### Twisty Puzzles

Learn cool stuff about various "twisty puzzles" on this section of my website.  Although there are many puzzle variations, I choose to focus on the "Platonic Solids".  Thus, further info can be found based on "geometric shape":
• Tetrahedron = 4 sides (each side is a triangle); example: Pyraminx
• Hexahedron = 6 sides (each side is a square); example: Rubik's Cube
• Octahedron = 8 sides (each side is a triangle); example: Skewb Diamond
• Dodecahedron = 12 sides (each side is a pentagon); example: Megaminx
• Icosahedron = 20 sides (each side is a triangle); example: Icosaminx

These are the only 5 "perfectly regular" solids possible in reality (3D-space).  Of course irregular (semi-regular) solids are possible, such as a hexahedron which is not a cube, like a "bi-pyramid".  Although a (regular) bi-pyramid hexahedron consists of sides which are all equilateral triangles (that sure sounds "regular"), some vertices consist of 3 triangles while others consist of 4 triangles (thus, "irregular").  So there are many other fun and interesting shapes which have been manufactured into twisty puzzles, but to keep this site manageable I prefer to focus only upon Platonic solids.  However, as a special exception, I do have a page about the "Buckyball" shape, more formally named a "truncated icosahedron", which is usually marketed in twisty-puzzle commerce as a "Tuttminx" (or similar).

It is generally (although not universally) agreed that the difficulty of a twisty-puzzle is highly (if not directly) related to the number of combinations possible. It is also generally agreed that overall orientation of the puzzle is not relevant (if you match the color of all facets on every side then the puzzle is considered "solved", no matter which color is on the top [front, left, right, etc.] face). With that in mind, I have tabulated the various "regular" puzzles (and "buckyballs") in order of "difficulty" (logarithm of combinations).

 Name Family Layers n Facets per Side Visible Facets Visible Blocks Total Blocks Combinations Difficulty Mini-minx F4:Tetrahedron (equilateral triangle facets) 1 2 4 (22) 16 (4*22) 5 (4+1) 5 81 (34) 1.908 Skewb Diamond F8:Octahedron (equilateral triangle facets) 1 2 4 (22) 32 (8*22) 14 (6+8) 14 138,240 (6!/2*25*4!/2) 5.141 Pocket Cube F6:Hexahedron (square facets) 1 2 4 (22) 24 (6*22) 8 (4*2) 8 (23) 3,674,160 (8!*37/24) 6.565 Dino Cube F6:Hexahedron (isosceles triangle facets) 1 2 4 (22) 24 (6*22) 12 (4*3) 21 (12+8+1) 19,958,400 (12!/2/12) 7.300 Pyraminx F4:Tetrahedron (equilateral triangle facets) 2 3 9 (32) 36 (4*32) 14 (4+4+6) 15 (14+1) 75,582,720 (34*34*6!/2*25) 7.878 Master Pyraminx F4:Tetrahedron (equilateral triangle facets) 3 4 16 (42) 64 (4*42) 30 (4+4+12+6+4) 34 (30+4) 217,225,462,874,112,000 (34*34*12!/2*6!/2*25*4!/2) 17.337 Helicopter Cube F6:Hexahedron (isosceles triangle facets) 1 2 8 (2*22) 48 (6*2*22) 32 (8+24) 45 (32+13) 493,694,233,804,800,000 (8!*37*[6!]4/2/24) 17.693 Void Cube F6:Hexahedron (square facets) 2 3 8 (32-1) 48 (6*8) 20 (8+12) 20 (20+0) 1,802,166,803,103,744,000 ? (8!*37*12!/2*211/24) 18.256 ? C-T ["Crystal"] Octahedron F8:Octahedron (equilateral triangle facets) 2 3 9 (32) 72 (8*32) 24 (6+6+2*3*2) 25 (24+1) 8,229,184,826,926,694,400 (46*46*12!/2*211) 18.915 Rubik's Cube F6:Hexahedron (square facets) 2 3 9 (32) 54 (6*32) 26 (8+12+6) 27 (33) 43,252,003,274,489,856,000 (8!*37*12!/2*211) 19.636 F-T Octahedron F8:Octahedron (equilateral triangle facets) 1 3 9 (32) 72 (8*32) 42 (6+12+3*8) 51 (42+9) 31,408,133,379,194,880,000,000 (6!/2*25*12!/2*[12!]2/[3!8]/12) 22.497 Kilominx (or Flowerminx) F12:Dodecahedron (quadrilateral facets) 1 2 5 (5*[1]) 60 (12*5) 20 (5*2*2) 52 (20+30) 23,563,902,142,421,896,679,424,000 (20!/2*319/60) 25.372 Professor's Pyraminx F4:Tetrahedron (equilateral triangle facets) 4 5 25 (52) 100 (4*52) 54 (4+4+18 +12+12+4) 63 ??? (54+9?) roughly 19.229 nonillion (1030) ? (34*34*6!/2*25*12!/2 *12!/2*12!/[3!4]*4!/2) 31.284 ? Rubik's Revenge F6:Hexahedron (square facets) 3 4 16 (42) 96 (6*42) 56 (8+24+4*6) 64 (43) roughly 7.401 quattuordecillion (1045) (8!*37*24!*24!/[4!6]/24) 45.869 Pyraminx Crystal F12:Dodecahedron (triangle and kite facets) 1 3 10 (5*2) 120 (12*10) 50 (20+30) 80 (50+30) roughly 1.678 unvigintillion (1066) (30!/2*228*20!*319/60) 66.225 Megaminx (or Supernova) F12:Dodecahedron (various polygon facets) 1 3 11 (5*2+1) 132 (12*11) 62 (20+30+12) 92 (62+30) roughly 100 unvigintillion (1066) (30!/2*228*20!*319) 68.003 Professor's Cube F6:Hexahedron (square facets) 4 5 25 (52) 150 (6*52) 98 (8+12+24 +24+24+6) 125 (53) roughly 282.87 trevigintillion (1072) (8!*37*12!/2*211*24! *24!/[4!6]*24!/[4!6]) 74.452 Dogic F20:Icosahedron (equilateral triangle facets) 1 2 4 (22) 80 (20*22) 80 (5*12+20) 112 ?? (80+30?) roughly 21.99 sesvigintillion (1081) (60!/[5!12]*20!/2*319/60) 82.342 V6 Cube F6:Hexahedron (square facets) 5 6 36 (62) 216 (6*62) 152 (8+24+24 +24+48+24) 216 (63) roughly 15.7 googol quadrillion (10115) (8!/2*37*24!*24! *24!/[4!6]*24!/[4!6]*24!/[4!6]/24) 116.20 Icosaix F20:Icosahedron (equilateral triangle facets) 1 3 9 (32) 180 (20*32) 102 (12+30+60) 132 (102+30) near 15.792 googol sextillion (10121) ? (12!/2*512*30!/2*229*60!/[3!20]/60) 122.20 ? Master Kilominx F12:Dodecahedron (various polygon facets) 2 4 20 (5*[3+1]) 240 (12*20) 140 (20+60+60) 170 (140+30) near 9.1494 googol vigintillion (10163) (20!/2*319*60!/2*60!/[5!12]/60) 163.96 Void Tuttminx F32:"Buckyball" (various polygon facets) 1 3 10 | 12 (5*2 | 6*2) 360 (12*10 +20*12) 150 (60+60+30) 150 (150+0) roughly 205.42 googol googol (10200) (60!/2*60!/2*30!/2*229/60) 202.31 Tuttminx F32:"Buckyball" (various polygon facets) 1 3 11 | 13 (5*2+1 | 6*2+1) 392 (12*11 +20*13) 182 (60+60+30 +32) ??? (182+??) roughly 12.325 googol googol  thousand (10203) (60!/2*60!/2*30!/2*229) 204.09 Gigaminx F12:Dodecahedron (various polygon facets) 2 5 31 (5*6+1) 372 (12*31) 242 (20+30+60 +60+60+12) 272 (242+30) roughly 3.6479*10263 (20!/2*319*30!*228 *60!/2*60!/[5!12]*60!/[5!12]) 263.56 Elite Kilominx F12:Dodecahedron (various polygon facets) 2 5 45 (5*[5+3+1]) 540 (12*45) 380 (20+60+60 +60+120+60) 410 (380+30) roughly 4.5580*10422 ? (20!/2*319*60!/2*60!/2 *60!/[5!12]*120!/[10!12]*60!/[5!12]/60) 422.66 ? Teraminx F12:Dodecahedron (various polygon facets) 3 7 61 (5*12+1) 732 (12*61) 542 (20+30+120 +60+60+120 +60+60+12) 572 (542+30) roughly 1.6960*10579 ? (20!/2*319*30!*228*120!/2 *60!/[5!12]*60!/[5!12]*120!/[10!12] *60!/[5!12]*60!/[5!12]) 579.23 ?

 Twisty Types

The regular (Platonic Solid) puzzles which I discuss on my site may be divided into 4 known types (or more, in theory) which affects the style of play used to solve them:

• Face-turning (F.T.) puzzles. Each face of the solid may be twisted either clockwise or counter-clockwise. Layer(s) parallel to a face may also be twisted in most puzzles. Importantly, the corners (and the layers parallel to the corners) may not be twisted. The most popular class of face-turning puzzles is the Rubik's Cube family (hexahedrons). The most common puzzles of the octahedron and dodecahedron ("Megaminx") are also Face-Turning puzzles.
• Corner-turning (C.T.) puzzles. Each corner may be twisted clockwise or counter-clockwise.  Layer(s) parallel to a corner may also be twisted in most puzzles.  Importantly, the faces of the solid may not be twisted.  The only examples I know for sure are the Dino Cube and the Crystal Octahedron (but there may be variants of the icosadhedron family which are strictly Corner-Turning puzzles). Importantly: no Platonic Solid puzzle which is strictly corner-turning can built from blocks (pieces) which are all Platonic Solids (hence, the overall puzzle would be compositionally irregular). For example, the Dino Cube is built from irregular tetrahedrons (the visible facets are not equilateral triangles) and the "crystal" octahedron (as another example) is built mainly from pentahedrons and decahedrons (neither shape is a Platonic Solid).
• Dual-turning (D.T.) puzzles allow each face to be turned either clockwise or counter-clockwise, and each corner to also be twisted clockwise or counter-clockwise.  Layer(s) parallel to each face and/or layer(s) parallel to each corner may also turn, depending on puzzle design.  The only puzzles that allow all layers to twist is the tetrahedron ("Pyraminx") family of puzzles.  The reason only the tetrahedron puzzles allow dual-turning of all layers is because each face is opposite a corner (and vice versa).  All other puzzles with a Platonic Solid shape have a face opposite a face (and a corner opposite a corner).  Please note that many (most?) people describe the Pyraminx family as "Corner-Turning (C.T.) puzzles", although they are really "Dual-Turning (D.T.) puzzles.  The only other dual-turning puzzles I know about are from the icosahedron family: the Dogic and the Icosaix.  However, they are both limited to a single layer.  This doesn't mean all icosahedron puzzles need to be dual-turning or single-layer.   Conceivably, an icosahedron puzzle could be made (if it hasn't already) which is strictly corner-turning in multiple layers (similar to a Crystal Octahedron).
• Edge-turning (E.T.) puzzles allow a line of adjacent blocks (all on the same edge) to be turned either clockwise or counter-clockwise.  These puzzles are uncommon, and the few I know about are single-layer.  In fact, I find it difficult to imagine a multi-layer Edge-Turning puzzle!  I also don't know of any puzzles that mix Edge-Turning with another method.  If any exist (or are later created), they would make my previous term (Dual-Turning) ambiguous (for example, an Edge-Turning and Corner-Turning puzzle would also be "Dual-Turning" but different from my definition of a Corner-Turning and Face-Turning puzzle).  Such exotic types would need new name(s)!
• Omni-turning (O.T.) puzzles don't exist, as far as I know.  Such things would allow 3 types of twisting: Corner-Turning, Edge-Turning, and Face-Turning.

Anyway, I hope you find the linked/sub pages give valuable information about twisting Platonic Solids!