View source for God's Algorithm

<seo title="God's Algorithm - Optimal solutions of the Rubik's Cube"/>

'''God's Algorithm''' is the name given to any algorithm which brings a puzzle from a scrambled state to the solved state in as few moves as possible. Such algorithms are more commonly referred to as '''optimal solutions'''. The term "God's Algorithm" may also be used to refer to a procedure for finding optimal solutions.

== God's Number ==
'''God's Number''' for a puzzle is the greatest number of moves that a state can be from the solved state. Equivalently, it is the maximum length of an optimal solution, and the minimum number of moves such that every state can be solved in at most that number of moves.

If the puzzle has the structure of a [[wikipedia:group|group]], then God's Number is also the largest number of moves required to move between any two states. In this case, God's Number may also be referred to as the '''diameter''' of the group. If the puzzle does not have a group structure, there may be pairs of states that are more than God's Number of moves apart.

God's Number has been known for small puzzles such as the [[2x2x2 cube]] since the 1980s, and for the [[3x3x3 cube]] since 2010, but remains unknown for most other puzzles.

=== Table of God's Numbers ===

==== WCA Puzzles ====

{| class=wikitable
|-
! Puzzle
! Metric
! style="width:10%;" | Lower Bound
! style="width:10%;" | Upper Bound
! style="width:10%;" | Conjectured Value
! State Realizing Lower Bound
! Source
|-
| rowspan=2 | 2x2x2
| HTM
| colspan=3 style="text-align: center;" | 11
| R U' R2 U' F' R' U F2 R U' F
| [https://www.jaapsch.net/puzzles/cube2.htm]
|-
| QTM
| colspan=3 style="text-align: center;" | 14
| U R U F' R' F R U F' R F U R' F
| [https://www.jaapsch.net/puzzles/cube2.htm]
|-
| rowspan=4 | 3x3x3
| HTM
| colspan=3 style="text-align: center;" | 20
| U R2 F B R B2 R U2 L B2 R U' D' R2 F R' L B2 U2 F2
| [http://cube20.org/]
|-
| QTM
| colspan=3 style="text-align: center;" | 26
| U2 F U2 R' L F2 U F' B' R L U2 R U D' R L' D R' L' D2
| [http://cube20.org/qtm]
|-
| STM
| style="text-align: center;" | 18
| style="text-align: center;" | 20
| style="text-align: center;" | 18
| D' M B2 R' E2 R' S2 R U S U R2 D F2 D S' U' B' x'
| [http://forum.cubeman.org/?q=node/view/200#comment-1612], [https://gist.github.com/rokicki/5163b0bc85bcdd8210eb4b8707fa1e30]
|-
| ATM
| style="text-align: center;" | 16
| style="text-align: center;" | 20
| style="text-align: center;" | 16
| R2 U' (F2 B') (U D') F' R' D B D' R' B U2 R' U2 F' R2
| [http://forum.cubeman.org/?q=node/view/200#comment-3081]
|-
| rowspan=3 | 4x4x4
| OBTM
| style="text-align: center;" | 35
| style="text-align: center;" | 55
|
|
| [http://forum.cubeman.org/?q=node/view/236], [http://forum.cubeman.org/?q=node/view/541#comment-3006]
|-
| SSTM
| style="text-align: center;" | 32
| style="text-align: center;" | 53
|
|
| [http://forum.cubeman.org/?q=node/view/236], [http://forum.cubeman.org/?q=node/view/541#comment-3009]
|-
| BTM
| style="text-align: center;" | 29
| style="text-align: center;" | 53
|
|
| [http://forum.cubeman.org/?q=node/view/236], [http://forum.cubeman.org/?q=node/view/541]
|-
| 5x5x5
| OBTM
| style="text-align: center;" | 52
| style="text-align: center;" | 130
|
|
| [http://forum.cubeman.org/?q=node/view/236], [https://www.speedsolving.com/threads/5x5x5-obtm-upper-bound.61270/]
|-
| 6x6x6
| OBTM
| style="text-align: center;" | 75
|
|
|
| [http://forum.cubeman.org/?q=node/view/236]
|-
| 7x7x7
| OBTM
| style="text-align: center;" | 99
|
|
|
| [http://forum.cubeman.org/?q=node/view/236]
|-
| Clock
|
| colspan=3 style="text-align: center;" | 12
| DDUU u d6 DUDU u5' d4 UDUD u2' d5 UUDD u d6 UDDU u d UUDU u2 DUUU u2
| [https://www.speedsolving.com/threads/gods-number-for-clock-found.47822/], [http://cube20.org/clock/]
|-
| rowspan=2 | Megaminx
| HTM
| style="text-align: center;" | 48
| style="text-align: center;" | 194
|
|
| [https://www.speedsolving.com/threads/lower-bound-for-megaminx-in-htm-and-qtm.35724/#post-1190421], [http://forum.cubeman.org/?q=node/view/328#comment-2657]
|-
| QTM
| style="text-align: center;" | 59
|
|
|
| <code>twsearch -q -C megaminx.tws</code>, [https://github.com/cubing/twsearch]
|-
| Pyraminx
|
| colspan=3 style="text-align: center;" | 11 (15 with tips)
| U R U R' U L' R' L' U' L B' u l r b
| [https://www.jaapsch.net/puzzles/pyraminx.htm]
|-
| Skewb
|
| colspan=3 style="text-align: center;" | 11
| U L U' R' U B' U L R' U B
| [https://www.jaapsch.net/puzzles/skewb.htm]
|-
| rowspan=2 | Square-1
| Twist Metric
| colspan=3 style="text-align: center;" | 13
| / (0,-3) / (0,-1) / (1,0) / (-4,-4) / (0,-5) / (0,-4) / (-5,2) / (-4,0) / (-5,-2) / (0,1) / (5,0) / (0,3) / (3,0)
| [http://forum.cubeman.org/?q=node/view/35]
|-
| FTM
| colspan=3 style="text-align: center;" | 31
|
| [https://www.speedsolving.com/threads/square-one-can-be-solved-in-31-moves-in-face-turn-metric.67363/]
|}

==== Big Cubes ====

{| class=wikitable style="width:100%;"
|-
! Puzzle
! Metric
! Lower Bound
! Source
|-
| 8x8x8
| OBTM
| style="text-align: center;" | 128
| [http://forum.cubeman.org/?q=node/view/236]
|-
| 9x9x9
| OBTM
| style="text-align: center;" | 158
| [http://forum.cubeman.org/?q=node/view/236]
|-
| 10x10x10
| OBTM
| style="text-align: center;" | 193
| [http://forum.cubeman.org/?q=node/view/236]
|-
| 11x11x11
| OBTM
| style="text-align: center;" | 229
| [http://forum.cubeman.org/?q=node/view/236]
|-
| 12x12x12
| OBTM
| style="text-align: center;" | 270
| [http://forum.cubeman.org/?q=node/view/236]
|-
| 13x13x13
| OBTM
| style="text-align: center;" | 312
| [http://forum.cubeman.org/?q=node/view/236]
|-
| 14x14x14
| OBTM
| style="text-align: center;" | 359
| [http://forum.cubeman.org/?q=node/view/236]
|-
| 15x15x15
| OBTM
| style="text-align: center;" | 406
| [http://forum.cubeman.org/?q=node/view/236]
|-
| 16x16x16
| OBTM
| style="text-align: center;" | 458
| [http://forum.cubeman.org/?q=node/view/236]
|-
| 17x17x17
| OBTM
| style="text-align: center;" | 511
| [http://forum.cubeman.org/?q=node/view/236]
|-
| 18x18x18
| OBTM
| style="text-align: center;" | 568
| [http://forum.cubeman.org/?q=node/view/236]
|-
| 19x19x19
| OBTM
| style="text-align: center;" | 626
| [http://forum.cubeman.org/?q=node/view/236]
|-
| 20x20x20
| OBTM
| style="text-align: center;" | 690
| [http://forum.cubeman.org/?q=node/view/236]
|-
| 21x21x21
| OBTM
| style="text-align: center;" | 753
|
|-
| 22x22x22
| OBTM
| style="text-align: center;" | 821
|
|-
| 30x30x30
| OBTM
| style="text-align: center;" | 1452
|
|-
| 33x33x33
| OBTM
| style="text-align: center;" | 1729
|
|-
| 40x40x40
| OBTM
| style="text-align: center;" | 2465
|
|-
| 50x50x50
| OBTM
| style="text-align: center;" | 3720
|
|-
| 60x60x60
| OBTM
| style="text-align: center;" | 5209
|
|-
| 70x70x70
| OBTM
| style="text-align: center;" | 6926
|
|-
| 80x80x80
| OBTM
| style="text-align: center;" | 8869
|
|-
| 90x90x90
| OBTM
| style="text-align: center;" | 11033
|
|-
| 100x100x100
| OBTM
| style="text-align: center;" | 13414
|
|-
| 111x111x111
| OBTM
| style="text-align: center;" | 16281
|
|-
| 121x121x121
| OBTM
| style="text-align: center;" | 19112
|
|-
| 128x128x128
| OBTM
| style="text-align: center;" | 21220
|
|-
| 150x150x150
| OBTM
| style="text-align: center;" | 28506
|
|-
| 200x200x200
| OBTM
| style="text-align: center;" | 48741
|
|-
| 256x256x256
| OBTM
| style="text-align: center;" | 77309
|
|-
| 500x500x500
| OBTM
| style="text-align: center;" | 271273
|
|-
| 1000x1000x1000
| OBTM
| style="text-align: center;" | 1001447
|
|-
| 10000x10000x10000
| OBTM
| style="text-align: center;" | 79634546
|
|}

{| class=wikitable
|-
! Puzzle
! Metric
! Lower Bound
! Upper Bound
! Conjectured Value
! Source
|-
| NxNxN, N<=1000
| OBTM
| style="text-align: center;" | N^2
|
|
|
|-
| NxNxN
| OBTM
| style="text-align: center;" | (1/4 log(24!/4!^6) + o(1)) N^2/log(N)
| style="text-align: center;" | (c + o(1)) N^2/log(N)
| style="text-align: center;" | (1/4 log(24!/4!^6) + o(1)) N^2/log(N)
| [https://arxiv.org/abs/1106.5736]
|}

==== Subsets/alternate generating sets ====

{| class=wikitable
|-
! Puzzle
! Subset
! Metric
! style="width:10%;" | Lower Bound
! style="width:10%;" | Upper Bound
! style="width:10%;" | Conjectured Value
! State Realizing Lower Bound
! Source
|-
| rowspan=2 | 2x2x2
| rowspan=2 | <U,R>
| HTM
| colspan=3 style="text-align: center;" | 14
| U R U' R' U R' U R' U2 R U2 R' U2 R2
|
|-
| QTM
| colspan=3 style="text-align: center;" | 17
| U R U R' U' R U' R2 U R' U R' U2 R U'
|
|-
| rowspan=16 | 3x3x3
| rowspan=2 | <U,R>
| HTM
| colspan=3 style="text-align: center;" | 20
| U R U R U R2 U2 R2 U2 R2 U' R U R U' R2 U2 R2 U2 R2
|
|-
| QTM
| colspan=3 style="text-align: center;" | 25
| U R U R' U R' U' R U R' U' R2 U R' U' R2 U R U' R U' R U
|
|-
| <U,F,R>
| HTM
| style="text-align: center;" | 21
|
| style="text-align: center;" | 21
| U R F2 R U2 F2 U' F' R U F U R' F R2 U F R U2 R F'
| [http://forum.cubeman.org/?q=node/view/566]
|-
| <U,F2,R>
| HTM
| colspan=3 style="text-align: center;" | 21
| U R U2 R U R' U2 R2 F2 R2 U R F2 R U2 R F2 R' U F2 R'
| [https://benwh1.github.io/web/twisty_puzzles/3x3/3x3_U_F2_R/index.html]
|-
| <U,L,R>
| HTM
| style="text-align: center;" | 24
|
|
| U L U' L2 R' U R U R2 U2 L' U2 L U R U' R2 U R2 U2 L R2 U' L'
|
|-
| <U2,L,R2>
| HTM
| colspan=3 style="text-align: center;" | 34
| U2 L U2 L U2 L U2 L2 U2 L' R2 U2 L U2 L U2 L' R2 U2 L2 U2 L U2 L U2 L U2 L U2 L2 R2 U2 L R2
|
|-
| <U,L,F,R,B>
| HTM
| style="text-align: center;" | 23
|
|
| F U F' L B2 F2 R2 B' U' R2 L' B2 R B' F2 U' R' F R' B2 L2 F2 R
|
|-
| <U,L,F2,R,B2,D>
| HTM
| style="text-align: center;" | 20
|
|
| D2 L D2 F2 D R' D' R' L2 B2 R2 U2 D R2 L D' L' D F2 B2
|
|-
| rowspan=2 | <U,M>
| STM
| colspan=3 style="text-align: center;" | 20
| U M U M U M U2 M' U2 M U M' U' M U' M' U' M U' M2
|
|-
| SQTM
| colspan=3 style="text-align: center;" | 23
| U M U M U M U M U' M U M' U M U M' U M U' M' U2 M
|
|-
| rowspan=2 | <U,r>
| HTM
| colspan=3 style="text-align: center;" | 30
| U r U r U' r2 U' r2 U2 r U r U r U2 r U2 r U2 r2 U2 r2 U2 r U2 r U2 r' U2 r2
| [https://benwh1.github.io/web/twisty_puzzles/3x3/3x3_U_r/index.html]
|-
| QTM
| colspan=3 style="text-align: center;" | 38
| r2 U' r U r' U' r' U r2 U r U r U r U r' U r U2 r U' r2 U2 r U2 r' U2 r' U2
| [https://benwh1.github.io/web/twisty_puzzles/3x3/3x3_U_r/index.html]
|-
| <U,R,r>
| HTM
| style="text-align: center;" | 22
|
|
| U2 R2 U R2 U r' U' r U' r' U2 R2 U r U R U2 r U R' U r2
|
|-
| <u,r>
| HTM
| style="text-align: center;" | 36
|
|
| u r u r u2 r u' r u' r2 u' r u r2 u r u r u' r2 u2 r' u' r' u' r2 u' r u r2 u2 r u2 r u2 r2
|
|-
| <u,f,r>
| HTM
| style="text-align: center;" | 27
|
|
| u2 f u r f' u2 r u2 f' r2 u r f u' r2 f r2 u' f2 r f u r' f2 u f2 r'
|
|-
| <U,L,F,R,B,D>, EO solved on all axes
| HTM
| colspan=3 style="text-align: center;" | 20
| U F R2 D' F' L2 U F' B' U F' B' U L2 B' D' F R2 U F'
| [https://cube20.org/distance20s/]
|-
| rowspan=4 | 4x4x4
| rowspan=2 | <U,2R>
| STM
| colspan=3 style="text-align: center;" | 28
| 2R2 U2 2R U2 2R2 U 2R' U2 2R' U2 2R U 2R' U' 2R2 U' 2R' U2 2R U' 2R U2 2R U 2R U' 2R' U'
| [https://benwh1.github.io/web/twisty_puzzles/4x4/4x4_2R_U/index.html]
|-
| SQTM
| colspan=3 style="text-align: center;" | 35
| U' 2R' U2 2R U' 2R' U2 2R U2 2R' U' 2R' U 2R' U' 2R' U 2R U 2R' U 2R U' 2R U 2R' U2 2R' U' 2R' U'
| [https://benwh1.github.io/web/twisty_puzzles/4x4/4x4_2R_U/index.html]
|-
| rowspan=2 | <U,r>
| OBTM
| style="text-align: center;" | 50
|
|
| 7 edge flip + swap DFr, DBr
|
|-
| OBQTM
| style="text-align: center;" | 62
|
|
| r U2 r U r' U' r2 U' r' U2 r2 U2 r U' r' U' r' U2 r' U r U' r' U r U2 r U r' U r' U r2 U2 r U' r2 U2 r U r U r' U2 r U' r U2 r'
|
|-
| rowspan=3 | Megaminx
| rowspan=2 | <U,R>
| HTM
| colspan=3 style="text-align: center;" | 26
| U R U' R2' U R U2' R2' U2' R' U' R U' R2 U R' U2' R' U2' R' U R2' U R2 U2 R'
| [https://benwh1.github.io/web/twisty_puzzles/megaminx/megaminx_RU/index.html]
|-
| QTM
| colspan=3 style="text-align: center;" | 34
| R2' U' R U2' R2' U R U' R' U R' U R2 U2' R2 U' R U R' U' R U' R U2' R' U2'
| [https://benwh1.github.io/web/twisty_puzzles/megaminx/megaminx_RU/index.html]
|-
| <U,F,R>
| HTM
| style="text-align: center;" | 27
|
|
| (F R2 U)9
|
|}

==== Steps/generation reduction ====

{| class=wikitable
|-
! Puzzle
! Subset
! Metric
! style="width:10%;" | Lower Bound
! style="width:10%;" | Upper Bound
! style="width:10%;" | Conjectured Value
! State Realizing Lower Bound
! Source
|-
| rowspan=3 | 2x2x2
| Face
| HTM
| colspan=3 style="text-align: center;" | 5
| 
| [https://www.lid.cubing.pt/2x2_fl.html]
|-
| Guimond Step 1
| HTM
| colspan=3 style="text-align: center;" | 3
| 
| [https://www.lid.cubing.pt/2x2_fl.html]
|-
| Guimond Step 1 & 2 (Reducing to <U, R2, F2>)
| HTM
| colspan=3 style="text-align: center;" | 6
| 
| [https://www.lid.cubing.pt/2x2_fl.html]
|-
| rowspan=5 | 3x3x3
| Cross
| HTM
| colspan=3 style="text-align: center;" | 8
| 
| [http://cubezone.be/crossstudy.html]
|-
| First Block (fixed)
| HTM
| colspan=3 style="text-align: center;" | 9
| 
| [https://docs.google.com/spreadsheets/d/1EectP3O_qwQp_2WohbDrlohxhs2ee17sjFfxZr_-D1g/edit]
|-
| EO
| HTM
| colspan=3 style="text-align: center;" | 7
| F D L U B R' F
| 
|-
| EOLine
| HTM
| colspan=3 style="text-align: center;" | 9
| B' R F' D' L' U' B' R' D'
| [https://media.discordapp.net/attachments/478546377262104607/736523655231569930/Screenshot_20200417-231004_Discord.png?ex=670308db&is=6701b75b&hm=a32f340199b32ee5b54863a3756c2958322baae7025bb54fac62a1963613056f&]
|-
| EOCross
| HTM
| colspan=3 style="text-align: center;" | 10
| B D2 R F U B D2 R F D
| [https://media.discordapp.net/attachments/478546377262104607/736523655231569930/Screenshot_20200417-231004_Discord.png?ex=670308db&is=6701b75b&hm=a32f340199b32ee5b54863a3756c2958322baae7025bb54fac62a1963613056f&]
|-
|}

==== Minxes ====

{| class=wikitable
|-
! Puzzle
! Metric
! style="width:10%;" | Lower Bound
! style="width:10%;" | Upper Bound
! style="width:10%;" | Conjectured Value
! State Realizing Lower Bound
! Source
|-
| rowspan=2 | Kilominx
| HTM
| style="text-align: center;" | 18
| style="text-align: center;" | 31
|
|
| [https://www.speedsolving.com/threads/kilominx-gods-number-bounds.67278/#post-1441983]
|-
| QTM
| style="text-align: center;" | 22
| style="text-align: center;" | 42
|
|
| [https://www.speedsolving.com/threads/kilominx-gods-number-bounds.67278/#post-1441983]
|-
| Master Kilominx
| OBTM
| style="text-align: center;" | 94
|
|
|
| [https://www.speedsolving.com/threads/dodecahedron-gods-numbers.87369/]
|-
| Gigaminx
| OBTM
| style="text-align: center;" | 152
|
|
|
| [https://www.speedsolving.com/threads/dodecahedron-gods-numbers.87369/]
|-
| Elite Kilominx
| OBTM
| style="text-align: center;" | 217
|
|
|
| [https://www.speedsolving.com/threads/dodecahedron-gods-numbers.87369/]
|-
| Teraminx
| OBTM
| style="text-align: center;" | 299
|
|
|
| [https://www.speedsolving.com/threads/dodecahedron-gods-numbers.87369/]
|-
| rowspan=2 | Pyraminx Crystal
| HTM
| style="text-align: center;" | 41
|
|
|
| [https://www.speedsolving.com/threads/dodecahedron-gods-numbers.87369/]
|-
| QTM
| style="text-align: center;" | 50
|
|
|
| [https://www.speedsolving.com/threads/dodecahedron-gods-numbers.87369/]
|}

==== Cuboids ====

{| class=wikitable
|-
! Puzzle
! Metric
! style="width:10%;" | Lower Bound
! style="width:10%;" | Upper Bound
! style="width:10%;" | Conjectured Value
! State Realizing Lower Bound
! Source
|-
| 1x2x2
| 
| colspan=3 style="text-align: center;" | 3
| R U R
|
|-
| 1x2x3
| 
| colspan=3 style="text-align: center;" | 6
| R U R U R U
|
|-
| 1x3x3
| HTM
| colspan=3 style="text-align: center;" | 8
| U L U R L U L D
| [https://www.jaapsch.net/puzzles/floppy.htm]
|-
| rowspan=2 | 2x2x3
| HTM
| colspan=3 style="text-align: center;" | 14
| U R2 U R2 U2 R2 F2 U F2 D2 R2 U R2 D
| [https://www.jaapsch.net/puzzles/cube223.htm]
|-
| QTM
| colspan=3 style="text-align: center;" | 15
| U R2 U R2 U' R2 F2 U' R2 U' R2 U F2 U' R2
| [https://www.jaapsch.net/puzzles/cube223.htm]
|-
| rowspan=2 | 2x3x3
| HTM
| colspan=3 style="text-align: center;" | 18
| U R2 U F2 U2 R2 U' B2 L2 U2 L2 U' L2 F2 U' L2 F2 L2
| [https://www.jaapsch.net/puzzles/domino.htm]
|-
| QTM
| colspan=3 style="text-align: center;" | 19
| U R2 U R2 U2 F2 U' F2 L2 B2 R2 D L2 U2 F2 U' R2
| [https://www.jaapsch.net/puzzles/domino.htm]
|}

==== Sliding Puzzles ====

{| class=wikitable
|-
! Puzzle
! Metric
! style="width:10%;" | Lower Bound
! style="width:10%;" | Upper Bound
! style="width:10%;" | Conjectured Value
! State Realizing Lower Bound
! Source
|-
| 2x2
|
| colspan=3 style="text-align: center;" | 6
| R D L U R D
| [https://web.archive.org/web/20140326092006/http://juropollo.xe0.ru/stp_results_mxn_a123_en.htm]
|-
| 3x2
| STM
| colspan=3 style="text-align: center;" | 21
| R2 D L U L D R2 U L D L U R2 D L U L D
| [https://web.archive.org/web/20140326092006/http://juropollo.xe0.ru/stp_results_mxn_a123_en.htm]
|-
| 4x2
| STM
| colspan=3 style="text-align: center;" | 36
| R D R U R D L U L D R U R D L3 U R2 D L2 U R3 D L3 U R3 D
| [https://web.archive.org/web/20140326092006/http://juropollo.xe0.ru/stp_results_mxn_a123_en.htm]
|-
| 5x2
| STM
| colspan=3 style="text-align: center;" | 55
| R2 D L U R2 D L3 U R3 D R U L D L3 U R3 D R U L D L2 U R3 D L4 U R2 D L2 U R4 D
| [https://web.archive.org/web/20140326092006/http://juropollo.xe0.ru/stp_results_mxn_a123_en.htm]
|-
| 6x2
| STM
| colspan=3 style="text-align: center;" | 80
| R2 D L U R D R2 U L2 D L2 U R2 D R3 U L3 D R3 U L2 D L3 U R3 D R2 U L2 D L3 U R4 D L4 U R5 D L5 U R5 D
| [https://web.archive.org/web/20140326092006/http://juropollo.xe0.ru/stp_results_mxn_a123_en.htm]
|-
| 7x2
| STM
| colspan=3 style="text-align: center;" | 108
| D R U R2 D R U R2 D L5 U L D R U R5 D L5 U L D R U R4 D L5 U R6 D L6 U R6 D L5 U R5 D L6 U R2 D L U R3 D L4 U R6
| [https://web.archive.org/web/20140326092006/http://juropollo.xe0.ru/stp_results_mxn_a123_en.htm]
|-
| 8x2
| STM
| colspan=3 style="text-align: center;" | 140
| D R3 U L2 D L U R D R5 U L5 D R5 U L4 D L2 U R2 D R5 U L5 D R5 U L4 D R4 U L3 D L4 U R4 D R3 U L3 D L4 U R4 D L4 U R5 D L5 U R6 D L6 U R7 D
| [https://web.archive.org/web/20140326092006/http://juropollo.xe0.ru/stp_results_mxn_a123_en.htm]
|-
| 3x3
| STM
| colspan=3 style="text-align: center;" | 31
| D2 R2 U2 L2 D R U L D2 R U R D L U R U L2 D2 R U2 L D
| [https://web.archive.org/web/20140326092006/http://juropollo.xe0.ru/stp_results_mxn_a123_en.htm]
|-
| 4x3
| STM
| colspan=3 style="text-align: center;" | 53
| D R D R U2 L2 D R D L U R2 D L2 U R3 D L2 U R2 U L3 D R3 U L3 D R2 U L D2 L U2 R3
| [https://web.archive.org/web/20140326092006/http://juropollo.xe0.ru/stp_results_mxn_a123_en.htm]
|-
| 5x3
| STM
| colspan=3 style="text-align: center;" | 84
| D R4 U L4 D R2 U L D2 R U L D L U R3 D L2 U R3 D L4 U R U L D R4 D L4 U R3 U L2 D R2 U R D L2 U R D L D L2 U R D L U2 R4
| [https://web.archive.org/web/20140326092006/http://juropollo.xe0.ru/stp_results_mxn_a123_en.htm]
|-
| 4x4
| STM
| colspan=3 style="text-align: center;" | 80
| D R2 D2 L U L U R U R2 D L2 D R U2 R D2 L D R U L U L D R D L2 U R D L U3 R D L U R3 D L D L U2 R D3 L U3 R2 D L D2 L U L U2 R3 D3
| [https://web.archive.org/web/20140326092006/http://juropollo.xe0.ru/stp_results_mxn_a123_en.htm]
|-
| 5x5
| STM
| style="text-align: center;" | 152
| style="text-align: center;" | 182
|
|
| [http://forum.cubeman.org/?q=node/view/559#comment-3116]
|-
| 6x6
| STM
| style="text-align: center;" | 230
| style="text-align: center;" | 382
|
|
|
|-
| 7x7
| STM
| style="text-align: center;" | 352
| style="text-align: center;" | 693
|
|
|
|}

{| class=wikitable
|-
! Puzzle
! Metric
! style="width:10%;" | Lower Bound
! style="width:10%;" | Upper Bound
! style="width:10%;" | Conjectured Value
! State Realizing Lower Bound
! Source
|-
| Nx2
| STM
| style="text-align: center;" | 2N^2 - N
| style="text-align: center;" | 2N^2 + O(N log(N))
|
| 180 degree rotation
|
|-
| NxN
| STM
| style="text-align: center;" | N^3 - 2N + 2 - (N if N is odd, 0 otherwise)
| style="text-align: center;" | 5N^3
| style="text-align: center;" | (4/3 + o(1))N^3
| 180 degree rotation
|
|}

==== Circle Puzzles ====

{| class=wikitable
|-
! Puzzle
! Metric
! style="width:10%;" | Lower Bound
! style="width:10%;" | Upper Bound
! style="width:10%;" | Conjectured Value
! State Realizing Lower Bound
! Source
|-
| rowspan=2 | Eliac
| HTM
| style="text-align: center;" | 71
|
| style="text-align: center;" | 71
| R2 L2 R2 L R2 L R2 L' R2 L2 R2 L R2 L' R2 L' R2 L' R2 L2 R2 L R2 L2 R2 L' R2 L R2 L' R2 L2 R2 L2 R2 L R2 L' R2 L2 R2 L R2 L2 R2 L R2 L' R2 L2 R2 L R2 L' R2 L2 R2 L' R2 L2 R2 L2 R2 L' R2 L2 R2 L R2 L' R2
| [http://forum.cubeman.org/?q=node/view/570]
|-
| QTM
| style="text-align: center;" | 80
|
| style="text-align: center;" | 80
| R2 L R2 L' R2 L' R2 L R2 L2 R2 L' R2 L R2 L' R2 L R2 L2 R2 L2 R2 L2 R2 L2 R2 L' R2 L' R2 L2 R2 L R2 L2 R2 L' R2 L R2 L2 R2 L2 R2 L' R2 L2 R2 L' R2 L2 R2 L' R2 L2 R2 L R2 L R2 L' R2 L' R2 L' R2 L'
| [http://forum.cubeman.org/?q=node/view/570]
|-
| rowspan=2 | Rashkey 1
| HTM
| colspan=3 style="text-align: center;" | 12
|
| [https://www.jaapsch.net/puzzles/rashkey.htm]
|-
| QTM
| colspan=3 style="text-align: center;" | 15
|
| [https://www.jaapsch.net/puzzles/rashkey.htm]
|-
| rowspan=2 | Rashkey 2
| HTM
| colspan=3 style="text-align: center;" | 19
|
| [https://www.jaapsch.net/puzzles/rashkey.htm]
|-
| QTM
| colspan=3 style="text-align: center;" | 23
|
| [https://www.jaapsch.net/puzzles/rashkey.htm]
|-
| rowspan=2 | Rashkey 3
| HTM
| colspan=3 style="text-align: center;" | 26
| L' R2 L2 R' L2 R2 L R L R' L' R' L2 R L' R' L R' L R' L' R L' R' L' R' 
| [https://benwh1.github.io/web/circle_puzzles/rashkey_3/index.html]
|-
| QTM
| colspan=3 style="text-align: center;" | 31
| L' R2 L2 R' L2 R2 L R L R' L' R' L2 R L' R' L R' L R' L' R L' R' L' R' 
| [https://benwh1.github.io/web/circle_puzzles/rashkey_3/index.html]
|}

==== Other ====

{| class=wikitable
|-
! Puzzle
! Metric
! style="width:10%;" | Lower Bound
! style="width:10%;" | Upper Bound
! style="width:10%;" | Conjectured Value
! State Realizing Lower Bound
! Source
|-
| rowspan=2 | 3x3x3 Supercube
| HTM
| style="text-align: center;" | 24
| style="text-align: center;" |
| style="text-align: center;" | 24
| U F U L D B2 R2 D2 L D2 L D2 B2 F U2 R' B U' R2 B L' D' U' R'
| [http://forum.cubeman.org/?q=node/view/536]
|-
| QTM
| style="text-align: center;" | 28
| style="text-align: center;" |
| style="text-align: center;" | 28
| R B' U2 R B D L B' R D' L F R2 D' F U R' B' U R' U D' R' F' U' F 
| [http://forum.cubeman.org/?q=node/view/536]
|-
| 3x3x3 2 color cube (U,F,B / L,R,D faces)
| HTM
| colspan=3 style="text-align: center;" | 13
| F' D2 R F2 U D' B' R' F' D F2 U' L'
| [https://benwh1.github.io/web/twisty_puzzles/3x3/2_color_cube/UFB/index.html]
|-
| 3x3x3 2 color cube (U,F,R / L,B,D faces)
| HTM
| colspan=3 style="text-align: center;" | 13
| U L2 U2 B F R' D2 B2 L U2 R2 B2 L'
| [https://benwh1.github.io/web/twisty_puzzles/3x3/2_color_cube/UFR/index.html]
|-
| rowspan=3 | 3x3x3 3 color cube (opposite faces the same color)
| HTM
| colspan=3 style="text-align: center;" | 15
| U F R2 B U' R F U2 D F R2 F D' B R
| [http://forum.cubeman.org/?q=node/view/248]
|-
| QTM
| colspan=3 style="text-align: center;" | 17
| U F2 B R B U' F B L U' F' L' B R' F D
| [http://forum.cubeman.org/?q=node/view/248]
|-
| STM
| colspan=3 style="text-align: center;" | 14
| U B L S R' U D2 R B' F2 M' F' U' R'
| [http://forum.cubeman.org/?q=node/view/248]
|-
| Redi Cube
|
| colspan=3 style="text-align: center;" | 19
|
| [https://www.speedsolving.com/threads/redi-cube-discussion-help-thread.65452/page-7#post-1263980]
|-
| Master Pyraminx
| OBTM
| style="text-align: center;" | 20
| style="text-align: center;" | 26
| style="text-align: center;" | 20
| Uw Rw Lw Bw' L R' Rw Bw U' Rw Uw' B' L' Uw' R' U' Rw' B' L' Bw
| [https://www.speedsolving.com/threads/master-pyraminx-gods-number-bounds.74056/]
|-
| rowspan=2 | Gear Cube
| HTM
| colspan=3 style="text-align: center;" | 6
|
| [https://www.jaapsch.net/puzzles/gearcube.htm]
|-
| QTM
| colspan=3 style="text-align: center;" | 12
|
| [https://www.jaapsch.net/puzzles/gearcube.htm]
|-
| Gear Cube Extreme
| HTM
| style="text-align: center;" | 17
| style="text-align: center;" | 25
| style="text-align: center;" | 17
|
| [http://forum.cubeman.org/?q=node/view/561]
|-
| rowspan=2 | Rubik's Ufo
| FTM
| colspan=3 style="text-align: center;" | 14
|
| [https://www.jaapsch.net/puzzles/rubufo.htm]
|-
| 6thTM
| colspan=3 style="text-align: center;" | 16
|
| [https://www.jaapsch.net/puzzles/rubufo.htm]
|-
| Face-Turning Octahedron
| 
| style="text-align: center;" | 21
| style="text-align: center;" |
| style="text-align: center;" |
|
| [https://www.youtube.com/watch?v=YKrwFoKw5O8]
|}

== External links ==
* [http://cube20.org God's number in HTM is 20]
* [http://cube20.org/qtm God's number in QTM is 26]
* [https://en.wikipedia.org/wiki/God%27s_algorithm God's algorithm] in Wikipedia

[[Category:Puzzle theory]]