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

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(Added Sigma_rev Scores obtained from running steps champions in BBchallange)
(→‎State-and-Symbol-Limited Busy Beaver functions: Updated BB(4,3) to slightly larger lower bound)
 
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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)==
=='''Original Busy Beaver Functions'''==
'''2 Symbols:'''
===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>
|<math>1</math>
|{{TM|1RZ---|halt}}
|{{TM|1RZ---|halt}}
|-
|-
|[[BB(2)]]
|[[BB(2)]]
|<math> 6 </math>
|<math>6</math>
|{{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>
|<math>21</math>
|{{TM|1RB1RZ_1LB0RC_1LC1LA|halt}}
|{{TM|1RB1RZ_1LB0RC_1LC1LA|halt}}
|-
|-
|[[BB(4)]]
|[[BB(4)]]
|<math> 107 </math>
|<math>107</math>
|{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}}
|{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}}
|-
|-
|[[BB(5)]]
|[[BB(5)]]
|<math> 47\,176\,870 </math>
|<math>47\,176\,870</math>
|{{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>
|<math>> 2\uparrow\uparrow 2\uparrow\uparrow 2\uparrow\uparrow 10</math>
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|-
|-
Line 42: Line 42:
|-
|-
|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)
|<math>1</math>
|{{TM|1RZ------|halt}}
|-
|-
|[[BB(2,3)]]
|[[BB(2,3)]]
|<math> 38 </math>
|<math>38</math>
|{{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^{2^{32}+1})</math>
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
|}
|}
'''4 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''4 Symbols:'''
!Runtime
!Runtime
!Champions
!Champions
|-
|BB(1,4)
|<math>1</math>
|{{TM|1RZ---------|halt}}
|-
|-
|[[BB(2,4)]]
|[[BB(2,4)]]
|<math> 3\,932\,964 </math>
|<math>3\,932\,964</math>
|{{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)
|<math>1</math>
|{{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)
|<math>1</math>
|{{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>
|<math>1</math>
|{{TM|1RZ---|halt}}
|{{TM|1RZ---|halt}}
|-
|-
|Σ(2)
|Σ(2)
|<math> 4 </math>
|<math>4</math>
|{{TM|1RB1LB_1LA1RZ|halt}}
|{{TM|1RB1LB_1LA1RZ|halt}}
|-
|-
|Σ(3)
|Σ(3)
|<math> 6 </math>
|<math>6</math>
|{{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>
|<math>13</math>
|{{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>
|<math>4098</math>
|{{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>
|<math>> 2\uparrow\uparrow 2\uparrow\uparrow 2\uparrow\uparrow 10</math>
|{{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)
|<math>1</math>
|{{TM|1RZ------|halt}}
|-
|-
|Σ(2,3)
|Σ(2,3)
|<math> 9 </math>
|<math>9</math>
|{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
|{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
|-
|-
|Σ(3,3)
|Σ(3,3)
|<math> \geq 374\,676\,383 </math>
|<math>\geq 374\,676\,383</math>
|{{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^{2^{32}+1})</math>
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
|}
|}
'''4 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''4 Symbols:'''
!Score
!Score
!Champions
!Champions
|-
|Σ(1,4)
|<math>1</math>
|{{TM|1RZ---------|halt}}
|-
|-
|Σ(2,4)
|Σ(2,4)
|<math> 2050 </math>
|<math>2050</math>
|{{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)
|<math>1</math>
|{{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)
|<math>1</math>
|{{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>
|<math>1</math>
|
|
|-
|-
|BBB(2)
|BBB(2)
|<math> 6 </math>
|<math>6</math>
|
|{{TM|1RB1LB_1LB1LA}}
|-
|-
|BBB(3)
|BBB(3)
|<math> 55 </math>
|<math>55</math>
|
|{{TM|1LB0RB_1RA0LC_1RC1RA}}
|-
|-
|BBB(4)
|BBB(4)
|<math> \geq 32\,779\,478 </math>
|<math>\geq 32\,779\,478</math>
|
|{{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}}
|}
|}
'''3 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''3 Symbols:'''
!Steps taken
!Steps taken
!Champions
!Champions
|-
|BBB(1,3)
|
|
|-
|-
|BBB(2,3)
|BBB(2,3)
|
|<math>59</math><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>
|
|
|-
|-
|BBB(3,3)
|BBB(3,3)
|<math> \geq 10 \uparrow\uparrow 6 </math>
|<math>\geq 10 \uparrow\uparrow 6</math>
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}
|}
Note: The BBB(2,3) champion might be {{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}.
{| class="wikitable"
|+
!'''4 Symbols:'''
!Steps taken
!Champions
|-
|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}}
|}
|}
='''Maximum Consecutive Ones Function ([[Num]])'''=
===Beeping Booping Busy Beaver ([[Beeping Busy Beaver#Beeping Booping Busy Beavers|BBBB]](n,m))===
'''2 Symbols:'''
There are currently no known/available Champions for this function.
=='''Maximum Consecutive Ones Function ([[Num]](n,m))'''==
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!'''2 Symbols:'''
!Number of Ones
!Number of Ones
!Champions
!Champions
|-
|-
|num(1)
|num(1)
|<math> 1 </math>
|<math>1</math>
|{{TM|1RZ---|halt}}
|{{TM|1RZ---|halt}}
|-
|-
|num(2)
|num(2)
|<math> 4 </math>
|<math>4</math>
|{{TM|1RB1LB_1LA1LZ|halt}}
|{{TM|1RB1LB_1LA1LZ|halt}}
|-
|-
|num(3)
|num(3)
|<math> 6 </math>
|<math>6</math>
|{{TM|1RB1LC_1RC1LZ_1LA0LB|halt}}
|{{TM|1RB1LC_1RC1LZ_1LA0LB|halt}}
|-
|-
|num(4)
|num(4)
|<math> 12 </math>
|<math>12</math>
|{{TM|1RB0LA_1RC1LB_1LB1RD_1RZ0RA|halt}}
|{{TM|1RB0LA_1RC1LB_1LB1RD_1RZ0RA|halt}}
|-
|-
|num(5)
|num(5)
|<math> 165 </math>
|<math>165</math>
|{{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 ([[Busy Beaver Functions#Other Busy Beaver functions|BB<sub>space</sub>]](n,m))'''==
{| class="wikitable"
|+
!'''2 Symbols:'''
!Cells visited
!Champions
|-
|BB<sub>space</sub>(1,2)
|<math>2</math>
|
|-
|BB<sub>space</sub>(2,2)
|<math>4</math>
|
|-
|BB<sub>space</sub>(3,2)
|<math>7</math>
|
|-
|BB<sub>space</sub>(4,2)
|<math>16</math>
|
|}
=='''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)
|<math>6</math>
|{{TM|0RB1RZ_1LA1RB|halt}}
|-
|BB<sub>rev</sub>(3)
|<math>17</math>
|{{TM|0RB1RZ_0LC1RA_1RB1LC|halt}}
|-
|BB<sub>rev</sub>(4)
|<math>48</math>
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC1RZ|halt}}
|-
|BB<sub>rev</sub>(5)
|<math>388</math>
|{{TM|1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA|halt}}
|-
|BB<sub>rev</sub>(6)
|<math>\geq 537\,556</math>
|{{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)
|<math>\geq 2</math>
|{{TM|0RB1RZ_1LA1RB|halt}}
|-
|Σ<sub>rev</sub>(3)
|<math>\geq 4</math>
|{{TM|0RB1RZ_0LC1RA_1RB1LC|halt}}
|-
|Σ<sub>rev</sub>(4)
|<math>\geq 6</math>
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC1RZ|halt}}
|-
|Σ<sub>rev</sub>(5)
|<math>\geq 16</math>
|{{TM|1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA|halt}}
|-
|Σ<sub>rev</sub>(6)
|<math>\geq 1161</math>
|{{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)
|inexistent
|inexistent
|-
|BLB(2)
|<math>\geq 8</math><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)
|<math>\geq 34</math><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)
|<math>\geq 32\,779\,477</math>
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
|}
{| class="wikitable"
|+
!'''3 Symbols:'''
!Steps
!Champions
|-
|BLB(1,3)
|inexistent
|inexistent
|-
|BLB(2,3)
|<math>\geq 77</math><ref name=":4" />
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|}
{| class="wikitable"
|+
!'''4 Symbols:'''
!Steps
!Champions
|-
|BLB(1,4)
|inexistent
|inexistent
|-
|BLB(2,4)
|<math>\geq 1\,367\,361\,263\,049</math><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
|<math>2</math>
|<math>7</math>
|<math>22</math>
|<math>72</math>
|<math>427</math>
|<math>8407</math>
|-
|3 Symbols
|<math>2</math>
|<math>23</math>
|<math>351</math>
|<math>189\,270</math>
|
|
|-
|4 Symbols
|<math>2</math>
|<math>93</math>
|<math>242\,789</math>
|
|
|
|-
|5 Symbols
|<math>2</math>
|<math>956</math>
|
|
|
|
|-
|6 Symbols
|<math>2</math>
|<math>33\,851</math>
|
|
|
|
|}
=='''Period-oriented Busy Beavers'''==
===Busy Preperiodic Beaver ([[BBS]](n,m))===
{| class="wikitable"
|+
!'''2 Symbols:'''
!Preperiod
!Champions
|-
|BBS(1,2)
|<math>0</math>
|{{TM|1RA---}}
|-
|BBS(2,2)
|<math>\geq 9</math>
|{{TM|1RB0LB_1LA0RB}} proven winner?
|-
|BBS(3,2)
|<math>101</math>
|{{TM|1RB1LB_0RC0LA_1LC0LA}}
|-
|BBS(4,2)
|<math>\geq 119\,120\,230\,102</math>
|{{TM|1RB1LC_0LA1RD_0RB0LC_1LA0RD}}
|}
{| class="wikitable"
|+
!'''3 Symbols:'''
!Preperiod
!Champions
|-
|BBS(1,3)
|<math>0</math>
|{{TM|1RA------}}
|}
{| class="wikitable"
|+
!'''4 Symbols:'''
!Preperiod
!Champions
|-
|BBS(1,4)
|<math>0</math>
|{{TM|1RA---------}}
|-
|BBS(2,4)
|<math>\geq 293\,225\,660\,896</math>
|{{TM|1RB2LA0RA3LA_1LA1LB3RB1RA}}
|}
===Busy Periodic Beaver ([[BBP]](n,m))===
{| class="wikitable"
|+
!'''2 Symbols:'''
!Period
!Champions
|-
|BBP(1,2)
|<math>1</math>
|{{TM|1RA---}}
|-
|BBP(2,2)
|<math>\geq 9</math>
|{{TM|1RB0RB_1LB1RA}} proven winner?
|-
|BBP(3,2)
|<math>92</math>
|{{TM|1RB0LA_0RC1LA_1LC0RB}}
|-
|BBP(4,2)
|<math>\geq 212\,081\,736</math>
|{{TM|1RB0LA_0RC1RD_1LD0RB_1LA1RB}}
|}
{| class="wikitable"
|+
!'''3 Symbols:'''
!Period
!Champions
|-
|BBP(1,3)
|<math>1</math>
|{{TM|1RA------}}
|}
{| class="wikitable"
|+
!'''4 Symbols:'''
!Period
!Champions
|-
|BBP(1,4)
|<math>1</math>
|{{TM|1RA---------}}
|-
|BBP(2,4)
|<math>\geq 33\,209\,131</math>
|{{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 307: Line 657:
|-
|-
|BBi(1)
|BBi(1)
|<math> 1 </math>
|<math>1</math>
|{{TM|0RH|halt}} {{TM|1RH---|halt}}
|{{TM|0RH|halt}} {{TM|1RH---|halt}}
|-
|-
|BBi(2)
|BBi(2)
|<math> 3 </math>
|<math>3</math>
|{{TM|0RB---_1LA---|halt}}
|{{TM|0RB---_1LA---|halt}}
|-
|-
|BBi(3)
|BBi(3)
|<math> 5 </math>
|<math>5</math>
|{{TM|1RB1LB_1LA---|halt}}
|{{TM|1RB1LB_1LA---|halt}}
|-
|-
|BBi(4)
|BBi(4)
|<math> 16 </math>
|<math>16</math>
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|{{TM|1RB---_0RC---_1LC0LA|halt}}
|-
|-
|BBi(5)
|BBi(5)
|<math> 37 </math>
|<math>37</math>
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|-
|BBi(6)
|BBi(6)
|<math> 123 </math>
|<math>123</math>
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|-
|BBi(7)
|BBi(7)
|<math> 3\,932\,963 </math>
|<math>3\,932\,963</math>
|{{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 346: Line 700:
|-
|-
|Σi(1)
|Σi(1)
|<math> 1 </math>
|<math>1</math>
|{{TM|1RH---|halt}}
|{{TM|1RH---|halt}}
|-
|-
|Σi(2)
|Σi(2)
|<math> 2 </math>
|<math>2</math>
|{{TM|1RB---_1LA---|halt}}
|{{TM|1RB---_1LA---|halt}}
|-
|-
|Σi(3)
|Σi(3)
|<math> 4 </math>
|<math>4</math>
|{{TM|1RB1LB_1LA---|halt}}
|{{TM|1RB1LB_1LA---|halt}}
|-
|-
|Σi(4)
|Σi(4)
|<math> 5 </math>
|<math>5</math>
|{{TM|1RB0LB---_1LA2RA---|halt}}
|{{TM|1RB0LB---_1LA2RA---|halt}}
|-
|-
|Σi(5)
|Σi(5)
|<math> 9 </math>
|<math>9</math>
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|{{TM|1RB2LB---_2LA2RB1LB|halt}}
|-
|-
|Σi(6)
|Σi(6)
|<math> 14 </math>
|<math>14</math>
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|{{TM|1RB3LA1RA0LA_2LA------3RA|halt}}
|-
|-
|Σi(7)
|Σi(7)
|<math> 2050 </math>
|<math>2050</math>
|{{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}}
|}
=='''Instruction-Limited Blanking Busy Beaver ([[Instruction-Limited Busy Beaver#Instruction-Limited Busy Beaver Variants|BLBi(n)]])'''==
{| class="wikitable"
|+
!
!Steps
!Champions
|-
|BLBi(1)
|inexistent
|inexistent
|-
|BLBi(2)
|inexistent
|inexistent
|-
|BLBi(3)
|<math>4</math>
|
|-
|BLBi(4)
|<math>12</math>
|
|-
|BLBi(5)
|<math>30</math>
|
|-
|BLBi(6)
|<math>77</math>
|{{TM|1RB2LA0RB_1LA0LB1RA}}
|-
|BLBi(7)
|<math>808</math>
|
|-
|BLBi(8)
|<math>\geq 1\,367\,361\,263\,049</math>
|{{TM|1RB2RA1RA2RB_2LB3LA0RB0RA}}
|}
|}
='''Reversible Turing Machines'''=
=='''Instruction-Limited Greedy Busy Beaver ([[Instruction-Limited Busy Beaver#Instruction-Limited Busy Beaver Variants|gBBi(n)]])'''==
==Maximum Shifts Function ([[Reversible Turing Machine|BB<sub>rev</sub>]])==
'''2 Symbols:'''
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 386: Line 782:
!Champions
!Champions
|-
|-
|BB<sub>rev</sub>(1)
|gBBi(1)
|<math>1</math>
|
|-
|gBBi(2)
|<math>3</math>
|
|-
|gBBi(3)
|<math>5</math>
|
|-
|gBBi(4)
|<math>13</math>
|
|-
|gBBi(5)
|<math>19</math>
|
|-
|gBBi(6)
|<math>25</math>
|
|-
|gBBi(7)
|<math>41</math>
|
|-
|gBBi(8)
|<math>55</math>
|
|-
|gBBi(9)
|<math>238</math>
|
|-
|gBBi(10)
|<math>941</math>
|
|-
|gBBi(11)
|<math>1341</math>
|
|-
|gBBi(12)
|<math>10465</math>
|
|
|-
|gBBi(13)
|<math>10675</math>
|
|
|-
|-
|BB<sub>rev</sub>(2)
|gBBi(14)
|<math> 6 </math>
|<math>\geq 9\,874\,580</math>
|{{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)
|<math>22</math>
|<code>\(\1 1) (1 (\2))</code>
|-
|BBλ(22)
|<math>24</math>
|<code>\(\1 1) (1 (\\1))\(\1 1 1) (1 1)</code>
|-
|BBλ(23)
|<math>26</math>
|<code>\(\1 1) (1 (\\2))</code>
|-
|BBλ(24)
|<math>30</math>
|<code>\(\1 1 1) (1 (\1))</code>
|-
|BBλ(25)
|<math>42</math>
|<code>\(\1 1) (\1 (2 1))</code>
|-
|BBλ(26)
|<math>52</math>
|<code>(\1 1) (\\2 (1 2))</code>
|-
|BBλ(27)
|<math>44</math>
|<code>\\(\1 1) (\1 (2 1))</code>
|-
|BBλ(28)
|<math>58</math>
|<code>\(\1 1) (\1 (2 (\2))))</code>
|-
|BBλ(29)
|<math>223</math>
|<code>\(\1 1) (\1 (1 (2 1)))</code>
|-
|BBλ(30)
|<math>160</math>
|<code>(\1 1 1) (\\2 (1 2))</code> and <code>(\1 (1 1)) (\\2 (1 2))</code>
|-
|BBλ(31)
|<math>267</math>
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|-
|BBλ(32)
|<math> 298 </math>
|<code>\(\1 1) (\1 (1 (2 (\2))))</code>
|-
|BBλ(33)
|<math>1812</math>
|<code>\(\1 1) (\1 (1 (1 (2 1))))</code>
|-
|BBλ(34)
|<math>327\,686</math>
|<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)
|<math>0</math>
|
|
|-
|BBλ<sub>1</sub>(2)
|<math>1</math>
|<code>1</code>
|-
|BBλ<sub>1</sub>(3)
|<math>0</math>
|
|
|-
|-
|Σ<sub>rev</sub>(2)
|BBλ<sub>1</sub>(4)
|<math> \geq 2 </math>
|<math>4</math>
|{{TM|0RB1RZ_1LA1RB|halt}}
|<code>\1</code>
|-
|BBλ<sub>1</sub>(5)
|<math>5</math>
|<code>\2</code>
|-
|BBλ<sub>1</sub>(6)
|<math>6</math>
|<code>\\1</code>
|-
|BBλ<sub>1</sub>(7)
|<math>7</math>
|<code>\\2</code>
|-
|BBλ<sub>1</sub>(8)
|<math>26</math>
|<code>1 (\1)</code>
|-
|BBλ<sub>1</sub>(9)
|<math>9</math>
|<code>\\2</code>
|-
|BBλ<sub>1</sub>(10)
|<math>36</math>
|<code>1 (\\1)</code>
|-
|BBλ<sub>1</sub>(11)
|<math>41</math>
|<code>1 (\\2)</code>
|-
|BBλ<sub>1</sub>(12)
|<math>266</math>
|<code>1 (1 (\1))</code>
|-
|BBλ<sub>1</sub>(13)
|<math>51</math>
|<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>
|-
|-
|Σ<sub>rev</sub>(3)
|BBλ<sub>1</sub>(22)
|<math> \geq 4 </math>
|<math>f^{52}(4) = f^{50}(266)</math>
|{{TM|0RB1RZ_0LC1RA_1RB1LC|halt}}
|<code>1 (1(\1)) 1(\1)</code>
|-
|-
|Σ<sub>rev</sub>(4)
|BBλ<sub>1</sub>(28)
|<math> \geq 6 </math>
|<math>\geq f^{BB \lambda(f^{3}(4))}(4)</math>
|{{TM|1RB0LD_0LC0RB_1LA1LD_1LC1RZ|halt}}
|<code>1 (\1) 1 (\1) 1 (\1)</code>
|-
|-
|Σ<sub>rev</sub>(5)
|BBλ<sub>1</sub>(29)
|<math> \geq 16 </math>
|<math>\geq f^{BB \lambda(f^{BB \lambda(f^{4}(4))+4}(4))+BB \lambda(f^{4}(4))+5}(4)</math>
|{{TM|1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA|halt}}
|<code>1(\1)(\1 2 1)(\1)</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>
|<math>\geq 487</math>
|{{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.
=References=

Latest revision as of 10:31, 18 August 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

Original Busy Beaver Functions

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

2 Symbols: Runtime Champions
BB(1) 1RZ--- (bbch)
BB(2) 1RB1LB_1LA1RZ (bbch) 1RB0LB_1LA1RZ (bbch) 1RB1RZ_1LB1LA (bbch) 1RB1RZ_0LB1LA (bbch) 0RB1RZ_1LA1RB (bbch)
BB(3) 1RB1RZ_1LB0RC_1LC1LA (bbch)
BB(4) 1RB1LB_1LA0LC_1RZ1LD_1RD0RA (bbch)
BB(5) 1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA (bbch)
BB(6) 1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE (bbch)
BB(7) 1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF (bbch)
BB(8)
BB(9) 1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH (bbch)
BB(10) 1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ (bbch)
BB(11) 1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RZ0LI_0LD1LE (bbch)
BB(12) 0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_1RZ1LA_1RF1LL (bbch)
BB(14) 1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ--- (bbch)
BB(15) 0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_1RZ0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN (bbch)
BB(16)
BB(18)
BB(20)
BB(21)
BB(40)
BB(41)
BB(51)
3 Symbols: Runtime Champions
BB(1,3) 1RZ------ (bbch)
BB(2,3) 1RB2LB1RZ_2LA2RB1LB (bbch)
BB(3,3) 0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC (bbch)
BB(4,3) 0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD (bbch)
4 Symbols: Runtime Champions
BB(1,4) 1RZ--------- (bbch)
BB(2,4) 1RB2LA1RA1RA_1LB1LA3RB1RZ (bbch)
BB(3,4) [1] 1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC (bbch)
5 Symbols: Runtime Champions
BB(1,5) 1RZ------------ (bbch)
BB(2,5) 1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ (bbch)
BB(3,5) 1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC (bbch)
6 Symbols: Runtime Champions
BB(1,6) 1RZ--------------- (bbch)
BB(2,6) [2] 1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA (bbch)

Maximum Score Function (Σ(n,m))

2 Symbols: Score Champions
Σ(1) 1RZ--- (bbch)
Σ(2) 1RB1LB_1LA1RZ (bbch)
Σ(3) 1RB1RZ_0RC1RB_1LC1LA (bbch) 1RB1RC_1LC1RZ_1RA0LB (bbch) 1RB1LC_1LA1RB_1LB1RZ (bbch) 1RB1RA_1LC1RZ_1RA1LB (bbch) 1RB1LC_1RC1RZ_1LA0LB (bbch)
Σ(4) 1RB1LB_1LA0LC_1RZ1LD_1RD0RA (bbch) 1RB0RC_1LA1RA_1RZ1RD_1LD0LB (bbch)
Σ(5) 1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA (bbch) 1RB1RA_1LC1LB_1RA1LD_1RA1LE_1RZ0LC (bbch)
Σ(6) 1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE (bbch)
Σ(7) 1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF (bbch)
3 Symbols: Score Champions
Σ(1,3) 1RZ------ (bbch)
Σ(2,3) 1RB2LB1RZ_2LA2RB1LB (bbch)
Σ(3,3) 0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC (bbch)
Σ(4,3) 0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD (bbch)
4 Symbols: Score Champions
Σ(1,4) 1RZ--------- (bbch)
Σ(2,4) 1RB2LA1RA1RA_1LB1LA3RB1RZ (bbch)
Σ(3,4) [1] 1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC (bbch)
5 Symbols: Score Champions
Σ(1,5) 1RZ------------ (bbch)
Σ(2,5) 1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ (bbch)
6 Symbols: Score Champions
Σ(1,6) 1RZ--------------- (bbch)
Σ(2,6) [2] 1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA (bbch)

Beeping Busy Beavers

Beeping Busy Beaver (BBB(n,m))

2 Symbols: Steps taken Champions
BBB(1)
BBB(2) 1RB1LB_1LB1LA (bbch)
BBB(3) 1LB0RB_1RA0LC_1RC1RA (bbch)
BBB(4) 1RB1LD_1RC1RB_1LC1LA_0RC0RD (bbch)
BBB(5) 1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA (bbch)
3 Symbols: Steps taken Champions
BBB(1,3)
BBB(2,3) [3]
BBB(3,3) 1RB0LB2LA_1LA0RC0LB_2RC2RB0LC (bbch)

Note: The BBB(2,3) champion might be 1RB0LB2LA_1LA0RC0LB_2RC2RB0LC (bbch).

4 Symbols: Steps taken Champions
BBB(1,4)
BBB(2,4) [4] 1RB2LA1RA1LB_0LB2RB3RB1LA (bbch)

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

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

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

2 Symbols: Number of Ones Champions
num(1) 1RZ--- (bbch)
num(2) 1RB1LB_1LA1LZ (bbch)
num(3) 1RB1LC_1RC1LZ_1LA0LB (bbch)
num(4) 1RB0LA_1RC1LB_1LB1RD_1RZ0RA (bbch)
num(5) 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)
BBspace(2,2)
BBspace(3,2)
BBspace(4,2)

Reversible Turing Machines

Maximum Shifts Function (BBrev(n,m))

2 Symbols: Steps Champions
BBrev(1)
BBrev(2) 0RB1RZ_1LA1RB (bbch)
BBrev(3) 0RB1RZ_0LC1RA_1RB1LC (bbch)
BBrev(4) 1RB0LD_0LC0RB_1LA1LD_1LC1RZ (bbch)
BBrev(5) 1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA (bbch)
BBrev(6) 1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA (bbch)
BBrev(7) 1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB1RZ (bbch)

Maximum Score Function (Σrev(n,m))

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

Blanking Busy Beaver (BLB(n,m))

2 Symbols: Steps Champions
BLB(1) inexistent inexistent
BLB(2) [4] 1RB0RA_1LB1LA (bbch)
BLB(3) [5] 1RB1LB_1LA1LC_1RC0LC (bbch)
BLB(4) 1RB1LD_1RC1RB_1LC1LA_0RC0RD (bbch)
3 Symbols: Steps Champions
BLB(1,3) inexistent inexistent
BLB(2,3) [5] 1RB2LA0RB_1LA0LB1RA (bbch)
4 Symbols: Steps Champions
BLB(1,4) inexistent inexistent
BLB(2,4) [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
3 Symbols
4 Symbols
5 Symbols
6 Symbols

Period-oriented Busy Beavers

Busy Preperiodic Beaver (BBS(n,m))

2 Symbols: Preperiod Champions
BBS(1,2) 1RA--- (bbch)
BBS(2,2) 1RB0LB_1LA0RB (bbch) proven winner?
BBS(3,2) 1RB1LB_0RC0LA_1LC0LA (bbch)
BBS(4,2) 1RB1LC_0LA1RD_0RB0LC_1LA0RD (bbch)
3 Symbols: Preperiod Champions
BBS(1,3) 1RA------ (bbch)
4 Symbols: Preperiod Champions
BBS(1,4) 1RA--------- (bbch)
BBS(2,4) 1RB2LA0RA3LA_1LA1LB3RB1RA (bbch)

Busy Periodic Beaver (BBP(n,m))

2 Symbols: Period Champions
BBP(1,2) 1RA--- (bbch)
BBP(2,2) 1RB0RB_1LB1RA (bbch) proven winner?
BBP(3,2) 1RB0LA_0RC1LA_1LC0RB (bbch)
BBP(4,2) 1RB0LA_0RC1RD_1LD0RB_1LA1RB (bbch)
3 Symbols: Period Champions
BBP(1,3) 1RA------ (bbch)
4 Symbols: Period Champions
BBP(1,4) 1RA--------- (bbch)
BBP(2,4) 1RB0RA3LB1RB_2LA0LB1RA2RB (bbch)

Instruction-Limited Busy Beaver

Instruction-Limited Classical Busy Beaver Functions

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

Steps Champions
BBi(1) 0RH (bbch) 1RH--- (bbch)
BBi(2) 0RB---_1LA--- (bbch)
BBi(3) 1RB1LB_1LA--- (bbch)
BBi(4) 1RB---_0RC---_1LC0LA (bbch)
BBi(5) 1RB2LB---_2LA2RB1LB (bbch)
BBi(6) 1RB3LA1RA0LA_2LA------3RA (bbch)
BBi(7) 1RB2LA1RA1RA_1LB1LA3RB--- (bbch) 1RB2LA1RA_1LC1LA2RB_---1LA--- (bbch)
BBi(8) 1RB1LA------_1RC3LB1RB---_2LA2LC---0LC (bbch)
BBi(9) 1RB3LA4RB0RB2LA_1LB2LA3LA1RA--- (bbch)

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

Score Champions
Σi(1) 1RH--- (bbch)
Σi(2) 1RB---_1LA--- (bbch)
Σi(3) 1RB1LB_1LA--- (bbch)
Σi(4) 1RB0LB---_1LA2RA--- (bbch)
Σi(5) 1RB2LB---_2LA2RB1LB (bbch)
Σi(6) 1RB3LA1RA0LA_2LA------3RA (bbch)
Σi(7) 1RB2LA1RA1RA_1LB1LA3RB--- (bbch) 1RB2LA1RA_1LC1LA2RB_---1LA--- (bbch)
Σi(8) 1RB1LA------_1RC3LB1RB---_2LA2LC---0LC (bbch)
Σi(9) 1RB3LA4RB0RB2LA_1LB2LA3LA1RA--- (bbch)

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

Steps Champions
BLBi(1) inexistent inexistent
BLBi(2) inexistent inexistent
BLBi(3)
BLBi(4)
BLBi(5)
BLBi(6) 1RB2LA0RB_1LA0LB1RA (bbch)
BLBi(7)
BLBi(8) 1RB2RA1RA2RB_2LB3LA0RB0RA (bbch)

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

Steps Champions
gBBi(1)
gBBi(2)
gBBi(3)
gBBi(4)
gBBi(5)
gBBi(6)
gBBi(7)
gBBi(8)
gBBi(9)
gBBi(10)
gBBi(11)
gBBi(12)
gBBi(13)
gBBi(14) 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 BBλ(n) = n.

BBλ(n) Champions
BBλ(21) \(\1 1) (1 (\2))
BBλ(22) \(\1 1) (1 (\\1))\(\1 1 1) (1 1)
BBλ(23) \(\1 1) (1 (\\2))
BBλ(24) \(\1 1 1) (1 (\1))
BBλ(25) \(\1 1) (\1 (2 1))
BBλ(26) (\1 1) (\\2 (1 2))
BBλ(27) \\(\1 1) (\1 (2 1))
BBλ(28) \(\1 1) (\1 (2 (\2))))
BBλ(29) \(\1 1) (\1 (1 (2 1)))
BBλ(30) (\1 1 1) (\\2 (1 2)) and (\1 (1 1)) (\\2 (1 2))
BBλ(31) (\1 1) (\\2 (2 (1 2)))
BBλ(32) \(\1 1) (\1 (1 (2 (\2))))
BBλ(33) \(\1 1) (\1 (1 (1 (2 1))))
BBλ(34) (\1 1 1 1) (\\2 (2 1)) and (\1 (1 1) 1) (\\2 (2 1))
BBλ(35) (\1 1 1) (\\2 (2 (2 1)))
BBλ(36) (\1 1) (\1 (1 (\\2 (2 1))))
BBλ(37) \(\1 1 1) (\\2 (2 (2 1)))
BBλ(38) (\1 1 1 1 1) (\\2 (2 1)) and (\1 (1 1) 1 1) (\\2 (2 1))
BBλ(39) (\1 1 1 1) (\\2 (2 (2 1)))
BBλ(40) (\1 1 1) (\1 (\\2 (2 1)) 1)
BBλ(41) (\1 (\1 1) 1) (\\2 (2 (2 1)))
BBλ(42) \(\1 1 1) (\1 (\\2 (2 1)) 1)
BBλ(43) (\1 1) (\1 (\1 (\\2 (2 1)) 2))
BBλ(44) (\1 1 1 1) (\1 (\\2 (2 1)) 1)
BBλ(45) \(\1 1) (\1 (\1 (\\2 (2 1)) 2))
BBλ(46) \(\1 1 1 1) (\1 (\\2 (2 1)) 1)
BBλ(47)
BBλ(48) (\1 1 1 1 1) (\1 (\\2 (2 1)) 1)
BBλ(49) (\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))
BBλ(1850) Too large for this list

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

Note that .

BBλ1(n) Champions
BBλ1(1)
BBλ1(2) 1
BBλ1(3)
BBλ1(4) \1
BBλ1(5) \2
BBλ1(6) \\1
BBλ1(7) \\2
BBλ1(8) 1 (\1)
BBλ1(9) \\2
BBλ1(10) 1 (\\1)
BBλ1(11) 1 (\\2)
BBλ1(12) 1 (1 (\1))
BBλ1(13) 1 (\\2)
BBλ1(14) 1 (1 (\\1))
BBλ1(15) 1 (1 (\\2))
BBλ1(16) 1 (1 (1 (\1)))
BBλ1(17) 1 (1 (\\\2))
BBλ1(18) 1 (\1) 1 (\1)
BBλ1(19) 1 (1 (1 (\\2)))
BBλ1(20) 1 (\\1) 1 (\1)
BBλ1(21) 1 (\\2) 1 (\1)
BBλ1(22) 1 (1(\1)) 1(\1)
BBλ1(28) 1 (\1) 1 (\1) 1 (\1)
BBλ1(29) 1(\1)(\1 2 1)(\1)

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)

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.

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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