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).
='''Original Busy Beaver Functions'''=
='''State-and-Symbol-Limited Busy Beaver functions'''=
==Maximum Shifts Function (BB)==
Busy Beaver functions where the programs size is limited by the amount of states and symbols.
'''2 Symbols:'''
=='''Original Busy Beaver Functions'''==
===Maximum Shifts Function ([[Busy Beaver Functions|S(n,m)]], also commonly called BB(n,m))===
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''2 Symbols:'''
!Runtime
!Runtime
!Champions
!Champions
|-
|-
|BB(1)
|BB(1)
|<math> 1 </math>
|1
|{{TM|1RZ---|halt}}
|{{TM|1RZ---|halt}}
|-
|-
|[[BB(2)]]
|[[BB(2)]]
|<math> 6 </math>
|6
|{{TM|1RB1LB_1LA1RZ|halt}} {{TM|1RB0LB_1LA1RZ|halt}} {{TM|1RB1RZ_1LB1LA|halt}} {{TM|1RB1RZ_0LB1LA|halt}} {{TM|0RB1RZ_1LA1RB|halt}}
|{{TM|1RB1LB_1LA1RZ|halt}} {{TM|1RB0LB_1LA1RZ|halt}} {{TM|1RB1RZ_1LB1LA|halt}} {{TM|1RB1RZ_0LB1LA|halt}} {{TM|0RB1RZ_1LA1RB|halt}}
|-
|-
|[[BB(3)]]
|[[BB(3)]]
|<math> 21 </math>
|21
|{{TM|1RB1RZ_1LB0RC_1LC1LA|halt}}
|{{TM|1RB1RZ_1LB0RC_1LC1LA|halt}}
|-
|-
|[[BB(4)]]
|[[BB(4)]]
|<math> 107 </math>
|107
|{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}}
|{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}}
|-
|-
|[[BB(5)]]
|[[BB(5)]]
|<math> 47\,176\,870 </math>
|47,176,870
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}
|-
|-
|[[BB(6)]]
|[[BB(6)]]
|<math> > 2\uparrow\uparrow 2\uparrow\uparrow 2\uparrow\uparrow 10 </math>
|> 10 ↑↑ 10 ↑↑ 10 ↑↑ 8.10237
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|-
|-
Line 42: Line 43:
|-
|-
|BB(9)
|BB(9)
|<math> > f_\omega(f_9(2)) </math>
|<math>> f_\omega(f_9(2))</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|-
|-
|BB(10)
|BB(10)
|<math> > f_\omega^2(25) </math>
|<math>> f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|-
|-
|BB(11)
|BB(11)
|<math> > f_\omega^2(2 \uparrow\uparrow 12) > f_\omega^2(f_3(9)) </math>
|<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}}
|{{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^4(2 \uparrow\uparrow\uparrow 4-3) > f_\omega^4(f_4(2))</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_1RZ1LA_1RF1LL|halt}}
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_1RZ1LA_1RF1LL|halt}}
|-
|-
|BB(14)
|BB(14)
|<math> > f_{\omega + 1}(65\,536) > g_{64} </math>
|<math>> f_{\omega + 1}(65\,536) > g_{64}</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ---|halt}}
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ---|halt}}
|-
|-
|BB(15)
|BB(15)
|<math> > f_{\omega + 1}(f_\omega(10^{57})) </math>
|<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}}
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_1RZ0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|-
|-
|BB(16)
|BB(16)
|<math> > f_{\omega + 1}^2(10^{10^{57}}) </math>
|<math>> f_{\omega + 1}^2(10^{10^{57}})</math>
|
|
|-
|-
|BB(18)
|BB(18)
|<math> > f_{\omega + 2}(f_{\omega + 1}^3(f_{\omega}^2(60))) </math>
|<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 + 2}^2(21)</math>
|
|
|-
|-
|BB(21)
|BB(21)
|<math> > f_{\omega^2}^2(4 \uparrow\uparrow 341) </math>
|<math>> f_{\omega^2}^2(4 \uparrow\uparrow 341)</math>
|
|
|-
|-
|BB(40)
|BB(40)
|<math> > f_{\omega^\omega}(75\,500) </math>
|<math>> f_{\omega^\omega}(75\,500)</math>
|
|
|-
|-
|BB(41)
|BB(41)
|<math> > f_{\omega^\omega}^4(32) </math>
|<math>> f_{\omega^\omega}^4(32)</math>
|
|
|-
|-
|BB(51)
|BB(51)
|<math> > f_{\varepsilon_0 + 1}(8) </math>
|<math>> f_{\varepsilon_0 + 1}(8)</math>
|
|
|}
|}
'''3 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''3 Symbols:'''
!Runtime
!Runtime
!Champions
!Champions
|-
|BB(1,3)
|1
|{{TM|1RZ------|halt}}
|-
|-
|[[BB(2,3)]]
|[[BB(2,3)]]
|<math> 38 </math>
|38
|{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
|{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
|-
|-
|[[BB(3,3)]]
|[[BB(3,3)]]
|<math> \geq 119\,112\,334\,170\,342\,541 > 10^{17} </math>
|<math>\geq 119\,112\,334\,170\,342\,541 > 10^{17}</math>
|{{TM|0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC|halt}}
|{{TM|0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC|halt}}
|-
|-
|[[BB(4,3)]]
|[[BB(4,3)]]
|<math> > 2 \uparrow\uparrow\uparrow (2^{2^{32}+1}-1)</math>
|<math>> 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow (7.92 \times 10^{28})</math>
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|}
|}
'''4 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''4 Symbols:'''
!Runtime
!Runtime
!Champions
!Champions
|-
|BB(1,4)
|1
|{{TM|1RZ---------|halt}}
|-
|-
|[[BB(2,4)]]
|[[BB(2,4)]]
|<math> 3\,932\,964 </math>
|3,932,964
|{{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
|{{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
|-
|-
|BB(3,4)
|BB(3,4)
|<math> > 2 \uparrow^{15} 5 </math>
|<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}}
|}
|}
'''5 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''5 Symbols:'''
!Runtime
!Runtime
!Champions
!Champions
|-
|BB(1,5)
|1
|{{TM|1RZ------------|halt}}
|-
|-
|[[BB(2,5)]]
|[[BB(2,5)]]
|<math> > 10^{10^{10^{3\,314\,360}}} </math>
|<math>> 10^{10^{10^{3\,314\,360}}}</math>
|{{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}}
|}
|}
'''6 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''6 Symbols:'''
!Runtime
!Runtime
!Champions
!Champions
|-
|BB(1,6)
|1
|{{TM|1RZ---------------|halt}}
|-
|-
|BB(2,6)
|BB(2,6)
|<math> > 10 \uparrow\uparrow 10 \uparrow\uparrow 10^{10^{115}} </math>
|<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}}
|}
|}
==Maximum Score Function (Σ)==
===Maximum Score Function ([[Busy Beaver Functions|Σ(n,m)]])===
'''2 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''2 Symbols:'''
!Score
!Score
!Champions
!Champions
|-
|-
|Σ(1)
|Σ(1)
|<math> 1 </math>
|1
|{{TM|1RZ---|halt}}
|{{TM|1RZ---|halt}}
|-
|-
|Σ(2)
|Σ(2)
|<math> 4 </math>
|4
|{{TM|1RB1LB_1LA1RZ|halt}}
|{{TM|1RB1LB_1LA1RZ|halt}}
|-
|-
|Σ(3)
|Σ(3)
|<math> 6 </math>
|6
|{{TM|1RB1RZ_0RC1RB_1LC1LA|halt}} {{TM|1RB1RC_1LC1RZ_1RA0LB|halt}} {{TM|1RB1LC_1LA1RB_1LB1RZ|halt}} {{TM|1RB1RA_1LC1RZ_1RA1LB|halt}} {{TM|1RB1LC_1RC1RZ_1LA0LB|halt}}
|{{TM|1RB1RZ_0RC1RB_1LC1LA|halt}} {{TM|1RB1RC_1LC1RZ_1RA0LB|halt}} {{TM|1RB1LC_1LA1RB_1LB1RZ|halt}} {{TM|1RB1RA_1LC1RZ_1RA1LB|halt}} {{TM|1RB1LC_1RC1RZ_1LA0LB|halt}}
|-
|-
|Σ(4)
|Σ(4)
|<math> 13 </math>
|13
|{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}} {{TM|1RB0RC_1LA1RA_1RZ1RD_1LD0LB|halt}}
|{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}} {{TM|1RB0RC_1LA1RA_1RZ1RD_1LD0LB|halt}}
|-
|-
|Σ(5)
|Σ(5)
|<math> 4098 </math>
|4098
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}} {{TM|1RB1RA_1LC1LB_1RA1LD_1RA1LE_1RZ0LC|halt}}
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}} {{TM|1RB1RA_1LC1LB_1RA1LD_1RA1LE_1RZ0LC|halt}}
|-
|-
|Σ(6)
|Σ(6)
|<math> > 2\uparrow\uparrow 2\uparrow\uparrow 2\uparrow\uparrow 10 </math>
|> 10 ↑↑ 10 ↑↑ 10 ↑↑ 8.10237
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|-
|Σ(7)
|<math>> 2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF|halt}}
|}
|}
'''3 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''3 Symbols:'''
!Score
!Score
!Champions
!Champions
|-
|Σ(1,3)
|1
|{{TM|1RZ------|halt}}
|-
|-
|Σ(2,3)
|Σ(2,3)
|<math> 9 </math>
|9
|{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
|{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
|-
|-
|Σ(3,3)
|Σ(3,3)
|<math> \geq 374\,676\,383 </math>
|≥ 374,676,383
|{{TM|0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC|halt}}
|{{TM|0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC|halt}}
|-
|-
|Σ(4,3)
|Σ(4,3)
|<math> > 2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
|<math>> 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow (7.92 \times 10^{28})</math>
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|}
|}
'''4 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''4 Symbols:'''
!Score
!Score
!Champions
!Champions
|-
|Σ(1,4)
|1
|{{TM|1RZ---------|halt}}
|-
|-
|Σ(2,4)
|Σ(2,4)
|<math> 2050 </math>
|2050
|{{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
|{{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
|-
|Σ(3,4)
|<math>\geq (2 \uparrow^{15} 5) + 14 </math><ref name=":0" />
|{{TM|1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|}
|}
'''5 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''5 Symbols:'''
!Score
!Score
!Champions
!Champions
|-
|Σ(1,5)
|1
|{{TM|1RZ------------|halt}}
|-
|-
|Σ(2,5)
|Σ(2,5)
|<math> > 10^{10^{10^{3\,314\,360}}} </math>
|<math>> 10^{10^{10^{3\,314\,360}}}</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}
|}
|}
='''Beeping Busy Beavers ([[BBB]])'''=
==Beeping Busy Beaver==
'''2 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''6 Symbols:'''
!Score
!Champions
|-
|Σ(1,6)
|1
|{{TM|1RZ---------------|halt}}
|-
|Σ(2,6)
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10^{10^{115}}</math><ref name=":1" />
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
|}
 
=='''Beeping Busy Beavers'''==
===Beeping Busy Beaver ([[BBB]](n,m))===
{| class="wikitable"
|+
!'''2 Symbols:'''
!Steps taken
!Steps taken
!Champions
!Champions
|-
|-
|BBB(1)
|BBB(1)
|<math> 1 </math>
|1
|
|
|-
|-
|BBB(2)
|BBB(2)
|<math> 6 </math>
|6
|
|{{TM|1RB1LB_1LB1LA}}
|-
|-
|BBB(3)
|BBB(3)
|<math> 55 </math>
|55
|
|{{TM|1RB0LB_1LA0RC_1LC1LA}}
|-
|-
|BBB(4)
|BBB(4)
|<math> \geq 32\,779\,478 </math>
|≥ 32,779,478
|
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|-
|-
|BBB(5)
|BBB(5)
|<math> \geq 10^{14006} </math>
|<math>\geq 10^{14006}</math>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA}}
|}
{| class="wikitable"
|+
!'''3 Symbols:'''
!Steps taken
!Champions
|-
|BBB(1,3)
|
|
|
|-
|BBB(2,3)
|59<ref name=":2">Nick Drozd. "[https://nickdrozd.github.io/2025/03/24/bbb-3-3.html BBB(3,3) > 10↑↑6]". Accessed 15 August 2025.</ref>
|{{TM|1RB2LB1LA_2LB2RA0RA}}
|-
|BBB(3,3)
|≥ 10 ↑↑ 6
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}
|}
|}
'''3 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''4 Symbols:'''
!Steps taken
!Steps taken
!Champions
!Champions
|-
|-
|BBB(2,3)
|BBB(1,4)
|
|
|
|-
|BBB(2,4)
|<math>\geq 205\,770\,076\,433\,044\,242\,247\,859 > 2\times 10^{23}</math><ref name=":3" />
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA}}
|}
===Beeping Booping Busy Beaver ([[Beeping Busy Beaver#Beeping Booping Busy Beavers|BBBB]](n,m))===
{| class="wikitable"
|+
!'''2 Symbols:'''
!Steps taken
!Champions
|-
|BBBB(1)
|2
|
|
|-
|-
|BBB(3,3)
|BBBB(2)
|<math> \geq 10 \uparrow\uparrow 6 </math>
|17
|
|
|}
|}
==Beeping Booping Busy Beaver==
 
There are currently no known/available Champions for this function.
=='''Maximum Consecutive Ones Function ([[Num]](n,m))'''==
='''Maximum Consecutive Ones Function ([[Num]])'''=
'''2 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''2 Symbols:'''
!Number of Ones
!Number of Ones
!Champions
!Champions
|-
|-
|num(1)
|num(1)
|<math> 1 </math>
|1
|{{TM|1RZ---|halt}}
|{{TM|1RZ---|halt}}
|-
|-
|num(2)
|num(2)
|<math> 4 </math>
|4
|{{TM|1RB1LB_1LA1LZ|halt}}
|{{TM|1RB1LB_1LA1LZ|halt}}
|-
|-
|num(3)
|num(3)
|<math> 6 </math>
|6
|{{TM|1RB1LC_1RC1LZ_1LA0LB|halt}}
|{{TM|1RB1LC_1RC1LZ_1LA0LB|halt}}
|-
|-
|num(4)
|num(4)
|<math> 12 </math>
|12
|{{TM|1RB0LA_1RC1LB_1LB1RD_1RZ0RA|halt}}
|{{TM|1RB0LA_1RC1LB_1LB1RD_1RZ0RA|halt}}
|-
|-
|num(5)
|num(5)
|<math> 165 </math>
|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}}
|}
|}
=='''Maximum Space Function ([[Maximum Space Function|BB<sub>space</sub>]](n,m))'''==
{| class="wikitable"
|+
!'''2 Symbols:'''
!Cells visited
!Champions
|-
|BB<sub>space</sub>(1,2)
|1
|{{TM|1RZ---|halt}}
|-
|BB<sub>space</sub>(2,2)
|4
|{{TM|1RB1LB_1LA1RZ|halt}} and {{TM|1RB0LB_1LA1RZ|halt}}
|-
|BB<sub>space</sub>(3,2)
|7
|{{TM|1RB1RC_1LC1RZ_1RA0LB|halt}} and {{TM|1RB0RC_1LC1RZ_1RA0LB|halt}}
|-
|BB<sub>space</sub>(4,2)
|16
|{{TM|1RB0RA_1LC0RD_0LD0LB_1RA1RZ|halt}}
|-
|BB<sub>space</sub>(5,2)
|12289
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}
|}
{| class="wikitable"
|+
!'''3 Symbols:'''
!Cells visited
!Champions
|-
|BB<sub>space</sub>(1,3)
|
|
|-
|BB<sub>space</sub>(2,3)
|9
|{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
|-
|BB<sub>space</sub>(2,4)
|2050
|{{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
|}
=='''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 ([[Reversible Turing Machine|BB<sub>rev</sub>]](n,m))===
{| class="wikitable"
|+
!'''2 Symbols:'''
!Steps
!Champions
|-
|BB<sub>rev</sub>(1)
|
|
|-
|BB<sub>rev</sub>(2)
|6
|{{TM|0RB1RZ_1LA1RB|halt}}
|-
|BB<sub>rev</sub>(3)
|17
|{{TM|0RB1RZ_0LC1RA_1RB1LC|halt}}
|-
|BB<sub>rev</sub>(4)
|48
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC1RZ|halt}}
|-
|BB<sub>rev</sub>(5)
|388
|{{TM|1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA|halt}}
|-
|BB<sub>rev</sub>(6)
|≥ 537,556
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA|halt}}
|-
|BB<sub>rev</sub>(7)
|<math>> 10^{19}</math>
|{{TM|1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB1RZ|halt}}
|}
===Maximum Score Function (Σ<sub>rev</sub>(n,m))===
{| class="wikitable"
|+
!'''2 Symbols:'''
!Score
!Champions
|-
|Σ<sub>rev</sub>(1)
|
|
|-
|Σ<sub>rev</sub>(2)
|≥ 2
|{{TM|0RB1RZ_1LA1RB|halt}}
|-
|Σ<sub>rev</sub>(3)
|≥ 4
|{{TM|0RB1RZ_0LC1RA_1RB1LC|halt}}
|-
|Σ<sub>rev</sub>(4)
|≥ 6
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC1RZ|halt}}
|-
|Σ<sub>rev</sub>(5)
|≥ 16
|{{TM|1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA|halt}}
|-
|Σ<sub>rev</sub>(6)
|≥ 1161
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA|halt}}
|}
=='''Blanking Busy Beaver ([[Busy Beaver Functions#Other Busy Beaver functions|BLB(n,m)]])'''==
{| class="wikitable"
|+
!'''2 Symbols:'''
!Steps
!Champions
|-
|BLB(1)
|nonexistent
|nonexistent
|-
|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>
|{{TM|1RB0RA_1LB1LA}}
|-
|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>
|{{TM|1RB1LB_1LA1LC_1RC0LC}}
|-
|BLB(4)
|≥ 32,779,477
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|}
{| class="wikitable"
|+
!'''3 Symbols:'''
!Steps
!Champions
|-
|BLB(1,3)
|nonexistent
|nonexistent
|-
|BLB(2,3)
|≥ 77<ref name=":4" />
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|}
{| class="wikitable"
|+
!'''4 Symbols:'''
!Steps
!Champions
|-
|BLB(1,4)
|nonexistent
|nonexistent
|-
|BLB(2,4)
|≥ 1,367,361,263,049<ref name=":4" />
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}
=='''Lazy Beaver'''==
===Shifts Function ([[Lazy Beaver|LB]](n,m))===
{| class="wikitable"
|+
!
!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))===
{| class="wikitable"
|+
!'''2 Symbols:'''
!Preperiod
!Champions
|-
|BBS(1,2)
|0
|{{TM|1RA---}}
|-
|BBS(2,2)
|≥ 9
|{{TM|1RB0LB_1LA0RB}} proven winner?
|-
|BBS(3,2)
|101
|{{TM|1RB1LB_0RC0LA_1LC0LA}}
|-
|BBS(4,2)
|≥ 119,120,230,102
|{{TM|1RB1LC_0LA1RD_0RB0LC_1LA0RD}}
|}
{| class="wikitable"
|+
!'''3 Symbols:'''
!Preperiod
!Champions
|-
|BBS(1,3)
|0
|{{TM|1RA------}}
|}
{| class="wikitable"
|+
!'''4 Symbols:'''
!Preperiod
!Champions
|-
|BBS(1,4)
|0
|{{TM|1RA---------}}
|-
|BBS(2,4)
|≥ 293,225,660,896
|{{TM|1RB2LA0RA3LA_1LA1LB3RB1RA}}
|}
===Busy Periodic Beaver ([[BBP]](n,m))===
{| class="wikitable"
|+
!'''2 Symbols:'''
!Period
!Champions
|-
|BBP(1,2)
|1
|{{TM|1RA---}}
|-
|BBP(2,2)
|≥ 9
|{{TM|1RB0RB_1LB1RA}} proven winner?
|-
|BBP(3,2)
|92
|{{TM|1RB0LA_0RC1LA_1LC0RB}}
|-
|BBP(4,2)
|≥ 212,081,736
|{{TM|1RB0LA_0RC1RD_1LD0RB_1LA1RB}}
|}
{| class="wikitable"
|+
!'''3 Symbols:'''
!Period
!Champions
|-
|BBP(1,3)
|1
|{{TM|1RA------}}
|}
{| class="wikitable"
|+
!'''4 Symbols:'''
!Period
!Champions
|-
|BBP(1,4)
|1
|{{TM|1RA---------}}
|-
|BBP(2,4)
|≥ 33,209,131
|{{TM|1RB0RA3LB1RB_2LA0LB1RA2RB}}
|}
='''Instruction-Limited Busy Beaver'''=
='''Instruction-Limited Busy Beaver'''=
==Maximum amount of steps ([[BBi]])==
=='''Instruction-Limited Classical Busy Beaver Functions'''==
===Instruction-Limited Maximum Shifts Function ([[BBi]](n))===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 309: Line 700:
|-
|-
|BBi(1)
|BBi(1)
|<math> 1 </math>
|1
|{{TM|0RH|halt}} {{TM|1RH---|halt}}
|{{TM|0RH|halt}} {{TM|1RH---|halt}}
|-
|-
|BBi(2)
|BBi(2)
|<math> 3 </math>
|3
|{{TM|0RB---_1LA---|halt}}
|{{TM|0RB---_1LA---|halt}}
|-
|-
|BBi(3)
|BBi(3)
|<math> 5 </math>
|5
|{{TM|1RB1LB_1LA---|halt}}
|{{TM|1RB1LB_1LA---|halt}}
|-
|-
|BBi(4)
|BBi(4)
|<math> 16 </math>
|16
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|-
|BBi(5)
|BBi(5)
|<math> 37 </math>
|37
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|-
|BBi(6)
|BBi(6)
|<math> 123 </math>
|123
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|-
|BBi(7)
|BBi(7)
|<math> 3\,932\,963 </math>
|3,932,963
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}} {{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}} {{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|-
|-
|BBi(8)
|BBi(8)
|<math> >6.889 \times 10^{1565} </math>
|<math>>6.889 \times 10^{1565}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|BBi(9)
|<math>>10^{10^{10^{3\,314\,360}}}</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|}
|}
==Maximum Score (Σi)==
===Instruction-Limited Maximum Score Function ([[Σi]](n))===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 348: Line 743:
|-
|-
|Σi(1)
|Σi(1)
|<math> 1 </math>
|1
|{{TM|1RH---|halt}}
|{{TM|1RH---|halt}}
|-
|-
|Σi(2)
|Σi(2)
|<math> 2 </math>
|2
|{{TM|1RB---_1LA---|halt}}
|{{TM|1RB---_1LA---|halt}}
|-
|-
|Σi(3)
|Σi(3)
|<math> 4 </math>
|4
|{{TM|1RB1LB_1LA---|halt}}
|{{TM|1RB1LB_1LA---|halt}}
|-
|-
|Σi(4)
|Σi(4)
|<math> 5 </math>
|5
|{{TM|1RB0LB---_1LA2RA---|halt}}
|{{TM|1RB0LB---_1LA2RA---|halt}}
|-
|-
|Σi(5)
|Σi(5)
|<math> 9 </math>
|9
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|-
|Σi(6)
|Σi(6)
|<math> 14 </math>
|14
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|-
|Σi(7)
|Σi(7)
|<math> 2050 </math>
|2050
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}} {{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|{{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}} {{TM|1RB2LA1RA_1LC1LA2RB_---1LA---|halt}}
|-
|-
|Σi(8)
|Σi(8)
|<math> >1.355 \times 10^{783} </math>
|<math>>1.355 \times 10^{783}</math>
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|{{TM|1RB1LA------_1RC3LB1RB---_2LA2LC---0LC|halt}}
|-
|Σi(9)
|<math>>10^{10^{10^{3\,314\,360}}}</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}
|}
|}
='''Reversible Turing Machines'''=
 
==Maximum Shifts Function ([[Reversible Turing Machine|BB<sub>rev</sub>]])==
=='''Instruction-Limited Blanking Busy Beaver ([[Instruction-Limited Busy Beaver#Instruction-Limited Busy Beaver Variants|BLBi(n)]])'''==
'''2 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 388: Line 786:
!Champions
!Champions
|-
|-
|BB<sub>rev</sub>(1)
|BLBi(1)
|nonexistent
|nonexistent
|-
|BLBi(2)
|nonexistent
|nonexistent
|-
|BLBi(3)
|4
|
|-
|BLBi(4)
|12
|
|-
|BLBi(5)
|30
|
|-
|BLBi(6)
|77
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLBi(7)
|808
|
|-
|BLBi(8)
|≥ 1,367,361,263,049
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}
 
=='''Instruction-Limited Greedy Busy Beaver ([[Instruction-Limited Busy Beaver#Instruction-Limited Busy Beaver Variants|gBBi(n)]])'''==
{| class="wikitable"
|+
!
!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
|
|
|-
|-
|BB<sub>rev</sub>(2)
|gBBi(14)
|<math> 6 </math>
|≥ 9,874,580
|{{TM|0RB1RZ_1LA1RB|halt}}
|{{TM|0RB6RB1LB---3LA1RB7RB2LB_1LA2RB3LA4RB5RB1LB5LA---|halt}}
|}
='''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 <math> 20 \geq n </math> BBλ(n) = n.
{| class="wikitable"
|+
!
!BBλ(n)
!Champions
|-
|BBλ(21)
|22
|<code>\(\1 1) (1 (\2))</code>
|-
|BBλ(22)
|24
|<code>\(\1 1) (1 (\\1))\(\1 1 1) (1 1)</code>
|-
|BBλ(23)
|26
|<code>\(\1 1) (1 (\\2))</code>
|-
|BBλ(24)
|30
|<code>\(\1 1 1) (1 (\1))</code>
|-
|BBλ(25)
|42
|<code>\(\1 1) (\1 (2 1))</code>
|-
|BBλ(26)
|52
|<code>(\1 1) (\\2 (1 2))</code>
|-
|BBλ(27)
|44
|<code>\\(\1 1) (\1 (2 1))</code>
|-
|BBλ(28)
|58
|<code>\(\1 1) (\1 (2 (\2))))</code>
|-
|BBλ(29)
|223
|<code>\(\1 1) (\1 (1 (2 1)))</code>
|-
|BBλ(30)
|160
|<code>(\1 1 1) (\\2 (1 2))</code> and <code>(\1 (1 1)) (\\2 (1 2))</code>
|-
|BBλ(31)
|267
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|-
|BBλ(32)
|298
|<code>\(\1 1) (\1 (1 (2 (\2))))</code>
|-
|BBλ(33)
|1812
|<code>\(\1 1) (\1 (1 (1 (2 1))))</code>
|-
|BBλ(34)
|327,686
|<code>(\1 1 1 1) (\\2 (2 1))</code> and <code>(\1 (1 1) 1) (\\2 (2 1))</code>
|-
|BBλ(35)
|<math>5 \times 3^{3^{3}} +6 = 38\,127\,987\,424\,941 > 3.8 \times 10^{13}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBλ(36)
|<math>5 \times 2^{2^{2^{3}}} +6 > 5.7 \times 10^{77}</math>
|<code>(\1 1) (\1 (1 (\\2 (2 1))))</code>
|-
|BBλ(37)
|<math>BB \lambda(35) +2 =5 \times 3^{3^{3}} +8 = 38\,127\,987\,424\,943 > 3.8 \times 10^{13}</math>
|<code>\(\1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBλ(38)
|<math>\geq 5 \times 2^{2^{2^{2^{2}}}} +6 > 10^{19729}</math>
|<code>(\1 1 1 1 1) (\\2 (2 1))</code> and <code>(\1 (1 1) 1 1) (\\2 (2 1))</code>
|-
|BBλ(39)
|<math>\geq 5 \times 3^{3^{3^{3}}} +6 > 10^{3\,638\,334\,640\,024}</math>
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|-
|BBλ(40)
|<math>> (2 \uparrow\uparrow)^{15}33 > 10 \uparrow\uparrow\uparrow 16</math>
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|BBλ(41)
|<math>\geq 5 \times 3^{3^{85}} +6 > 10^{1.7 \times 10^{40}}</math>
|<code>(\1 (\1 1) 1) (\\2 (2 (2 1)))</code>
|-
|BBλ(42)
|<math>\geq BB \lambda(40) + 2 > (2 \uparrow\uparrow)^{15}33 > 10 \uparrow\uparrow\uparrow 16</math>
|<code>\(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|BBλ(43)
|<math>> 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|-
|BBλ(44)
|<math>> 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
|<code>(\1 1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|BBλ(45)
|<math>\geq BB \lambda(43) + 2 > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
|<code>\(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|-
|-
|BB<sub>rev</sub>(3)
|BBλ(46)
|<math> 17 </math>
|<math>\geq BB \lambda(44) + 2 > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
|{{TM|0RB1RZ_0LC1RA_1RB1LC|halt}}
|<code>\(\1 1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|-
|BB<sub>rev</sub>(4)
|BBλ(47)
|<math> 48 </math>
|
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC1RZ|halt}}
|
|-
|-
|BB<sub>rev</sub>(5)
|BBλ(48)
|<math> 388 </math>
|<math>> 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
|{{TM|1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA|halt}}
|<code>(\1 1 1 1 1) (\1 (\\2 (2 1)) 1)</code>
|-
|-
|BB<sub>rev</sub>(6)
|BBλ(49)
|<math> \geq 537\,556 </math>
|<math>> f_{\omega+1}(\frac{2 \uparrow\uparrow 6}{2}) > \text{Graham's Number}</math>
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA|halt}}
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|-
|-
|BB<sub>rev</sub>(7)
|BBλ(1850)
|<math> >10^{19} </math>
|<math>> \text{Loader's Number}</math>
|{{TM|1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB1RZ|halt}}
|<code>Too large for this list</code>
|}
|}
==Maximum Score Function (Σ<sub>rev</sub>)==
===Oracle Busy Beaver for Lambda Calculus ([[Busy Beaver for lambda calculus#Oracle Busy Beaver|BBλ<sub>1</sub>(n)]])===
'''2 Symbols:'''
Note that <math>f(n) = 6 + 5 \times BB \lambda(n)</math>.
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!
!Score
!BBλ<sub>1</sub>(n)
!Champions
!Champions
|-
|-
|Σ<sub>rev</sub>(1)
|BBλ<sub>1</sub>(1)
|0
|
|
|-
|BBλ<sub>1</sub>(2)
|1
|<code>1</code>
|-
|BBλ<sub>1</sub>(3)
|0
|
|
|-
|-
|Σ<sub>rev</sub>(2)
|BBλ<sub>1</sub>(4)
|<math> \geq 2 </math>
|4
|{{TM|0RB1RZ_1LA1RB|halt}}
|<code>\1</code>
|-
|BBλ<sub>1</sub>(5)
|5
|<code>\2</code>
|-
|BBλ<sub>1</sub>(6)
|6
|<code>\\1</code>
|-
|BBλ<sub>1</sub>(7)
|7
|<code>\\2</code>
|-
|BBλ<sub>1</sub>(8)
|26
|<code>1 (\1)</code>
|-
|BBλ<sub>1</sub>(9)
|9
|<code>\\2</code>
|-
|BBλ<sub>1</sub>(10)
|36
|<code>1 (\\1)</code>
|-
|BBλ<sub>1</sub>(11)
|41
|<code>1 (\\2)</code>
|-
|BBλ<sub>1</sub>(12)
|266
|<code>1 (1 (\1))</code>
|-
|BBλ<sub>1</sub>(13)
|51
|<code>1 (\\2)</code>
|-
|BBλ<sub>1</sub>(14)
|<math>f(36) = 25 \times 2^{2^{2^{3}}}+36 > 2.85 \times 10^{78}</math>
|<code>1 (1 (\\1))</code>
|-
|BBλ<sub>1</sub>(15)
|<math>f(41) \geq 25 \times 3^{3^{85}}+36 > 10^{1.7 \times 10^{40}}</math>
|<code>1 (1 (\\2))</code>
|-
|BBλ<sub>1</sub>(16)
|<math>f(266)</math>
|<code>1 (1 (1 (\1)))</code>
|-
|BBλ<sub>1</sub>(17)
|<math>f(51)</math>
|<code>1 (1 (\\\2))</code>
|-
|BBλ<sub>1</sub>(18)
|<math>f^{4}(4) = f(f(266))</math>
|<code>1 (\1) 1 (\1)</code>
|-
|BBλ<sub>1</sub>(19)
|<math>f^{3}(7) = f(f(41))</math>
|<code>1 (1 (1 (\\2)))</code>
|-
|BBλ<sub>1</sub>(20)
|<math>f^{6}(4) = f^{4}(266)</math>
|<code>1 (\\1) 1 (\1)</code>
|-
|BBλ<sub>1</sub>(21)
|<math>f^{7}(4) = f^{5}(266)</math>
|<code>1 (\\2) 1 (\1)</code>
|-
|BBλ<sub>1</sub>(22)
|<math>f^{52}(4) = f^{50}(266)</math>
|<code>1 (1(\1)) 1(\1)</code>
|-
|BBλ<sub>1</sub>(28)
|<math>\geq f^{BB \lambda(f^{3}(4))}(4)</math>
|<code>1 (\1) 1 (\1) 1 (\1)</code>
|-
|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>
|<code>1(\1)(\1 2 1)(\1)</code>
|}
 
='''Fractran ([[Fractran|BBf]](n))'''=
{| class="wikitable"
|+
!n
!BBf(n)
!Example Champion
!Vector Representation
|-
| 2 || 1 || <code>[1/2]</code>
|<math>\begin{bmatrix}
  -1
\end{bmatrix}</math>
|-
| 3 || 1 || <code>[3/2]</code>
|<math>\begin{bmatrix}
  -1 & 1
\end{bmatrix}</math>
|-
| 4 || 1 || <code>[9/2]</code>
|<math>\begin{bmatrix}
  -1 & 2
\end{bmatrix}</math>
|-
| 5 || 2 || <code>[3/2, 1/3]</code>
|<math>\begin{bmatrix}
  -1 &  1 \\
  0 & -1
\end{bmatrix}</math>
|-
| 6 || 3 || <code>[9/2, 1/3]</code>
|<math>\begin{bmatrix}
  -1 &  2 \\
  0 & -1
\end{bmatrix}</math>
|-
| 7 || 4 || <code>[27/2, 1/3]</code>
|<math>\begin{bmatrix}
  -1 &  3 \\
  0 & -1
\end{bmatrix}</math>
|-
| 8 || 5 || <code>[81/2, 1/3]</code>
|<math>\begin{bmatrix}
  -1 &  4 \\
  0 & -1
\end{bmatrix}</math>
|-
| 9 || 6 || <code>[243/2, 1/3]</code>
|<math>\begin{bmatrix}
  -1 &  5 \\
  0 & -1
\end{bmatrix}</math>
|-
| 10 || 7 || <code>[729/2, 1/3]</code>
|<math>\begin{bmatrix}
  -1 &  6 \\
  0 & -1
\end{bmatrix}</math>
|-
| 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>
|-
| 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>
|-
| 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>
|-
| 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>
|-
| 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>
|-
| 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>
|-
| 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>
|-
| 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>
|-
|-
|Σ<sub>rev</sub>(3)
| 19 || 370 || <code>[5/6, 49/2, 3/5, 160/7]</code>
|<math> \geq 4 </math>
|<math>\begin{bmatrix}
|{{TM|0RB1RZ_0LC1RA_1RB1LC|halt}}
  -1 & -1 &  1 &  0 \\
  -1 &  0 &  0 &  2 \\
    0 &  1 & -1 &  0 \\
    5 &  0 &  1 & -1
\end{bmatrix}</math>
|-
|-
|Σ<sub>rev</sub>(4)
|20
|<math> \geq 6 </math>
|≥ 746
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC1RZ|halt}}
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|<math>\begin{bmatrix}
      &    -1 &    -1 &    +1 &      \\
  +1 &    -1 &      &      &    +1 \\
  +1 &    +1 &      &    -1 &    -1 \\
  -1 &      &    +1 &      &      \\
      &    +2 &    -1 &      &
\end{bmatrix}</math>
|-
|-
|Σ<sub>rev</sub>(5)
|21
|<math> \geq 16 </math>
|≥  31,957,632
|{{TM|1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA|halt}}
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|
|}
 
='''Doodle Function ([[Doodle function|doodle(c,n)]])'''=
doodle(1,n) = 1 and doodle(2,n) = n. Also note that doodle(c) = doodle(c,2).
{| class="wikitable"
|+
!'''2 Symbols:'''
!Runtime
!Champions
|-
|-
|Σ<sub>rev</sub>(6)
|doodle(3,2)
|<math> \geq 1161 </math>
|≥ 487
|{{TM|1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA|halt}}
|
|}
|}
='''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(2,2) = 2
<pre>
####
#S-#
#.F#
####
</pre>
Bug(3,3) = 8
<pre>
#####
#S..#
#.#.#
#.#F#
#####
</pre>
Bug(4,4) = 20
<pre>
######
#S...#
#..###
#....#
#..#F#
######
</pre>
Bug(5,5) = 42
<pre>
#######
#S...##
#.....#
#.....#
#.#.###
#.#..F#
#######
</pre>
Bug(6,6) = 96
<pre>
########
#S.....#
#.###.##
#.#...##
#.#....#
#..#.###
#..#..F#
########
</pre>
Bug(7,7) = 218
<pre>
#########
#S.###..#
#......##
#.#.##..#
#..#...##
#..#....#
#...#.###
#..#-..F#
#########
</pre>
Bug(8,8) = 506
<pre>
##########
#S.#.....#
#.#..##.##
#.#.##..##
#....##..#
#.#.#...##
#..##....#
#....#.###
#....#..F#
##########
</pre>
=References=

Latest revision as of 20:33, 17 November 2025

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

State-and-Symbol-Limited Busy Beaver functions

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

Original Busy Beaver Functions

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

2 Symbols: Runtime Champions
BB(1) 1 1RZ--- (bbch)
BB(2) 6 1RB1LB_1LA1RZ (bbch) 1RB0LB_1LA1RZ (bbch) 1RB1RZ_1LB1LA (bbch) 1RB1RZ_0LB1LA (bbch) 0RB1RZ_1LA1RB (bbch)
BB(3) 21 1RB1RZ_1LB0RC_1LC1LA (bbch)
BB(4) 107 1RB1LB_1LA0LC_1RZ1LD_1RD0RA (bbch)
BB(5) 47,176,870 1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA (bbch)
BB(6) > 10 ↑↑ 10 ↑↑ 10 ↑↑ 8.10237 1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE (bbch)
BB(7) >2↑112↑113 1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF (bbch)
BB(8)
BB(9) >fω(f9(2)) 1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH (bbch)
BB(10) >fω2(25) 1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ (bbch)
BB(11) >fω2(2↑↑12)>fω2(f3(9)) 1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RZ0LI_0LD1LE (bbch)
BB(12) >fω4(2↑↑↑4−3)>fω4(f4(2)) 0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_1RZ1LA_1RF1LL (bbch)
BB(14) >fω+1(65536)>g64 1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ--- (bbch)
BB(15) >fω+1(fω(1057)) 0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_1RZ0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN (bbch)
BB(16) >fω+12(101057)
BB(18) >fω+2(fω+13(fω2(60)))
BB(20) >fω+22(21)
BB(21) >fω22(4↑↑341)
BB(40) >fωω(75500)
BB(41) >fωω4(32)
BB(51) >fε0+1(8)
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)
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[1] 1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC (bbch)
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)
6 Symbols: Runtime Champions
BB(1,6) 1 1RZ--------------- (bbch)
BB(2,6) >10↑↑10↑↑1010115[2] 1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA (bbch)

Maximum Score Function (Σ(n,m))

2 Symbols: Score Champions
Σ(1) 1 1RZ--- (bbch)
Σ(2) 4 1RB1LB_1LA1RZ (bbch)
Σ(3) 6 1RB1RZ_0RC1RB_1LC1LA (bbch) 1RB1RC_1LC1RZ_1RA0LB (bbch) 1RB1LC_1LA1RB_1LB1RZ (bbch) 1RB1RA_1LC1RZ_1RA1LB (bbch) 1RB1LC_1RC1RZ_1LA0LB (bbch)
Σ(4) 13 1RB1LB_1LA0LC_1RZ1LD_1RD0RA (bbch) 1RB0RC_1LA1RA_1RZ1RD_1LD0LB (bbch)
Σ(5) 4098 1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA (bbch) 1RB1RA_1LC1LB_1RA1LD_1RA1LE_1RZ0LC (bbch)
Σ(6) > 10 ↑↑ 10 ↑↑ 10 ↑↑ 8.10237 1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE (bbch)
Σ(7) >2↑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[1] 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[2] 1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA (bbch)

Beeping Busy Beavers

Beeping Busy Beaver (BBB(n,m))

2 Symbols: Steps taken Champions
BBB(1) 1
BBB(2) 6 1RB1LB_1LB1LA (bbch)
BBB(3) 55 1RB0LB_1LA0RC_1LC1LA (bbch)
BBB(4) ≥ 32,779,478 1RB1LD_1RC1RB_1LC1LA_0RC0RD (bbch)
BBB(5) ≥1014006 1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA (bbch)
3 Symbols: Steps taken Champions
BBB(1,3)
BBB(2,3) 59[3] 1RB2LB1LA_2LB2RA0RA (bbch)
BBB(3,3) ≥ 10 ↑↑ 6 1RB0LB2LA_1LA0RC0LB_2RC2RB0LC (bbch)
4 Symbols: Steps taken Champions
BBB(1,4)
BBB(2,4) ≥205770076433044242247859>2×1023[4] 1RB2LA1RA1LB_0LB2RB3RB1LA (bbch)

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

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

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

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

Maximum Space Function (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)
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[4] 1RB0RA_1LB1LA (bbch)
BLB(3) ≥ 34[5] 1RB1LB_1LA1LC_1RC0LC (bbch)
BLB(4) ≥ 32,779,477 1RB1LD_1RC1RB_1LC1LA_0RC0RD (bbch)
3 Symbols: Steps Champions
BLB(1,3) nonexistent nonexistent
BLB(2,3) ≥ 77[5] 1RB2LA0RB_1LA0LB1RA (bbch)
4 Symbols: Steps Champions
BLB(1,4) nonexistent nonexistent
BLB(2,4) ≥ 1,367,361,263,049[5] 1RB2RA1RA2RB_2LB3LA0RB0RA (bbch)

Lazy Beaver

Shifts Function (LB(n,m))

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

Period-oriented Busy Beavers

Busy Preperiodic Beaver (BBS(n,m))

2 Symbols: Preperiod Champions
BBS(1,2) 0 1RA--- (bbch)
BBS(2,2) ≥ 9 1RB0LB_1LA0RB (bbch) proven winner?
BBS(3,2) 101 1RB1LB_0RC0LA_1LC0LA (bbch)
BBS(4,2) ≥ 119,120,230,102 1RB1LC_0LA1RD_0RB0LC_1LA0RD (bbch)
3 Symbols: Preperiod Champions
BBS(1,3) 0 1RA------ (bbch)
4 Symbols: Preperiod Champions
BBS(1,4) 0 1RA--------- (bbch)
BBS(2,4) ≥ 293,225,660,896 1RB2LA0RA3LA_1LA1LB3RB1RA (bbch)

Busy Periodic Beaver (BBP(n,m))

2 Symbols: Period Champions
BBP(1,2) 1 1RA--- (bbch)
BBP(2,2) ≥ 9 1RB0RB_1LB1RA (bbch) proven winner?
BBP(3,2) 92 1RB0LA_0RC1LA_1LC0RB (bbch)
BBP(4,2) ≥ 212,081,736 1RB0LA_0RC1RD_1LD0RB_1LA1RB (bbch)
3 Symbols: Period Champions
BBP(1,3) 1 1RA------ (bbch)
4 Symbols: Period Champions
BBP(1,4) 1 1RA--------- (bbch)
BBP(2,4) ≥ 33,209,131 1RB0RA3LB1RB_2LA0LB1RA2RB (bbch)

Instruction-Limited Busy Beaver

Instruction-Limited Classical Busy Beaver Functions

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

Steps Champions
BBi(1) 1 0RH (bbch) 1RH--- (bbch)
BBi(2) 3 0RB---_1LA--- (bbch)
BBi(3) 5 1RB1LB_1LA--- (bbch)
BBi(4) 16 1RB---_0RC---_1LC0LA (bbch)
BBi(5) 37 1RB2LB---_2LA2RB1LB (bbch)
BBi(6) 123 1RB3LA1RA0LA_2LA------3RA (bbch)
BBi(7) 3,932,963 1RB2LA1RA1RA_1LB1LA3RB--- (bbch) 1RB2LA1RA_1LC1LA2RB_---1LA--- (bbch)
BBi(8) >6.889×101565 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
BLBi(4) 12
BLBi(5) 30
BLBi(6) 77 1RB2LA0RB_1LA0LB1RA (bbch)
BLBi(7) 808
BLBi(8) ≥ 1,367,361,263,049 1RB2RA1RA2RB_2LB3LA0RB0RA (bbch)

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

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

Program-Limited Busy Beaver

Busy Beaver for Lambda Calculus

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

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

BBλ(n) Champions
BBλ(21) 22 \(\1 1) (1 (\2))
BBλ(22) 24 \(\1 1) (1 (\\1))\(\1 1 1) (1 1)
BBλ(23) 26 \(\1 1) (1 (\\2))
BBλ(24) 30 \(\1 1 1) (1 (\1))
BBλ(25) 42 \(\1 1) (\1 (2 1))
BBλ(26) 52 (\1 1) (\\2 (1 2))
BBλ(27) 44 \\(\1 1) (\1 (2 1))
BBλ(28) 58 \(\1 1) (\1 (2 (\2))))
BBλ(29) 223 \(\1 1) (\1 (1 (2 1)))
BBλ(30) 160 (\1 1 1) (\\2 (1 2)) and (\1 (1 1)) (\\2 (1 2))
BBλ(31) 267 (\1 1) (\\2 (2 (1 2)))
BBλ(32) 298 \(\1 1) (\1 (1 (2 (\2))))
BBλ(33) 1812 \(\1 1) (\1 (1 (1 (2 1))))
BBλ(34) 327,686 (\1 1 1 1) (\\2 (2 1)) and (\1 (1 1) 1) (\\2 (2 1))
BBλ(35) 5×333+6=38127987424941>3.8×1013 (\1 1 1) (\\2 (2 (2 1)))
BBλ(36) 5×2223+6>5.7×1077 (\1 1) (\1 (1 (\\2 (2 1))))
BBλ(37) BBλ(35)+2=5×333+8=38127987424943>3.8×1013 \(\1 1 1) (\\2 (2 (2 1)))
BBλ(38) ≥5×22222+6>1019729 (\1 1 1 1 1) (\\2 (2 1)) and (\1 (1 1) 1 1) (\\2 (2 1))
BBλ(39) ≥5×3333+6>103638334640024 (\1 1 1 1) (\\2 (2 (2 1)))
BBλ(40) >(2↑↑)1533>10↑↑↑16 (\1 1 1) (\1 (\\2 (2 1)) 1)
BBλ(41) ≥5×3385+6>101.7×1040 (\1 (\1 1) 1) (\\2 (2 (2 1)))
BBλ(42) ≥BBλ(40)+2>(2↑↑)1533>10↑↑↑16 \(\1 1 1) (\1 (\\2 (2 1)) 1)
BBλ(43) >2↑↑↑2↑↑↑2↑↑8 (\1 1) (\1 (\1 (\\2 (2 1)) 2))
BBλ(44) >10↑↑↑10↑↑↑16 (\1 1 1 1) (\1 (\\2 (2 1)) 1)
BBλ(45) ≥BBλ(43)+2>2↑↑↑2↑↑↑2↑↑8 \(\1 1) (\1 (\1 (\\2 (2 1)) 2))
BBλ(46) ≥BBλ(44)+2>10↑↑↑10↑↑↑16 \(\1 1 1 1) (\1 (\\2 (2 1)) 1)
BBλ(47)
BBλ(48) >10↑↑↑10↑↑↑10↑↑↑16 (\1 1 1 1 1) (\1 (\\2 (2 1)) 1)
BBλ(49) >fω+1(2↑↑62)>Graham's Number (\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))
BBλ(1850) >Loader's Number Too large for this list

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 \1
BBλ1(5) 5 \2
BBλ1(6) 6 \\1
BBλ1(7) 7 \\2
BBλ1(8) 26 1 (\1)
BBλ1(9) 9 \\2
BBλ1(10) 36 1 (\\1)
BBλ1(11) 41 1 (\\2)
BBλ1(12) 266 1 (1 (\1))
BBλ1(13) 51 1 (\\2)
BBλ1(14) f(36)=25×2223+36>2.85×1078 1 (1 (\\1))
BBλ1(15) f(41)≥25×3385+36>101.7×1040 1 (1 (\\2))
BBλ1(16) f(266) 1 (1 (1 (\1)))
BBλ1(17) f(51) 1 (1 (\\\2))
BBλ1(18) f4(4)=f(f(266)) 1 (\1) 1 (\1)
BBλ1(19) f3(7)=f(f(41)) 1 (1 (1 (\\2)))
BBλ1(20) f6(4)=f4(266) 1 (\\1) 1 (\1)
BBλ1(21) f7(4)=f5(266) 1 (\\2) 1 (\1)
BBλ1(22) f52(4)=f50(266) 1 (1(\1)) 1(\1)
BBλ1(28) ≥fBBλ(f3(4))(4) 1 (\1) 1 (\1) 1 (\1)
BBλ1(29) ≥fBBλ(fBBλ(f4(4))+4(4))+BBλ(f4(4))+5(4) 1(\1)(\1 2 1)(\1)

Fractran (BBf(n))

n BBf(n) Example Champion Vector Representation
2 1 [1/2] [−1]
3 1 [3/2] [−11]
4 1 [9/2] [−12]
5 2 [3/2, 1/3] [−110−1]
6 3 [9/2, 1/3] [−120−1]
7 4 [27/2, 1/3] [−130−1]
8 5 [81/2, 1/3] [−140−1]
9 6 [243/2, 1/3] [−150−1]
10 7 [729/2, 1/3] [−160−1]
11 10 [27/2, 25/3, 1/5] [−1300−1200−1]
12 13 [81/2, 25/3, 1/5] [−1400−1200−1]
13 17 [81/2, 125/3, 1/5] [−1400−1300−1]
14 21 [243/2, 125/3, 1/5] [−1500−1300−1]
15 28 [1/45, 4/5, 3/2, 25/3] [0−2−120−1−1100−12]
16 53 [1/45, 4/5, 3/2, 125/3] [0−2−120−1−1100−13]
17 107 [5/6, 49/2, 3/5, 40/7] [−1−110−100201−10301−1]
18 211 [5/6, 49/2, 3/5, 80/7] [−1−110−100201−10401−1]
19 370 [5/6, 49/2, 3/5, 160/7] [−1−110−100201−10501−1]
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]
21 ≥ 31,957,632 [7/15, 4/3, 27/14, 5/2, 9/5]

Doodle Function (doodle(c,n))

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

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

Turmites

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

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

2D Turmites (turNing machines)

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

Bug Function

Bug(2,2) = 2 

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

Bug(3,3) = 8 

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

Bug(4,4) = 20 

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

Bug(5,5) = 42 

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

Bug(6,6) = 96 

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

Bug(7,7) = 218 

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

Bug(8,8) = 506 

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

References

  1. ↑ 1.0 1.1 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.
  2. ↑ 2.0 2.1 S. Ligocki, "BB(2,6) > 10↑↑10↑↑10↑↑3". Blog post, 2023. Accessed 15 August 2025.
  3. ↑ Nick Drozd. "BBB(3,3) > 10↑↑6". Accessed 15 August 2025.
  4. ↑ 4.0 4.1 Nick Drozd. "Blanking Beavers". Accessed 15 August 2025.
  5. ↑ 5.0 5.1 5.2 Nick Drozd. "Latest Beeping Busy Beaver Results". Accessed 15 August 2025.
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