
|≥ 2↑2097156*13-2
|SSS(SSK)(SS(SK)(S(KS)K))
|-
|Ξ₀<sub>SK(n)</sub>(15)
|≥ 2↑2↑2097156*3-2
|SSS(SSK)(SS(SK)(S(KS)K))S
|-
|Ξ₀<sub>SK(n)</sub>(16)
|> 2↑2↑2↑8193
|S(SS(SS))(SSK)(SS(SK)(S(KS)K))
|-
|Ξ₀<sub>SK(n)</sub>(17)
|> 2↑↑2↑↑2↑258
|SSS(SKS)(SSK)(SS(SK)(S(KS)K))
|-
|Ξ₀<sub>SK(n)</sub>(22)
|> 2↑↑↑19
|S(S(S(S(SK))))(SKS)(S(SSK)(K(SS(SK)(S(KS)K)))
|-
|Ξ₀<sub>SK(n)</sub>(23)
|> 2↑↑↑↑2↑↑↑17
|SS(S(SK))(SSK(SS(SK)(S(KS)K)))(S(K(S(SSK)))K)
|-
|Ξ₀<sub>SK(n)</sub>(24)
|> 2↑↑↑↑2↑↑↑19
|S(SS(S(SK))(SSK(SS(SK)(S(KS)K)))(S(K(S(SSK)))K))
|-
|Ξ₀<sub>SK(n)</sub>(25)
|> 3↑<sup>7</sup>3
|SS(S(SK))(SSK(SS(SS(SK))(S(KS)K)))(S(K(S(SSK)))K)
|-
|Ξ₀<sub>SK(n)</sub>(26)
|> f<sub>ω+1</sub>(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 ===
{| class="wikitable"
! !! Ξ₀<sub>BCKW</sub>(n) !! Champion
|-
| Ξ₀<sub>BCKW</sub>(1) || 1 || B
|-
| Ξ₀<sub>BCKW</sub>(2) || 2 || BB
|-
| Ξ₀<sub>BCKW</sub>(3) || 3 || BBB
|-
| Ξ₀<sub>BCKW</sub>(4) || 11 || WW(WB)
|-
| Ξ₀<sub>BCKW</sub>(5) || 47 || W(WW)(WB)
|-
|Ξ₀<sub>BCKW</sub>(6)
|≥ 2↑256*6-1
|WW(WC(WB))
|-
|Ξ₀<sub>BCKW</sub>(7)
|> 2↑↑↑17
|WW(C(WC)(WB))
|-
|Ξ₀<sub>BCKW</sub>(8)
|> 2↑↑↑2↑↑↑17
|W(WW)(C(WC)(WB))
|-
|Ξ₀<sub>BCKW</sub>(9)
|> 2↑<sup>5</sup>2↑↑↑↑2↑↑↑17
|WW(WB(C(WC))(WB))
|-
|Ξ₀<sub>BCKW</sub>(10)
|> f<sub>ω+1</sub>(2↑↑6/2-1)
|WW(C(WB)(C(CC(WB))))
|-
|Ξ₀<sub>BCKW</sub>(15)
|> f<sub>ω<sup>2^^18-1</sup></sub>(2)
|WW(C(WB)(B(BC(CC(WB)))(C(BC))))
|}
 
== [[Brainfuck]] ==
{| class="wikitable"
!
!BB_brainf(n)
!Champion
|-
|BB_brainf(1)
|1
|<code>+</code>
|-
|BB_brainf(2)
|2
|<code>++</code>
|-
|BB_brainf(3)
|3
|<code>+++</code>
|-
|BB_brainf(4)
|4
|<code>++++</code>
|-
|BB_brainf(5)
|5
|<code>+++++</code>
|-
|BB_brainf(6)
|≥ 6
|<code>++++++</code>
|-
|BB_brainf(7)
|≥ 7
|<code>+++++++</code>
|-
|BB_brainf(8)
|≥ 8
|<code>++++++++</code>
|-
|BB_brainf(9)
|≥ 9
|<code>+++++++++</code>
|-
|BB_brainf(10)
|≥ 10
|<code>++++++++++</code>
|-
|BB_brainf(11)
|≥ 11
|<code>+++++++++++</code>
|-
|BB_brainf(12)
|≥ 12
|<code>++++++++++++</code>
|-
|BB_brainf(13)
|≥ 16
|<code>++++[->++++<]</code>
|-
|BB_brainf(14)
|≥ 20
|<code>+++++[->++++<]</code>
|-
|BB_brainf(15)
|≥ 25
|<code>+++++[->+++++<]</code>
|-
|BB_brainf(16)
|≥ 30
|<code>++++++[->+++++<]</code>
|-
|BB_brainf(17)
|≥ 36
|<code>++++++[->++++++<]</code>
|-
|BB_brainf(18)
|≥ 42
|<code>+++++++[->++++++<]</code>
|-
|BB_brainf(19)
|≥ 49
|<code>+++++++[->+++++++<]</code>
|-
|BB_brainf(20)
|≥ 56
|<code>++++++++[->+++++++<]</code>
|-
|BB_brainf(21)
|≥ 64
|<code>++++++++[->++++++++<]</code>
|-
|BB_brainf(22)
|≥ 72
|<code>+++++++++[->++++++++<]</code>
|-
|BB_brainf(23)
|≥ 81
|<code>+++++++++[->+++++++++<]</code>
|-
|BB_brainf(24)
|≥ 90
|<code>+++++++++[->++++++++++<]</code>
|-
|BB_brainf(25)
|≥ 340
|<code>++++[>+[->++<]>[-<++>]<<]</code>
|-
|BB_brainf(26)
|≥ 1364
|<code>+++++[>+[->++<]>[-<++>]<<]</code>
|}
 
== [[Boolfuck]] ==
{| class="wikitable"
!
!BB_boolf(n)
!Champion
|-
|BB_boolf(1)
|1
|<code>*</code>
|-
|BB_boolf(2)
|1
|<code>*></code>
|-
|BB_boolf(3)
|2
|<code>*>*</code>
|-
|BB_boolf(4)
|2
|<code>*>*></code>
|-
|BB_boolf(5)
|3
|<code>*>*>*</code>
|-
|BB_boolf(6)
|3
|<code>*>*>*></code>
|-
|BB_boolf(7)
|4
|<code>*>*>*>*</code>
|}
 
== [[CounterScript]] ==
Macros for compression:
{| class="wikitable"
!
!Macro
!Definition
!Size
!Function
|-
|Constant Addition
|<code>A+=n;</code>
|increment <code>A</code> by <code>n</code>
|<code>n</code>
|<code>A++; repeated n times</code>
|-
|Transfer
|<code>A>>B*n;</code>
|increment <code>B</code> by <code>A*n</code> then set <code>A</code> to 0
|<code>n+2</code>
|<code>while A {A--; (B++; repeated n times)}</code>
|}
 
{| class="wikitable"
!
!BBCS(n)
!Champion
|-
|BBCS(1)
|1
|<code>A++;</code>
|-
|BBCS(2)
|2
|<code>A++; A++;</code>
|-
|BBCS(3)
|3
|<code>A++; A++; A++;</code>
|-
|BBCS(4)
|4
|<code>A++; A++; A++; A++;</code>
|-
|BBCS(5)
|5
|<code>A++; A++; A++; A++; A++;</code>
|-
|BBCS(6)
|6
|<code>A++; A++; A++; A++; A++; A++;</code>
|-
|BBCS(7)
|7
|<code>A++; A++; A++; A++; A++; A++; A++;</code>
|-
|BBCS(8)
|9
|<code>A++; A++; A++; while A {A--; B++; B++; B++;}</code>
|-
|BBCS(9)
|12
|<code>A++; A++; A++; A++; while A {A--; B++; B++; B++;}</code>
|-
|BBCS(10)
|16
|<code>A++; A++; A++; A++; while A {A--; B++; B++; B++; B++;}</code>
|-
|BBCS(11)
|20
|<code>A++; A++; A++; A++; A++; while A {A--; B++; B++; B++; B++;}</code>
|-
|BBCS(12)
|≥ 25
|<code>A++; A++; A++; A++; A++; while A {A--; B++; B++; B++; B++; B++;}</code>
|-
|BBCS(13)
|≥ 42
|<code>A++; while A {A++; while B {A--; B--; C++; C++;} C>>B*2; C++;}</code>
|-
|BBCS(14)
|≥ 129
|<code>A++; while A {A++; while B {A--; B--; C++; C++;} C>>B*3; C++;}</code>
|-
|BBCS(15)
|≥ 340
|<code><nowiki>A++; A++; A++; A++; while A {A--; B++; while B {B--; C++; C++;} while C {C--; B++; B++;}}</nowiki></code>
|-
|BBCS(16)
|≥ 1,554
|<code><nowiki>A++; A++; A++; A++; while A {A--; B++; while B {B--; C++; C++; C++;} while C {C--; B++; B++;}}</nowiki></code>
|-
|BBCS(17)
|≥ 9,330
|<code><nowiki>A++; A++; A++; A++; A++; while A {A--; B++; while B {B--; C++; C++; C++;} while C {C--; B++; B++;}}</nowiki></code>
|-
|BBCS(18)
|≥ 66,429
|<code><nowiki>A++; A++; A++; A++; A++; while A {A--; B++; while B {B--; C++; C++; C++;} while C {C--; B++; B++; B++;}}</nowiki></code>
|-
|BBCS(19)
|≥ (3^122 - 3)/2
|<code>A++; while A {A++; while B {A--; B--; C++;} while C {B++; C--; B>>D*3; D>>B;} C++;}</code>
|-
|BBCS(20)
|≥ 2^65536 - 2
|<code>A+=4; while A {A--; B++; while B {B--; C++; C>>D*2; D>>C;} C>>B;}</code>
|-
|BBCS(21)
|≥ 2^2^65536 - 2
|<code>A+=5; while A {A--; B++; while B {B--; C++; C>>D*2; D>>C;} C>>B;}</code>
|-
|BBCS(22)
|≥ 2^^7 - 2
|<code>A+=6; while A {A--; B++; while B {B--; C++; C>>D*2; D>>C;} C>>B;}</code>
|-
|BBCS(23)
|≥ 2^^8 - 2
|<code>A+=7; while A {A--; B++; while B {B--; C++; C>>D*2; D>>C;} C>>B;}</code>
|-
|BBCS(24)
|≥ 2^^65536 - 2
|<code>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++;}</code>
|-
|BBCS(25)
|> 2^^2^^6
|<code>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++;}</code>
|-
|BBCS(26)
|≥ 2^^2^^65536 - 2
|<code><nowiki>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++;}}</nowiki></code>
|-
|BBCS(64)
|> Graham's number
|[https://discord.com/channels/960643023006490684/1484108791636033659/1509179336136327310 too large to show]
|-
|BBCS(85)
|> q(5)
|too large to show, see [https://discord.com/channels/960643023006490684/1484108791636033659/1514987825127358615 Discord]
|}
 
== Fractran ([[Fractran|BBf]](n)) ==
{| class="wikitable"
|+
!n
!BBf(n)
!Example Champion
!Vector Representation
|-
| BBf(2) || 1 || <code>[1/2]</code>
|<math>\begin{bmatrix}
  -1
\end{bmatrix}</math>
|-
| BBf(3) || 1 || <code>[3/2]</code>
|<math>\begin{bmatrix}
  -1 & 1
\end{bmatrix}</math>
|-
| BBf(4) || 1 || <code>[9/2]</code>
|<math>\begin{bmatrix}
  -1 & 2
\end{bmatrix}</math>
|-
| BBf(5) || 2 || <code>[3/2, 1/3]</code>
|<math>\begin{bmatrix}
  -1 &  1 \\
  0 & -1
\end{bmatrix}</math>
|-
| BBf(6) || 3 || <code>[9/2, 1/3]</code>
|<math>\begin{bmatrix}
  -1 &  2 \\
  0 & -1
\end{bmatrix}</math>
|-
| BBf(7) || 4 || <code>[27/2, 1/3]</code>
|<math>\begin{bmatrix}
  -1 &  3 \\
  0 & -1
\end{bmatrix}</math>
|-
| BBf(8) || 5 || <code>[81/2, 1/3]</code>
|<math>\begin{bmatrix}
  -1 &  4 \\
  0 & -1
\end{bmatrix}</math>
|-
| BBf(9) || 6 || <code>[243/2, 1/3]</code>
|<math>\begin{bmatrix}
  -1 &  5 \\
  0 & -1
\end{bmatrix}</math>
|-
| BBf(10) || 7 || <code>[729/2, 1/3]</code>
|<math>\begin{bmatrix}
  -1 &  6 \\
  0 & -1
\end{bmatrix}</math>
|-
| BBf(11) || 10 || <code>[27/2, 25/3, 1/5]</code>
|<math>\begin{bmatrix}
  -1 &  3 &  0 \\
  0 & -1 &  2 \\
  0 &  0 & -1
\end{bmatrix}</math>
|-
| BBf(12) || 13 || <code>[81/2, 25/3, 1/5]</code>
|<math>\begin{bmatrix}
  -1 &  4 &  0 \\
  0 & -1 &  2 \\
  0 &  0 & -1
\end{bmatrix}</math>
|-
| BBf(13) || 17 || <code>[81/2, 125/3, 1/5]</code>
|<math>\begin{bmatrix}
  -1 &  4 &  0 \\
  0 & -1 &  3 \\
  0 &  0 & -1
\end{bmatrix}</math>
|-
| BBf(14) || 21 || <code>[243/2, 125/3, 1/5]</code>
|<math>\begin{bmatrix}
  -1 &  5 &  0 \\
  0 & -1 &  3 \\
  0 &  0 & -1
\end{bmatrix}</math>
|-
| BBf(15) || 28 || <code>[1/45, 4/5, 3/2, 25/3]</code>
|<math>\begin{bmatrix}
  0 & -2 & -1 \\
  2 &  0 & -1 \\
  -1 &  1 &  0 \\
  0 & -1 &  2
\end{bmatrix}</math>
|-
| BBf(16) || 53 || <code>[1/45, 4/5, 3/2, 125/3]</code>
|<math>\begin{bmatrix}
  0 & -2 & -1 \\
  2 &  0 & -1 \\
  -1 &  1 &  0 \\
  0 & -1 &  3
\end{bmatrix}</math>
|-
| BBf(17) || 107 || <code>[5/6, 49/2, 3/5, 40/7]</code>
|<math>\begin{bmatrix}
  -1 & -1 &  1 &  0 \\
  -1 &  0 &  0 &  2 \\
    0 &  1 & -1 &  0 \\
    3 &  0 &  1 & -1
\end{bmatrix}</math>
|-
| BBf(18) || 211 || <code>[5/6, 49/2, 3/5, 80/7]</code>
|<math>\begin{bmatrix}
  -1 & -1 &  1 &  0 \\
  -1 &  0 &  0 &  2 \\
    0 &  1 & -1 &  0 \\
    4 &  0 &  1 & -1
\end{bmatrix}</math>
|-
| BBf(19) || 370 || <code>[5/6, 49/2, 3/5, 160/7]</code>
|<math>\begin{bmatrix}
   -1 & -1 &  1 &  0 \\
   -1 & -1 &  1 &  0 \\
   -1 &  0 &  0 &  2 \\
   -1 &  0 &  0 &  2 \\
     0 &  1 & -1 &  0 \\
     0 &  1 & -1 &  0 \\
     5 &  0 &  1 & -1
     5 &  0 &  1 & -1
\end{bmatrix}</math>
\end{bmatrix}</math>
|-
|-
|20
|BBf(20)
|≥ 746
| 746
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|<math>\begin{bmatrix}
|<math>\begin{bmatrix}
       &    -1 &    -1 &    +1 &      \\
       &    -1 &    -1 &    +1 &      \\
   +1 &    -1 &      &      &    +1 \\
   +1 &    -1 &      &      &    +1 \\
   +1 &    +1 &      &    -1 &    -1 \\
   +1 &    +1 &      &    -1 &    -1 \\
   -1 &      &    +1 &      &      \\
   -1 &      &    +1 &      &      \\
       &    +2 &    -1 &      &
       &    +2 &    -1 &      &
\end{bmatrix}</math>
\end{bmatrix}</math>
|-
|-
|21
|BBf(21)
|≥ 31,957,632
|≥ 31,957,632
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|<math>\begin{bmatrix}
|<math>\begin{bmatrix}
     0 &    -1 &    -1 &    1 \\
     0 &    -1 &    -1 &    1 \\
     2 &    -1 &    0 &    0 \\
     2 &    -1 &    0 &    0 \\
   -1 &    3 &    0 &    -1 \\
   -1 &    3 &    0 &    -1 \\
   -1 &    0 &    1 &    0 \\
   -1 &    0 &    1 &    0 \\
     0 &    2 &    -1 &    0
     0 &    2 &    -1 &    0
\end{bmatrix}</math>
\end{bmatrix}</math>
|}
|-
|BBf(22)
|<math>> 1.146 \times 10^{62}</math>
|<code>[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]</code>
|<math>\begin{bmatrix}
  -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{bmatrix}</math>
|-
|BBf(23)
|<math>> 4.393 \times 10^{124}</math>
|<code>[10/3, 9/14, 5/4, 121/2, 7/5, 3/11]</code>
|<math>\begin{bmatrix}
    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{bmatrix}</math>
|-
|BBf(24)
|<math>> 9.000 \times 10^{14448}</math><ref>https://discord.com/channels/960643023006490684/1438019511155691521/1547182811713380383</ref>
|<code>[14/15, 1/6, 11/3, 441/2, 5/7, 15/11]</code>
|<math>\begin{bmatrix}
    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{bmatrix}</math>
|}
 
== Cyclic Tree Rewriting System ([[Cyclic Tree Rewriting System|CTBB(n)]]) ==
{| class="wikitable"
|+
!
!Runtime
!Champions
|-
|CTBB(2)
|5<ref>https://discord.com/channels/960643023006490684/960643023530762341/1440163057463726233</ref>
|
|-
|CTBB(3)
|<u>></u> 38<ref>https://discord.com/channels/960643023006490684/1438694294042181742/1440193579006951517</ref>
|
|-
|CTBB(4)
|≥ 672<ref>https://discord.com/channels/960643023006490684/1438694294042181742/1443246244142252043</ref>
|
|-
|CTBB(5)
|≥ 2^2^2^2^182<ref>https://discord.com/channels/960643023006490684/1438694294042181742/1443298934217900063</ref>
|
|-
|CTBB(6)
|<u>></u> 2↑↑↑131<ref>https://discord.com/channels/960643023006490684/1438694294042181742/1442847677883875479</ref>
|
|-
|CTBB(7)
|<u>></u> 4↑↑↑↑(4↑↑↑3)<ref>https://discord.com/channels/960643023006490684/1438694294042181742/1442950825545564281</ref> <ref>https://discord.com/channels/960643023006490684/1438694294042181742/1442819117735346217</ref>
|
|}
 
== Cyclic Tag ([[Cyclic Tag|BBCT(n)]]) ==
{| class="wikitable"
|+
!BBCT(n)
!Value
!Champion
|-
|BBCT(1)
|1
|<code>empty</code>
|-
|BBCT(2)
|2
|<code>0</code>
|-
|BBCT(3)
|3
|<code>00</code>
|-
|BBCT(4)
|4
|<code>000</code>
|-
|BBCT(5)
|7
|<code>010_</code>
|-
|BBCT(6)
|8
|<code>010_0</code>
|-
|BBCT(7)
|= 13
|<code>0110__</code>
|-
|BBCT(8)
|≥ 28
|<code>101_10_</code>
|-
|BBCT(9)
|≥ 717
|<code>1011_00_</code>
|-
|BBCT(10)
|≥ 1,626,862
|<code>1001_1_00</code>
|-
|BBCT(11)
|≥ 1,859,166
|<code>10_10010_0</code>
|-
|BBCT(12)
|≥ 6,328
|<code>0111__010_0</code>
|-
|BBCT(13)
|≥ 5,217,735
|<code>001110_1_0_0</code>
|-
|BBCT(14)
|≥ 154,171,565
|<code>01011__0_0100</code>
|}
 
== [[Tiny Tag]] ==
{| class="wikitable"
|+
!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 ([[Register machine|MBB(n)]]) ==
 
{| class="wikitable"
!Domain
!Halting Time
!Champion
|-
|MBB(1)
|1
|<code>0+*</code>
|-
|MBB(2)
|3
|<code>0+B_0-B*</code>
|-
|MBB(3)
|5
|<code>0+B_0+C_0-C*</code>
|-
|MBB(4)
|10
|<code>0+B_1+C_0-BD_1-C*</code>
|-
|MBB(5)
|24
|<code>0-DB_0+C_1-ED_1+A_1-B*</code>
|-
|MBB(6)
|49
|<code>0+B_1-FC_1+D_0-CE_0+A_1-A*</code>
|-
|MBB(7)
|231
|<code>0+B_0+C_0+D_1-GE_1+F_0-EC_1-A*</code>
|-
|MBB(8)
|≥ 3,394
|<code>0+B_0+C_1-GD_1+E_0-DF_2-HG_2+A_2-D*</code>
|-
|MBB(9)
|> 4.493 × 10<sup>74</sup>
|<code>0+B_0+C_1-HD_0-EG_0-FI_1+D_2+H_0+A_2-B*</code>
|-
|MBB(10)
|> 10↑↑4.03
|<code>0+B_1-AC_2-ED_1+E_1+F_0-DG_2-JH_2+I_1-HE_2-A*</code>
|-
|MBB(11)
|> 10↑↑17.74
|<code>0-KB_1-IC_1+D_1+E_2-CF_0-AG_0+H_1-GI_2+J_2+B_0-B*</code>
|}
 
== Pebble Automation ([[Pebble Automaton|peBBle(n)]]) ==
{| class="wikitable"
|+
!n
!peBBle(n)
!Champion
|-
|peBBle(1)
|0
|<code>00</code>
|-
|peBBle(2)
|1
|<code>00_01</code>
|-
|peBBle(3)
|3
|<code>00_01_20</code>
|-
|peBBle(4)
|6
|<code>00_01_20_03</code>
|-
|peBBle(5)
|11<ref>https://discord.com/channels/960643023006490684/1535695107448381591/1536023084975525999</ref>
|<code>00_11_03_21_04</code>
|-
|peBBle(6)
|17<ref name="PA67">https://discord.com/channels/960643023006490684/1535695107448381591/1536608195156381747</ref>
|<code>00_11_03_21_40_15</code>
|-
|peBBle(7)
|28<ref name="PA67" />
|<code>00_11_03_21_10_33_61</code>
|-
|peBBle(8)
|≥ 63<ref>https://discord.com/channels/960643023006490684/1535695107448381591/1537528571394793563</ref>
|<code>00_11_03_21_50_30_24_70</code>
|-
|peBBle(9)
|≥ 63577<ref>https://discord.com/channels/960643023006490684/1535695107448381591/1539048792400203796</ref>
|<code>00_02_01_40_14_12_00_52_81</code>
|}
 
== Wang Tiles ([[Wang Tiles|BB<sub>WT</sub>(n)]]) ==
{| class="wikitable"
|+
!n
!BB<sub>WT</sub>(n)
!Champion
!Image
|-
|BB<sub>WT</sub>(1)
|1
|<code>[1,0,0,0]</code>
|[[File:BusyBeaverWangTiles CurrentChampion 1.png|alt=The definite champion for the largest valid square patch (1 by 1) made by a set of 1 Wang tile.|border|center|frameless]]
|-
|BB<sub>WT</sub>(2)
|2<ref name="WT23">https://discord.com/channels/960643023006490684/1529586687926210691/1530032750579417198</ref>
|<code>[1,0,2,0] [2,0,3,0]</code>
|[[File:BusyBeaverWangTiles CurrentChampion 2.png|alt=The definite champion for the largest valid square patch (2 by 2) made by a set of 2 Wang tiles.|border|center|frameless]]
|-
|BB<sub>WT</sub>(3)
|3<ref name="WT23" />
|<code>[1,0,2,0] [2,0,3,0] [3,0,4,0]</code>
|[[File:BusyBeaverWangTiles CurrentChampion 3.png|alt=The definite champion for the largest valid square patch (3 by 3) made by a set of 3 Wang tiles.|border|center|frameless]]
|-
|BB<sub>WT</sub>(4)
|4<ref>https://discord.com/channels/960643023006490684/1529586687926210691/1530697443820437645</ref>
|<code>[1,0,2,0] [2,0,3,0] [3,0,4,0] [4,0,5,0]</code>
|[[File:BusyBeaverWangTiles CurrentChampion 4.png|alt=The definite champion for the largest valid square patch (4 by 4) made by a set of 4 Wang tiles.|border|center|frameless]]
|-
|BB<sub>WT</sub>(5)
|≥ 6<ref>https://discord.com/channels/960643023006490684/1529586687926210691/1530722442761474218</ref>
|<code>[1,0,2,0] [2,0,3,0] [3,0,4,0] [4,6,1,5] [4,5,2,6]</code>
|[[File:BusyBeaverWangTiles CurrentChampion 5.png|alt=The best known champion for the largest valid square patch (6 by 6) made by a set of 5 Wang tiles.|border|center|frameless]]
|-
|BB<sub>WT</sub>(6)
|≥ 8
|<code>[1,0,2,0] [2,0,3,0] [3,0,4,0] [4,0,5,0] [5,7,1,6] [5,6,2,7]</code>
|[[File:BusyBeaverWangTiles CurrentChampion 6.png|alt=The best known champion for the largest valid square patch (8 by 8) made by a set of 6 Wang tiles.|border|center|frameless]]
|-
|BB<sub>WT</sub>(7)
|≥ 11
|<code>[1,0,2,0] [2,0,3,0] [3,0,4,0] [4,0,5,0] [5,7,1,6] [5,6,2,7] [4,6,3,7]</code>
|[[File:BusyBeaverWangTiles CurrentChampion 7.png|alt=The best known champion for the largest valid square patch (11 by 11) made by a set of 7 Wang tiles.|border|center|frameless]]
|-
|BB<sub>WT</sub>(8)
|≥ 14
|<code>[1,0,2,0] [2,0,3,0] [3,0,4,0] [4,0,5,0] [5,0,6,0] [6,8,1,7] [6,7,2,8] [5,7,3,8]</code>
|[[File:BusyBeaverWangTiles CurrentChampion 8.png|alt=The best known champion for the largest valid square patch (14 by 14) made by a set of 8 Wang tiles.|border|center|frameless]]
|-
|BB<sub>WT</sub>(9)
|≥ 17
|<code>[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]</code>
|[[File:BusyBeaverWangTiles CurrentChampion 9.png|alt=The best known champion for the largest valid square patch (17 by 17) made by a set of 9 Wang tiles.|border|center|frameless]]
|-
|BB<sub>WT</sub>(10)
|≥ 212
|
|
|}
 
== Branching Beaver ([[Branching Beaver|BrB(n)]]) ==
=== Maximum Steps Function (BrB(n)) ===
{| class="wikitable"
|+
!Domain
!Steps
!Champions
|-
|BrB(1)
|3
|<code>1RA1RZ</code>
|-
|BrB(2)
|10
|<code>1RA0GB_1GA1RZ</code>
|-
|BrB(3)
|≥ 52
|<code>1RB0GB_0GC1RZ_1GC0RA</code>
|-
|BrB(4)
|≥ 4636
|<code>1RA0GB_1RC1RD_1GA0GC_1RZ0GA</code>
|-
|BrB(2,3)
|≥ 68
|<code>1RA2GA0BB_1BA2BA1RZ</code><code>1RA2GA0BB_2BA1RZ1BA</code>
|-
|BrB(2,4)
|≥ 332,372
|<code>1RB2RB1GB1RZ_2GB3GA3RA1RA</code>
|}
 
=== Maximum Space Function (BrB<sub>space</sub>(n)) ===
{| class="wikitable"
|+
!Domain
!Max Space
!Champions
|-
|BrB<sub>space</sub>(1)
|2
|<code>1RA1RZ</code>
|-
|BrB<sub>space</sub>(2)
|4
|<code>1RA0GB_1GA1RZ</code> (and 10 others)
|-
|BrB<sub>space</sub>(3)
|≥ 22
|<code>1RB1RZ_0GC0BB_1GA0RA</code>
|-
|BrB<sub>space</sub>(4)
|≥ 384
|<code>1RB0BB_0RC1RZ_0GA1GD_1RC0RA</code>
|-
|BrB<sub>space</sub>(2,3)
|≥ 20
|<code>1RA2GA0BB_1BA2BA1RZ</code><code>1RA2GA0BB_2BA1RZ1BA</code>
|-
|BrB<sub>space(2,4)</sub>
|≥ 5996
|<code>1RB2GA3BA0RA_1GB2BB1RA1RZ</code>
|}
 
= '''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) ===
{| class="wikitable"
!Domain
!Runtime
!Champions
|-
|TT(2)
|≥ 13
|<code>1TB---_1PA0PB</code>
|-
|TT(3)
|≥ 82
|<code>1PB0PA_1TA0PC_1PA---</code>
|-
|TT(4)
|≥ 48,186
|<code>1TB1PA_1PC0PA_1TA0PD_---1TA</code>
|-
|TT(2,3)
|≥ 223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TT(3,3)
|> <math>2.27 \times 10^{248}</math><ref>https://discord.com/channels/960643023006490684/1243312334907375676/1535305073259974728</ref>
|<code>1PB0TC1PA_2PC1PZ2PA_2TC0PA2PC</code>
|-
|TT(2,4)
|> 3.467*10<sup>15</sup>
|<code>1TA2PB3TB---_3TA1PB1TA1PA</code>
|}
 
=== Maximum score for TT(n,k) ===
{| class="wikitable"
!Domain
!Score
!Champion
|-
|TT(3,3)
|≥ 32,778<ref name="TT(3,3) 715M">https://discord.com/channels/960643023006490684/1243312334907375676/1533503518906912778</ref>
|<code>1PB2PA0PC_2TB1PC2PB_2PB1PZ0PA</code>
|}
 
== 2D Turmites ([[Terminating Turmite|turNing machines]]) ==
There are currently no known/available Champions for this function.


='''Doodle Function ([[Doodle function|doodle(c,n)]])'''=
= '''Doodle Function ([[Doodle function|doodle(c,n)]])''' =
doodle(1,n) = 1 and doodle(2,n) = n. Also note that doodle(c) = doodle(c,2).
doodle(1,n) = 1 and doodle(2,n) = n. Also note that doodle(c) = doodle(c,2).
{| class="wikitable"
{| class="wikitable"
Line 1,279: Line 2,589:
|
|
|}
|}
='''Turmites'''=
==Terminating Turmites ([[TT]](n,k), 1D Turmites)==
Where n is the amount of states and k is the amount of symbols. There are currently no known/available Champions for this function.
==2D Turmites ([[Terminating Turmite|turNing machines]])==
There are currently no known/available Champions for this function.


='''Bug Function'''=
= '''Bug Function ([[Bug Game]])''' =
Bug(2,2) = 2
Bug(2,2) = 2
<pre>
<pre>
Line 1,363: Line 2,668:
</pre>
</pre>


=References=
= '''References''' =

Latest revision as of 13:27, 19 October 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
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↑112↑113 1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF (bbch)
BB(8) >2↑373[1] 1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_1LC0RH_1RG1LF (bbch)
BB(9) >fω(2↑133)>fω2(12)[2] 1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LI0LC_1RG1LD_0RH0RF_0RG0RE_0LH1RZ (bbch)
BB(10) >fω+1(2↑363)>g64[3] 1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LI0LC_0RG1LD_1RH1LG_1LC0RG_0LJ0LI_1RZ0LC (bbch)
BB(11)
BB(12) >fω+12(2↑363)[4] 1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LI0LC_0RG1LD_1RH1LG_1LC0RG_0LJ0LI_1RK0LC_0RL---_0RF1RZ (bbch)
BB(13) >fω+2(fω(20))[5] 1LB1LA_1RC0LA_1RD1RC_1RE1RK_1RF0LJ_0LG1RI_1RE0RH_1LF0RC_1RM0LE_1LA1LE_0LL1LB_0RM1LL_1RZ1RH (bbch)
BB(14) >fω+23(15)[6] 1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LN1RF_1LJ0RA_1RA0LH_1LG1LH_1RK0LM_1RZ0LE_0LE0RJ_1LH1LM_0LH0LL_0LG1LN (bbch)
BB(20) >fω+62(4) 1RR---_0LJ0RC_1RB1LD_0LD1LE_1LF1RG_1LI0LE_0RE1LD_1RB1RH_1RH1RN_1LK1LJ_0LL1LJ_1RM1RA_0RM0RH_0LO1RN_1RH0LP_1RH1LQ_1RH1LO_1LS1RR_1RZ1LT_0LH0LS (bbch)
BB(21) >fω22(4↑↑341)[7] 0LI0LF_0RJ1RG_0RD1LC_1RH1RE_1LO0RH_1LA1LF_0LC1LA_1RB1RH_1RD0LF_1LP0RK_0LM1LL_0LL1LM_1LE1LN_0RQ0LM_0RP1LO_1LR0RH_1LF1RQ_1RZ0LS_1LH0LT_1LH1LU_0LE1LR (bbch)
BB(40) >fωω(75500)
BB(41) >fωω4(32)
BB(51) >fε0+1(8)
BB(150) >flim(BMS)(10↑↑15)[8] too large to show
BB(1015) >D5(99) = Loader's Number
3 Symbols: Runtime Champions
BB(1,3) 1 1RZ------ (bbch)
BB(2,3) 38 1RB2LB1RZ_2LA2RB1LB (bbch)
BB(3,3) ≥119112334170342541>1017 0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC (bbch)
BB(4,3) >2↑↑↑2↑↑↑2↑↑↑(7.92×1028) 1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD (bbch)
BB(6,3) >fω2(10↑↑↑6) 1LE2LD2LC_2RB1RB1RA_2RB1LC0LC_1RB1LD2LD_2RD2LE2LF_1RZ---1LE (bbch)
BB(8,3) >fω+1(fω(f7(2))) 1LD1LE2RA_1RE2LE2RB_1RD2RA2LC_1RB1RD2RB_1LH2RF1LC_1RZ0LF1LG_1RD2LG2LG_0LD1LH1LH (bbch)
BB(11,3) >fω+32(4) 1LD2RG2LA_2LE2LH2RB_2LE1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RI---_0RC1LH2RG_1RC2LH1LA_1RI1RI1LJ_0LA2LK---_1RZ2LJ--- (bbch)
BB(36,3) >fε0(7)
BB(37,3) >fε0(373676378)
4 Symbols: Runtime Champions
BB(1,4) 1 1RZ--------- (bbch)
BB(2,4) 3,932,964 1RB2LA1RA1RA_1LB1LA3RB1RZ (bbch)
BB(3,4) >(2↑155)+14[9] 1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC (bbch)
BB(5,4) >fω2(10↑410↑↑65533) 1LD2RA3RB1LC_1LD2LA3RB---_2RA1LA3RC2LC_2RA---2LD2LE_------1RZ1LD (bbch)
BB(31,4) >fε0(373676378)
5 Symbols: Runtime Champions
BB(1,5) 1 1RZ------------ (bbch)
BB(2,5) >1010103314360 1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ (bbch)
BB(3,5) >fω(2↑155)>fω2(15) 1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC (bbch)
BB(19,5) >fε0(1.7⋅10352) 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) >fψ0(Ω2)(1.7⋅10352)
6 Symbols: Runtime Champions
BB(1,6) 1 1RZ--------------- (bbch)
BB(2,6) >10↑↑10↑↑1010115[10] 1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA (bbch)
BB(61,6) >fψ0(Ω2)(1.7⋅10352)
7 Symbols: Runtime Champions
BB(18,7) >fε0(1.7⋅10352) 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) >fψ0(Ω2)(1.7⋅10352)
BB(134,7) >fψ0(Ωω)(2050)
8 Symbols or more: Runtime Champions
BB(17,8) >fε0(1.7⋅10352) 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(42,9) >fψ0(Ω2)(1.7⋅10352)

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↑112↑113 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↑↑↑2↑↑↑2↑↑↑(7.92×1028) 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) ≥(2↑155)+14[9] 1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC (bbch)
5 Symbols: Score Champions
Σ(1,5) 1 1RZ------------ (bbch)
Σ(2,5) >1010103314360 1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ (bbch)
6 Symbols: Score Champions
Σ(1,6) 1 1RZ--------------- (bbch)
Σ(2,6) >10↑↑10↑↑1010115[10] 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[11] 1RB1LD_1RC1RB_1LC1LA_0RC0RD (bbch)
BBB(5) ≥1014006 1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA (bbch)
3 Symbols: Steps taken Champions
BBB(1,3)
BBB(2,3) 59[12] 1RB2LB1LA_2LB2RA0RA (bbch)
BBB(3,3) ≥ 10 ↑↑ 6.74[13] 1RB0LB2LA_1LA0RC0LB_2RC2RB0LC (bbch)
4 Symbols: Steps taken Champions
BBB(1,4)
BBB(2,4) ≥205770076433044242247859>2×1023[14] 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(6) ≥ ≈ 10↑↑15.60465 1RB0LD_1RC0RF_1LC1LA_0LE0RZ_1LF0RB_0RC0RE (bbch)
num(2,3) 6 1RB1LA1LB_0LA2RA1RZ (bbch)
num(3,3) ≥ 12 1RB1RA1RZ_1LC1LC2LA_2RA1LB1LA (bbch)
num(4,3) >10↑44 1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD (bbch)

Maximum Space Function (BBspace(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) 1 1RZ------ (bbch)
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 (BBrev(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) >1019 1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB1RZ (bbch)

Maximum Score Function (Σrev(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[14] 1RB0RA_1LB1LA (bbch)
BLB(3) 34[15] 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[16] 1RB0RF_1RC1RA_1LD0LA_0RG0RE_0RA1LF_0LE0LC_1RZ0RD (bbch)
3 Symbols: Steps Champions
BLB(1,3) nonexistent nonexistent
BLB(2,3) 77[15] 1RB2LA0RB_1LA0LB1RA (bbch)
BLB(3,3) > 10 ↑↑ 6.54[13] 1RB1RC2RA_0RC2RA0RB_2LC2LB0LA (bbch)
4 Symbols: Steps Champions
BLB(1,4) nonexistent nonexistent
BLB(2,4) ≥ 1,367,361,263,049[15] 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[17] 1RB0LA---_1LB2LA0RB (bbch)
BBS(3,3) > 10 ↑↑ 6 1RB0LB2LA_1LA0RC0LB_2RC2RB0LC (bbch)
BBS(4,3) > 10↑44 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) >6.889×101565 1RB1LA------_1RC3LB1RB---_2LA2LC---0LC (bbch)
BBi(9) >1010103314360 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) >1.355×10783 1RB1LA------_1RC3LB1RB---_2LA2LC---0LC (bbch)
Σi(9) >1010103314360 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≥n BBλ(n) = n.

BBλ(n) Champions
BBλ(4) 4 λ1
BBλ(6) 6 λλ1
BBλ(7) 7 λλ2
BBλ(8) 8 λλλ1
BBλ(9) 9 λλλ2
BBλ(10) 10 λλλλ1
BBλ(11) 11 λλλλ2
BBλ(12) 12 λλλλλ1
BBλ(13) 13 λλλλλ2
BBλ(14) 14 λλλλλλ1
BBλ(15) 15 λλλλλλ2
BBλ(16) 16 λλλλλλλ1
BBλ(17) 17 λλλλλλλ2
BBλ(18) 18 λλλλλλλλ1
BBλ(19) 19 λλλλλλλλ2
BBλ(20) 20 λλλλλλλλλ1
BBλ(21) 22 λ(1(λ2))(1(λ2))
BBλ(22) 24 λ(11)(11)(11)
BBλ(23) 26 λ(1(λλ2))(1(λλ2))
BBλ(24) 30 λ(1(λ1))(1(λ1))(1(λ1))
BBλ(25) 42 λ1(λ1(21))(1(1(λ1(21))))
BBλ(26) 52 λλ2(λλ2(12))(1(2(λλ2(12))))
BBλ(27) 44 λλ1(λ1(21))(1(1(λ1(21))))
BBλ(28) 58 λ1(λλ1(3(λ2)))(1(λ2(λλ1(4(λ2)))))
BBλ(29) 223 λ(λ11)(λ1(1(21)))
BBλ(30) 160 (λ111)(λλ2(12))
BBλ(31) 267 (λ11)(λλ2(2(12)))
BBλ(32) 298 λ(λ11)(λ1(1(2(λ2))))
BBλ(33) 1812 λ(λ11)(λ1(1(1(21))))
BBλ(34) 327,686 (λ1111)(λλ2(21))
BBλ(35) 5×333+6=38127987424941>3.8×1013 (λ111)(λλ2(2(21)))
BBλ(36) 5×2223+6>5.7×1077 (λ11)(λ1(1(λλ2(21))))
BBλ(37) BBλ(35)+2=5×333+8=38127987424943>3.8×1013 λ(λ111)(λλ2(2(21)))
BBλ(38) 5×22222+6>1019729 (λ11111)(λλ2(21))
BBλ(39) 5×3333+6>103638334640024 (λ1111)(λλ2(2(21)))
BBλ(40) ≈(2↑↑)1533>10↑↑↑16 (λ111)(λ1(λλ2(21))1)
BBλ(41) 5×3385+6>101.7×1040 (λ1(λ11)1)(λλ2(2(21)))
BBλ(42) ≥BBλ(40)+2>(2↑↑)1533>10↑↑↑16 λ(λ111)(λ1(λλ2(21))1)
BBλ(43) >2↑↑↑2↑↑↑2↑↑8 (λ11)(λ1(λ1(λλ2(21))2))
BBλ(44) >10↑↑↑10↑↑↑16 (λ1111)(λ1(λλ2(21))1)
BBλ(45) ≥BBλ(43)+2>2↑↑↑2↑↑↑2↑↑8 λ(λ11)(λ1(λ1(λλ2(21))2))
BBλ(46) ≥BBλ(44)+2>10↑↑↑10↑↑↑16 λ(λ1111)(λ1(λλ2(21))1)
BBλ(47) >fω(f5(2)) (λ111)(λλ1(12)(λλ2(21)))
BBλ(48) >10↑↑↑10↑↑↑10↑↑↑16 (λ11111)(λ1(λλ2(21))1)
BBλ(49) >fω+1(2↑↑62)>Graham's Number (λ11)(λ1(1(λλ12(λλ2(21)))))
BBλ(61) >fω2↑↑18−1(2) (λ111)(λ1(1(λλλ132(λλ2(21)))))
BBλ(70) >fωω2(fωω(fωω(3))) (λ11(λ1(λλλ1325)2)1)(λλ2(21))
BBλ(74) >fωω2+1(7) (λ111(λ1(λλλ1325)2)1)(λλ2(21))
BBλ(76) >fωω3(2) (λ11(λλλλ14321)1111)(λλ2(21))
BBλ(77) >fωω2(3) (λ1(λλλλ14321)1111)(λλ2(2(21)))
BBλ(80) >fωω14(2) (λ111(λλλλ14321)1111)(λλ2(21))
BBλ(82) >fωωω(2↑↑↑↑18) (λ1(λ1(λλλλ14321)1111)1)(λλ2(21))
BBλ(86) >fωω65536(2) (λ1(λλλλ1444321)1111)(λλ2(21))
BBλ(90) >fζ0(15) (λ11(λλλλ1444321)1111)(λλ2(21))
BBλ(94) >fψ(Ωω)(12) (λ111(λλλλ1444321)1111)(λλ2(21))
BBλ(95) >fψ(Ωω)(23) (λ11(λλλλ1444321)1111)(λλ2(2(21)))
BBλ(96) >fψ(Ωω)(fωω2(2)) (λ1(λ1(λλλλ1444321)1111)1)(λλ2(21))
BBλ(100) >fψ(Ωω)+1(4) (λ11(λ1(λλλλ1444321)1111)1)(λλ2(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) >flim⁡(BMS⁡)3(5) too large to show
BBλ(1797) >D5(99) (Loader's number) too large to show

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

Note that f(n)=6+5×BBλ(n).

BBλ1(n) Champions
BBλ1(1) 0
BBλ1(2) 1 1
BBλ1(3) 0
BBλ1(4) 4 λ1
BBλ1(5) 5 λ2
BBλ1(6) 6 λλ1
BBλ1(7) 7 λλ2
BBλ1(8) 26 1(λ1)
BBλ1(9) 9 λλλ2
BBλ1(10) 36 1(λλ1)
BBλ1(11) 41 1(λλ2)
BBλ1(12) 266 1(1(λ1))
BBλ1(13) 51 1(λλ2)
BBλ1(14) f(36)=25×2223+36>2.85×1078 1(1(λλ1))
BBλ1(15) f(41)=25×3385+36>101.7×1040 1(1(λλ2))
BBλ1(16) f(266) 1(1(1(λ1)))
BBλ1(17) f(51) 1(1(λλλ2))
BBλ1(18) f4(4)=f(f(266)) 1(λ1)1(λ1)
BBλ1(19) f3(7)=f(f(41)) 1(1(1(λλ2)))
BBλ1(20) f6(4)=f4(266) 1(λλ1)1(λ1)
BBλ1(21) f7(4)=f5(266) 1(λλ2)1(λ1)
BBλ1(22) f26(4)=f24(266) 1(1(λ1))1(λ1)
BBλ1(28) ≥fBBλ(f3(4))(4) 1(λ1)1(λ1)1(λ1)
BBλ1(29) ≥fBBλ(fBBλ(f4(4))+4(4))+BBλ(f4(4))+5(4) 1(λ1)(λ121)(λ1)

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 >7.625×1012 (\1 1 1) (\\2 (2 (2 1)))
11 >107.625×1012 (\1 1 1 1) (\\2 (2 (2 1)))
12 >10↑316 (\1 1 1) (\1 (\\2 (2 1)) 1)
13 >10↑310↑310↑26 (\1 1) (\1 (\1 (\\2 (2 1)) 2))
14 >fω(f5(2)) (\1 1 1) (\\1 (1 2) (\\2 (2 1)))
15 >fω+1(101019,727) (\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))
18 >fωω(2↑↑18) (\1 1 1) (\1 (1 (\\\1 3 2 (\\2 (2 1)))))
19 >fωω2(fωω2(3)) (\1 1 (\1 (\\1 3 2 5) 2) 1) (\2 (2 1)))
20 >fωω2+1(7) (\1 1 1 (\1 (\\1 3 2 5) 2) 1) (\2 (2 1)))
21 >fωω2+ω+2(2) (\1 1 (\\\\1 4 3 2 1) 1 1 1 1) (\\2 (2 1))
22 >fωω14(2) (\1 1 1 (\\1 4 3 2 1) 1 1 1 1) (\2 (2 1))
23 >fζ0(15) (\1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))
24 >fψ(Ωω)(12) (\1 1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))
25 >fψ(Ωω)(fωω+2(2)) (\1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))
26 >fψ(Ωω+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↑↑↑17 SSI(SSK(S(S(KS)K)I))(S(K(S(SSK)))K)
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)(22) > 2↑↑↑19 S(S(S(S(SK))))(SKS)(S(SSK)(K(SS(SK)(S(KS)K)))
Ξ₀SK(n)(23) > 2↑↑↑↑2↑↑↑17 SS(S(SK))(SSK(SS(SK)(S(KS)K)))(S(K(S(SSK)))K)
Ξ₀SK(n)(24) > 2↑↑↑↑2↑↑↑19 S(SS(S(SK))(SSK(SS(SK)(S(KS)K)))(S(K(S(SSK)))K))
Ξ₀SK(n)(25) > 3↑73 SS(S(SK))(SSK(SS(SS(SK))(S(KS)K)))(S(K(S(SSK)))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↑52↑↑↑↑2↑↑↑17 WW(WB(C(WC))(WB))
Ξ₀BCKW(10) > fω+1(2↑↑6/2-1) WW(C(WB)(C(CC(WB))))
Ξ₀BCKW(15) > fω2^^18-1(2) WW(C(WB)(B(BC(CC(WB)))(C(BC))))

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] [−1]
BBf(3) 1 [3/2] [−11]
BBf(4) 1 [9/2] [−12]
BBf(5) 2 [3/2, 1/3] [−110−1]
BBf(6) 3 [9/2, 1/3] [−120−1]
BBf(7) 4 [27/2, 1/3] [−130−1]
BBf(8) 5 [81/2, 1/3] [−140−1]
BBf(9) 6 [243/2, 1/3] [−150−1]
BBf(10) 7 [729/2, 1/3] [−160−1]
BBf(11) 10 [27/2, 25/3, 1/5] [−1300−1200−1]
BBf(12) 13 [81/2, 25/3, 1/5] [−1400−1200−1]
BBf(13) 17 [81/2, 125/3, 1/5] [−1400−1300−1]
BBf(14) 21 [243/2, 125/3, 1/5] [−1500−1300−1]
BBf(15) 28 [1/45, 4/5, 3/2, 25/3] [0−2−120−1−1100−12]
BBf(16) 53 [1/45, 4/5, 3/2, 125/3] [0−2−120−1−1100−13]
BBf(17) 107 [5/6, 49/2, 3/5, 40/7] [−1−110−100201−10301−1]
BBf(18) 211 [5/6, 49/2, 3/5, 80/7] [−1−110−100201−10401−1]
BBf(19) 370 [5/6, 49/2, 3/5, 160/7] [−1−110−100201−10501−1]
BBf(20) 746 [7/15, 22/3, 6/77, 5/2, 9/5] [−1−1+1+1−1+1+1+1−1−1−1+1+2−1]
BBf(21) ≥ 31,957,632 [7/15, 4/3, 27/14, 5/2, 9/5] [0−1−112−100−130−1−101002−10]
BBf(22) >1.146×1062 [1/12, 9/10, 14/3, 11/2, 5/7, 3/11] [−2−1000−12−1001−1010−10001001−100100−1]
BBf(23) >4.393×10124 [10/3, 9/14, 5/4, 121/2, 7/5, 3/11] [1−1100−120−10−20100−1000200−1100100−1]
BBf(24) >9.000×1014448[18] [14/15, 1/6, 11/3, 441/2, 5/7, 15/11] [1−1−110−1−10000−1001−12020001−100110−1]

Cyclic Tree Rewriting System (CTBB(n))

Runtime Champions
CTBB(2) 5[19]
CTBB(3) > 38[20]
CTBB(4) ≥ 672[21]
CTBB(5) ≥ 2^2^2^2^182[22]
CTBB(6) > 2↑↑↑131[23]
CTBB(7) > 4↑↑↑↑(4↑↑↑3)[24] [25]

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))

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*

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[26] 00_11_03_21_04
peBBle(6) 17[27] 00_11_03_21_40_15
peBBle(7) 28[27] 00_11_03_21_10_33_61
peBBle(8) ≥ 63[28] 00_11_03_21_50_30_24_70
peBBle(9) ≥ 63577[29] 00_02_01_40_14_12_00_52_81

Wang Tiles (BBWT(n))

n BBWT(n) Champion Image
BBWT(1) 1 [1,0,0,0]
The definite champion for the largest valid square patch (1 by 1) made by a set of 1 Wang tile.
BBWT(2) 2[30] [1,0,2,0] [2,0,3,0]
The definite champion for the largest valid square patch (2 by 2) made by a set of 2 Wang tiles.
BBWT(3) 3[30] [1,0,2,0] [2,0,3,0] [3,0,4,0]
The definite champion for the largest valid square patch (3 by 3) made by a set of 3 Wang tiles.
BBWT(4) 4[31] [1,0,2,0] [2,0,3,0] [3,0,4,0] [4,0,5,0]
The definite champion for the largest valid square patch (4 by 4) made by a set of 4 Wang tiles.
BBWT(5) ≥ 6[32] [1,0,2,0] [2,0,3,0] [3,0,4,0] [4,6,1,5] [4,5,2,6]
The best known champion for the largest valid square patch (6 by 6) made by a set of 5 Wang tiles.
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]
The best known champion for the largest valid square patch (8 by 8) made by a set of 6 Wang tiles.
BBWT(7) ≥ 11 [1,0,2,0] [2,0,3,0] [3,0,4,0] [4,0,5,0] [5,7,1,6] [5,6,2,7] [4,6,3,7]
The best known champion for the largest valid square patch (11 by 11) made by a set of 7 Wang tiles.
BBWT(8) ≥ 14 [1,0,2,0] [2,0,3,0] [3,0,4,0] [4,0,5,0] [5,0,6,0] [6,8,1,7] [6,7,2,8] [5,7,3,8]
The best known champion for the largest valid square patch (14 by 14) made by a set of 8 Wang tiles.
BBWT(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]
The best known champion for the largest valid square patch (17 by 17) made by a set of 9 Wang tiles.
BBWT(10) ≥ 212

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 (BrBspace(n))

Domain Max Space Champions
BrBspace(1) 2 1RA1RZ
BrBspace(2) 4 1RA0GB_1GA1RZ (and 10 others)
BrBspace(3) ≥ 22 1RB1RZ_0GC0BB_1GA0RA
BrBspace(4) ≥ 384 1RB0BB_0RC1RZ_0GA1GD_1RC0RA
BrBspace(2,3) ≥ 20 1RA2GA0BB_1BA2BA1RZ1RA2GA0BB_2BA1RZ1BA
BrBspace(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(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) > 2.27×10248[33] 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[34] 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 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. ↑ https://github.com/jeff87654/busy-beaver/tree/main/BB8
  2. ↑ [1]
  3. ↑ https://github.com/jeff87654/busy-beaver/tree/main/BB10
  4. ↑ https://github.com/jeff87654/busy-beaver/tree/main/BB12
  5. ↑ https://github.com/jeff87654/busy-beaver/tree/main/BB13
  6. ↑ https://github.com/jeff87654/busy-beaver/tree/main/BB14
  7. ↑ https://discord.com/channels/960643023006490684/1026577255754903572/1274471360206344213
  8. ↑ https://discord.com/channels/960643023006490684/1026577255754903572/1328863966688182345
  9. ↑ 9.0 9.1 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.
  10. ↑ 10.0 10.1 S. Ligocki, "BB(2,6) > 10↑↑10↑↑10↑↑3". Blog post, 2023. Accessed 15 August 2025.
  11. ↑ Coq-BBB4
  12. ↑ Nick Drozd. "BBB(3,3) > 10↑↑6". Accessed 15 August 2025.
  13. ↑ 13.0 13.1 Nick Drozd. "A Tetrational Self-Cleaning Turing Machine". Blog post. Accessed 17 October 2026.
  14. ↑ 14.0 14.1 Nick Drozd, "Blanking Beavers". Accessed 15 August 2025.
  15. ↑ 15.0 15.1 15.2 Nick Drozd, "Latest Beeping Busy Beaver Results". Accessed 15 August 2025.
  16. ↑ https://discord.com/channels/960643023006490684/1369339127652159509/1374531970830307409
  17. ↑ Nick Drozd. Blanking Beavers
  18. ↑ https://discord.com/channels/960643023006490684/1438019511155691521/1547182811713380383
  19. ↑ https://discord.com/channels/960643023006490684/960643023530762341/1440163057463726233
  20. ↑ https://discord.com/channels/960643023006490684/1438694294042181742/1440193579006951517
  21. ↑ https://discord.com/channels/960643023006490684/1438694294042181742/1443246244142252043
  22. ↑ https://discord.com/channels/960643023006490684/1438694294042181742/1443298934217900063
  23. ↑ https://discord.com/channels/960643023006490684/1438694294042181742/1442847677883875479
  24. ↑ https://discord.com/channels/960643023006490684/1438694294042181742/1442950825545564281
  25. ↑ https://discord.com/channels/960643023006490684/1438694294042181742/1442819117735346217
  26. ↑ https://discord.com/channels/960643023006490684/1535695107448381591/1536023084975525999
  27. ↑ 27.0 27.1 https://discord.com/channels/960643023006490684/1535695107448381591/1536608195156381747
  28. ↑ https://discord.com/channels/960643023006490684/1535695107448381591/1537528571394793563
  29. ↑ https://discord.com/channels/960643023006490684/1535695107448381591/1539048792400203796
  30. ↑ 30.0 30.1 https://discord.com/channels/960643023006490684/1529586687926210691/1530032750579417198
  31. ↑ https://discord.com/channels/960643023006490684/1529586687926210691/1530697443820437645
  32. ↑ https://discord.com/channels/960643023006490684/1529586687926210691/1530722442761474218
  33. ↑ https://discord.com/channels/960643023006490684/1243312334907375676/1535305073259974728
  34. ↑ https://discord.com/channels/960643023006490684/1243312334907375676/1533503518906912778
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