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F12: Dodecahedron

A "Dodecahedron" twisty-puzzle is, overall, a regular dodecahedron (no surprise here!) which has the following key properties:
  • 12 faces (each an identical regular pentagon, except for color) -- a regular pentagon is a polygon of 5 equal sides
  • 30 edges (each a line where two faces meet)
  • 20 vertices (each a point where three edges meet)
Like all twisty puzzles discussed on my site, each face is divided into multiple facets which are colored polygons of various shapes.  The goal is to arrange the facets (by twisting layers) such that every face shows a single color.  Most puzzles discussed on this page have twelve unique colors (one for each face), but there are some which duplicate colors in pairs and thus have only six unique colors.

There are several variations of this puzzle, most described in some detail further below.  But a quick summary is:
  • Kilominx / Flower Minx [n=2] (22+20*1=5 quadrilaterals [kites] per face * 12 faces = 60 facets; [{2}*5]*2 hexahedrons = 20 surface blocks)
  • Pyraminx Crystal [n=3] (10 polygons [5 triangles + 5 kites] per face * 12 = 120 facets; [{2+3}*5]*2 = 50 surface blocks)
  • Megaminx [n=3] (32+21*1=11 [various] polygons per face * 12 faces = 132 facets; [{2+3}*5+6]*2 = 62 surface blocks)
  • Master Kilominx [n=4] (5*[3+1]=20 [various] polygons per face * 12 faces = 240 facets; [{2+6+6}*5]*2 = 140 surface blocks)
  • Gigaminx [n=5] (52+21*3=31 [various] polygons per face * 12 faces = 3672 facets; [{2+3+6+12}*5+6]*2 = 242 surface blocks)
  • Elite Kilominx [n=6] (5*[5+3+1]=45 [various] polygons per face * 12 faces = 540 facets; [{2+12+18+6}*5]*2 = 380 surface blocks)
  • Teraminx [n=7] (72+22*3=61 [various] polygons per face * 12 faces = 732 facets; [{2+3+6+6+6+6+12+12}*5+6]*2 = 542 surface blocks)
An important thing to note about the Dodecahedron twisty-puzzles is that there is no way to divide up a face (pentagon shape) into equal-sized regular polygons, which is contrary to both the "lower-class" puzzle families (Pyraminx, Cube, Octahedron) and the "upper-class" puzzle family (Icosahedron).  However, the Kilominx (as a special exception of the Dodecahedron family) does have each face divided into identical (but irregular) quadrilaterals... all other Dodecahedron puzzles have each face divided into various types of polygons (like kite, pentagon, rhombus, trapezoid and/or triangle).

The Dodecahedron puzzles are all "face-turning" puzzles.  This is because the irregular corners can not possibly be twisted (they're "locked" in place by surrounding blocks [like the Cube and "true" Octahedron puzzles].

Trivia: for Dodecahedrons with an "odd n" [Megaminx / Gigaminx / Teraminx] each face has a regular pentagon as a center facet.  In contrast, Dodecahedrons with an "even n" (Kilominx, Master Kilominx, Elite Kilominx) have no "true" center facet, although most (all but Kilominx) have a set of 5 identical facets which are called "centers" (none are truly in the center, but all five in the set surround a central point).

Dodecahedron (12-sided solid)
Dodecahedron

  Kilominx (Flower Minx)  
This is a face-turning [n=2] twisty puzzle.  The Kilominx is the only Dodecahedron puzzle with identical facets (colored polygons) on each face (although the Pyraminx Crystal appears to have identical facets at first sight).  Each facet is, almost, a kite (a bi-isosceles quadrilateral).  In particular, two adjacent sides of each facet/kite (the "exterior" [2D] sides which align with the 3D edges) are both straight and of equal length, while the other two sides of each facet/kite (the "interior" sides) are not straight but are of equal length (but different in length from the "straight/exterior" sides).  Note if the facets had all "straight" sides, this puzzle could not rotate its face layers.  Even with its slightly distorted (2 of 4 sides in each facet) design, it looks like it could not twist, but  it actually (amazingly) can because the corners "overlap"  the lower layer (and avoid "collision" because of the non-straight "interior" sides). .I apologize for the poorly written description... take a look at the photo (on the right) to gain a better understanding.

The Kilominx has only 1 type of surface block, which seems to be a distorted cube (irregular hexahedron), but this is not quite accurate because the hidden facets are curved and not flat (which is a requirement to be, technically, a polyhedron).  Anyway, this puzzle has 5*2 + 2*5 = 20 visible blocks.  I've never disassembled one, but I'm willing to bet that the "core" isn't a polyhedron block either.  It has 5*12 = 60 visible facets (colored polygons).  Thus it has exactly 3 visible facets per surface block.

Flower-Minx

The Kilominx has 20!/2*319 ("corner" blocks) / 60 (no central orientation) = 23,563,902,142,421,896,679,424,000 (about 23.56 septillion) combinations. Although this puzzle has roughly a million more combinations than the classic Rubik's Cube, many (most?) players consider this puzzle to be easier!  Presumably this is because there are only "corner" blocks (no "edge" blocks, nor "center" blocks).


  Megaminx / Supernova
 
This face-turning puzzle [n=3] has three different types of colored polygons ("facets") per face:
  • 5 Rhombus (diamond) "corners" (unlike the Kilominx, each corner facet is a true polygon).
  • 5 "edges" -- the Megaminx versions have an Isosceles Trapazoid shape as facets on the edge blocks, while the Supernova versions have an Isosceles Triangle as facets on the edge blocks (the Supernova has larger "corners" and smaller "centers" to compensate).
  • 1 Regular Pentagon "center"

So each face is divided into 11 facets (colored polygons).  There are a total of 11*12 = 132 visible facets which are distributed across 11*2 + 4*5*2 = 62 surface blocks (approximately 2.129 visible facets per surface block).

The Megaminx has 3 types of blocks:
  • 20 "corner" blocks (with 3 visible facets, each a rhombus)
  • 30 "edge" blocks (with 2 visible facets, each is a trapezoid [if Megaminx] or a triangle [if Supernova])
  • 12 "center" blocks (with 1 visible facet, each is a pentagon)

Note the "Megaminx" and "Supernova " play exactly the same way.  The main difference is in the shape ("artistic expression") of the "edge" blocks.

In either case, the puzzle has 20!/2*319 ("corner" blocks) * 30!/2*229 ("edge" blocks) = 100,669,616,553,523,347,122,516,032,313,645,505,168,688,116,411,019,768,627,200,000,000,000 (about 100.7 unvigintillion [100.7*1066]) combinations.  Trivia: this puzzle has about 2.3 trillion trillion trillion trillion (oh my God!) more combinations than  a Rubik's Cube... yet many players find this puzzle to be "about" the same difficulty as a Rubik's Cube (these players usually acknowledge it takes longer to solve a Megaminx, but mainly because it has 132 [instead of 54]  facets, and not because of the "intrinsic" difficulty).

Although the shape of the Megaminx is radically different than a Rubik's Cube, it has a lot of similarities (with some features simply increased): 3 kinds of blocks ("centers", "corners", and "edges" -- same as the Cube), 6*2=12 faces (instead of 6) and 11 (instead of 9) polygons per face.  If you can solve a Rubik's Cube with the "layer by layer" method, then you should easily be able to solve all but the final layer of the Megaminx / Supernova.  The final layer can be solved with the same strategy as the cube (first solve the corners then the edges), but the algorithms needed are different.

Photo of "Megaminx" dodecahedron
Megaminx (note each "edge" facet is a trapezoid and not a triangle)
 
Photo of "Supernova" dodecahedron
Supernova (note each "edge" facet is an isosceles triangle)

  Pyraminx Crystal
 

This is a face-turning [n=3] puzzle, but each "twist" will make a "deeper" cut than the similar Megaminx.  Each twist will rotate 15 blocks, which is 50% more than twisting a face of the Megaminx (disregarding the Megaminx's center block which simply rotates in-place)!  Because of this, the Pyraminx Crystal plays differently than most of the others on this page (for example, a face-twist of the Megaminx will move only 10 surface blocks. 

At first glance, it appears each facet is the same (triangular) but a short amount of scrutiny reveals that 40% of the facets (those on the corner blocks) are actually kite-shaped.  Thus it is similar to the Kilominx which has truly identical [but distorted] kites as facets (and also lacks center facets).  The name, "Pyraminx Crystal", is unfortunate because it has little in common with the Pyraminx except that all the facets resemble triangles (only 60% of this puzzle's facets are truly triangles).

Each face is divided into ten facets (colored polygons): five kites and five isosceles triangles.  There are two types of surface blocks:
  • 20 irregular heptahedron (a.k.a. septahedron) "corners".  These 7-faced blocks have three visible, kite-shaped facets and four hidden, triangular facets.  Three of the triangular facets ("wide" isosceles) are partially visible when twisting and the fourth triangular facet (equilateral) is completely hidden (attaches to a face of the "core").
  • 30 irregular tetrahedron "edges".  These 4-faced blocks have isosceles triangular facets on all sides, but they are irregular because the two visible facets are "narrow" (short uncommon side) while the two hidden facets are "wide" (long uncommon side).  These blocks have one hidden edge which coincides with an edge of the "core".

Thus, there are a total of 10*12 = 120 visible facets which are distributed across 20+30 = 50 surface blocks (exact average of 2.4 visible facets per surface block).  The diagram on the righted is an "exploded" view of the blocks which compose a Pyraminx Crystal.  The "core" is shown as a transparent (white) icosahedron.  In theory, this would be a corner-turning mini-puzzle (and not a solid block like the core of a Rubik's Cube or Pyraminx).  In reality, disassembling this puzzle will probably reveal a sphere with movable "holes" (where the corner blocks attach and which move in groups of five) -- see the photo of the one I disassembled below.

Importantly, there are no "center" blocks.  This requires the player to use a bit more mental effort to orient the puzzle.  So like the (original/Master/Elite) Kilominx, the easy solution is to use corner blocks with "matching" colors for orientation.  The central point of a face (where all 10 facets converge) is a vertex of the "core".

The Pyraminx Crystal has 20!/2*319 ("corner" blocks) * 30!/2*229 ("edge" blocks) / 60 (no fixed orientation) = 1,677,826,942,558,722,452,041,933,871,894,091,752,811,468,606,850,329,477,120,000,000,000 (about 1.678 unvigintillion [1.678*1066]) combinations.  This puzzle has only 60 (5*12) times fewer combinations than a Megaminx / Supernova due to lack center pieces (no defined orientation of the whole puzzle).  Based on number of combinations, this puzzle should be the easier of the two, but because of the deeper cuts and missing centers (which can create "parity errors") many people consider this puzzle more difficult than a Megaminx.  I don't own a Void Megaminx so I don't discuss it separately, but I imagine it would have the same number of combinations as the Pyraminx Crystal and be considered "more difficult" too.  Trivia: the Void Cube has 24 (4*6) times fewer combinations than a Rubik's Cube, but is also often considered "more difficult" for one of the same reasons (no centers can lead to "parity errors").


Pyraminx Crystal

Diagram of the 50 surface blocks (and the "core") which compose a Pyraminx Crystal

  Master Kilominx
 
This face-turning puzzle [n=4] (also known as "Professor Kilominx" or "4x4 Megaminx") is an expanded form of the [n=3] Megaminx.  Not only does each face twist, but also the layer "below" a face.  Thus it is a "2-layer" puzzle (like the Gigaminx).  This puzzle has three different types of colored polygons ("facets") per face:
  • 5 Rhombus (diamond) "corners"  -- these are regular, like the Megaminx "corners" [unlike Kilominx "corners"]
  • 10 Right Trapazoid "edges" -- there are actually 2 types (5 each) and the two types are mirror images of each other
  • 5 Quadrilateral "centers" -- each is a regular kite in the MF8 version, or a "curved kite" in the ShengShou version (like the "corners" of the Kilominx)

So each face is divided into 20 facets (colored polygons).  There are a total of 20*12 = 240 visible facets which are distributed across [{2+6+6}*5]*2 = 140 surface blocks (approximately 1.714285 visible facets per surface block).  I have seen two variants (I don't own either): one by "ShengShou" [first photo at right] and another by "MF8" [second photo at right].  Both versions allow not only a face to be twisted, but also the layer below that one.  This explains the curved centers of ShengShou's version, but interestingly the MF8 version (with kite-shaped centers) can also twist two layers.  It doesn't look possible, but if you watch a video (or play with one yourself), you can see how the center blocks sneak under/behind edge blocks... at least in theory.  The MF8 version is notorious for center facets popping of the puzzle (due to collisions).  So I think the MF8 version looks "better" (all facets are true polygons, and no big center "gap"), it seems ShengShou's version is easier and more reliable to twist.

The Master Kilominx has 3 types of surface blocks:
  • 20 "corner" blocks (with 3 visible facets, each a rhombus)
  • 60 "edge" blocks (with 2 visible facets, each is a trapezoid)
  • 60 "center" blocks (with 1 visible facet, each is a kite [approximately for "ShengShou" and literally for "MF8"])
I believe (can't find any confirmation) the Master Kilominx has 20!/2*319 ("corner" blocks) * 60!/2 ("edge" blocks) * 60!/[5!12] ("center" blocks) / 60 ("orientation") = (approximately) 9.1493862271246433856073350438306 * 10163 combinations.  Note the "edge" blocks come in two different (yet symmetrical) types, so any position of an "edge" block has only one orientation, which explains why the factor for "edge" blocks (given above) doesn't include multiplication by a power of two (contrary to Megaminx "edges").  Trivia: this puzzle has about 9.149 googol vigintillion combinations (it probably sounds like I made this up, but do the research to see I am telling the truth)!

Although the shape of the Master Kilominx is radically different than a Rubik's Revenge, it has a lot of similarities.  In particular, "parity" errors are a problem for many new players of Rubik's Revenge because any two adjacent "edge" blocks appear identical in shape (and thus seem interchangeable).  The same "error" can occur with adjacent "edge" blocks of the Master Kilominx, but their non-identical (yet symmetrical) shapes should make it obvious that two adjacent blocks can not be simply swapped "in-place" -- reversing their position also requires "flipping" their orientation!  So this puzzle doesn't have the same type of parity error as a Rubik's Revenge, but it does have parity errors.  In particular, correctly paired edges can be arranged in sets to form patterns impossible on the lower-order Megaminx.

Photo of a "ShengShou" Master Kilominx (n=4 dodecahedron)
"ShengShou" Master Kilominx

(note irregular "centers"
around big pentagram "gaps")

Photo of a "MF8" Master Kilominx (n=4 dodecahedron)
"MF8" Master Kilominx

(note kite "centers"
around tiny pentagon "gaps")

  Gigaminx  
This face-turning [n=5] puzzle is an expanded form of the Megaminx [n=3].  Not only can each face twist, but also the layer below a face.  Thus it is a "2-layer" puzzle (like the Master Kilominx). It has four different types of colored polygons ("facets") per face:
  • 20 Rhombus (diamond) facets of 3 types: 5 "corners", 10 "wings" (each polygon sharing a side with both a "corner" and an "edge"), and 5 "central corners" (each polygon sharing a corner-point with both a "corner" and the "center")
  • 5 Isosceles Triangle "edges"
  • 5 Isosceles Trapazoid "central edges"
  • 1 Regular Pentagon "center"

So each face is divided into 31 facets (colored polygons).  There are a total of 31*12 = 372 visible facets which are distributed across 31*2 + 4*5*2 + 10*5*2 + 4*5*2 = 242 surface blocks (approximately 1.537 visible facets / surface block).

The Gigaminx has six different block types (and these are only the surface blocks):
  • 20 "corner" blocks (with 3 visible facets, each a rhombus)
  • 30 "edge" blocks (with 2 visible facets, each an isosceles triangle)
  • 60 "wing" blocks (with 2 visible facets, each a rhombus)
  • 60 "central corners" (with 1 visible facet: a rhombus)
  • 60 "central edges" (with 1 visible facet: an isosceles trapezoid)
  • 12 "center" blocks (with 1 visible facet: a regular pentagon)
This puzzle has 20!/2*319 ("corner" blocks) * 30!/2*229 ("edge" blocks) * 60!/2 ("wing" blocks) * 60!/(5!12) ["central corner" blocks] * 60!/(5!12) ["central edge" blocks"] ≈ 3.6479011530733075986308161037232 * 10263 combinations.  This is an unnameable (unholy) number!  The number of combinations is far bigger in size than "astronomical" (the count of all electrons and quarks in the visible universe).  I can't really describe the insanity of this ultra-huge number, but I can name it: approximately 3.6479 googol googol vigintillion.

Tangent topic: the number of combinations is also far greater than possible with 512-bit encryption.  So, if you are serious about data security, consider an n=5 dodecahedron encryption scheme!

Tangent topic 2: the largest confirmed atom [as of 2017] has only 117 electrons and 117 protons for a total of 234 charged particles (which is about 3.3% less than the count of Gigaminx surface blocks).  The nucleus of such an atom (Tennessine), on the other hand, has 294 nucleons (protons and neutrons) which is about 25% more than the number of Gigaminx surface blocks.  In summary, one might be able to model the electron orbits with a Gigaminx, but this puzzle (at least its surface blocks) aren't sufficient to model the nucleus of the largest known atom.
Photograph of a Gigaminx puzzle
Gigaminx

  Elite Kilominx  
This face-turning [n=6] puzzle (also called a "6x6 Megaminx") is an expanded form of the Master Kilominx [n=4].  It can twist not only a face layer, but also the next two layers (independently) below each face (like the Teraminx).  Thus it is a "3-layer" puzzle.  The Elite Kilominx has four "different" types of colored polygons ("facets") per face:
  • 20 Rhombus (diamond) facets of 3 types: 5 "corners", 10 "wings" (each polygon sharing a side with a "corner"), and 5 "[outer] central corners" (each polygon sharing a corner-point with both a "corner" and a "center")
  • 10 Right Trapezoid "outer edges" -- these may be divided into 2 sets (of 5), each a mirror image of the other
  • 10 distorted trapezoid (almost Kite) "central edges" -- similar to the "outer edges" but one side is slightly curved
  • 5  almost Kite "[inner] centers" -- two sides are slightly curved

Note that similarly-shaped facets (like "Rhombus") are not identical in this puzzle.  This is because, like the "higher" version of this puzzle [the Teraminx], it can not be built from identical facets!  In this puzzle's case, the polygons are distorted in size but remain flat (while in the "higher" Teraminx the "polygons" are regular in size but curved ["pillowed"] in shape).

Each face is thus divided into 45 facets (colored polygons).  There are a total of 45*12 = 540 visible facets which are distributed across [{2+12+18+6}*5]*2 = 380 surface blocks (approximately 1.421052631578947368 visible facets / surface block).

The Elite Kilominx has six different block types (and these are only the surface blocks):
  • 20 "corner" blocks (with 3 visible facets, each a rhombus)
  • 60 "wing (corner-edge)" blocks (with 2 visible facets, each a rhombus)
  • 60 "[middle] edge" blocks (with 2 visible facets, each a right trapezoid)
  • 60 "[outer] central corner" blocks (with 2 visible facets, each a rhombus)
  • 120 "[outer] central edge" blocks (with 1 visible facet: a distorted trapezoid resembling both a kite and a rhombus)
  • 60 "[inner] center [corner]" blocks (with 1 visible facet: a distorted kite)
This puzzle has 20!/2*319 ("corner" blocks) * 60!/2 ("wing" blocks) * 60!/2 ("edge" blocks) * 60!/(5!12) ["central corner" blocks] * 120!/(10!12) ["central edge" blocks] * 60!/(5!12) ["center" blocks"] / 60 (orientation) ≈ 4.5579983569512343039431875039395 * 10422 combinations.  This is a virtually unnameable (unholy) number!  The number of combinations is far bigger in size than "astronomical" (the count of all electrons and quarks in the visible universe).  I can't really describe the insanity of this ultra-huge number!  Trivia: the number of combinations can technically be named as about 45.57998 googol googol googol googol sextillion.

Tangent topic: the largest confirmed atom [as of 2017], Tennessine, has only 117 protons and 177 neutrons for a total of 294 nucleons.  This is obviously less than the 380 surface blocks of an Elite Kilominx (and the total number of blocks is unknown by me, but much larger than 380).  In other words, this puzzle is more complex (in some sense[s]) than the most complex element known in the universe!
Photo of an "Elite" Kilominx (n=6 dodecahedron)
Elite Kilominx

n=6 dodecahedron

  Teraminx  
This face-turning [n=7] puzzle is an expanded form of the Gigaminx [n=5].  Not only does each face twist, but also the next two layers below a face can twist (independently).  Thus the Teraminx is a "3-layer" puzzle, like the Elite Kilominx.  I don't own one, and information on the internet about this one is lacking details.  But here is what I have learned:
  • This puzzle is "pillowed" -- the faces are curved (not flat) and thus this puzzle isn't technically a polyhedron -- also the facets aren't technically polygons (because a polygon is flat)
  • Each face has 9*5 = 45 apparently identical facets (each a rhombus), but the corners (in particular) are distorted from a "true" rhombus shape
  • Each face has 6*5 + 4*5 + 2*5 + 1 = 61 visible facets (colored polygons)
  • This puzzle has a total of 61 * 12 = 732 visible facets

The Teraminx has ten different types of blocks (and these are only the surface blocks):

  • 20 "corner" blocks, each with 3 visible facets (of distorted rhombus shape)
  • 30 "[middle] edge" blocks", each with 2 visible facets (of isosceles triangle shape)
  • 60 "middle wing" blocks, each with 2 visible facets (of rhombus shape)
  • 60 "corner wing" blocks, each with 2 visible facets (of rhombus shape)
  • 60 "outer center corner" blocks, each with 1 visible facet (rhombus shape)
  • 120 "outer center wing" blocks, each with 1 visible facet (rhombus shape)
  • 60 "outer center edge" blocks, each with 1 visible facet (isosceles trapezoid shape)
  • 60 "inner center corner" blocks, each with 1 visible facet (rhombus shape)
  • 60 "inner center edge" blocks, each with 1 visible facet (isosceles trapezoid shape)
  • 12 "center" blocks, each with 1 visible facet (regular pentagon)

Thus, this puzzle has a total of 542 surface blocks.  Consequently, the Teraminx has an average of (approximately) 1.35055 visible facets per surface block (732/542).

I haven't found any website which documents the number of combinations of this puzzle (but I haven't looked very hard).  Because I'm bored, and you might be interested, I will guess that the number of combinations is about 20!/2*319 ("corner" blocks) * 30!/2*229 ("edge" blocks) * 60!/2 ("middle wing" blocks) * 60!/2 ("corner wing" blocks) * 60!/(5!12) ["outer central corner" blocks] * 60!/(5!12) ["outer central edge" blocks] * 120!/(10!12) ["outer central wing" blocks] * 60!/(5!12) ["inner central corner" blocks] * 60!/(5!12) ["inner central edge" blocks] ≈ 1.6959973206910656736480025259487 * 10579Oh My God!!  Even if I am wrong by a factor of a thousand million billion trillion, I would be still be close to the true value.

Now I am truly at a loss for words... the only analogy I can imagine is from the movie "Spaceballs" where the imperial starship had 4 speeds:
  • Cruising Speed
  • Light Speed
  • Ridiculous Speed
  • Ludicrous Speed

The number of Teraminx combinations surely falls into the "Ludicrous" category!  If my calculation is correct, then the Teraminx has about 16.9599732 googol googol googol googol googol quinquavigintillion combinations...

(Solved) Teraminx puzzle (pillowed)
Teraminx

n=7 dodecahedron

© 2017, H2Obsession
Diagram of Dodecahedron from Wikipedia © 2005 Cyp (http://en.wikipedia.org/wiki/Regular_dodecahedron#/media/File:Dodecahedron.jpg)
Photograph of Kilominx ("Flower-Minx") from Wikipedia © 2011Hellbus (http://en.wikipedia.org/wiki/Megaminx#/media/File:Flowerminx_black.jpg)
Photograph of Megaminx © 1970-2017 Uwe Meffert (http://www.mefferts.com)
Photograph of "ShengShou" Master Kilominx © Aircee (http://www.amazon.com)
Photograph of "MF8" Master Kilominx © MF8 (http://www.amazon.com)
Photograph of Gigaminx from Wikipedia © 2009 Tetracube (http://en.wikipedia.org/wiki/Megaminx#/media/File:C4U-Gigaminx-01.jpg)
Photograph of Elite Kilominx © AliExpress (http://www.aliexpress.com/w/wholesale-6x6-cube.html)
Photograph of Teraminx from Wikipedia © 2010 Tetracube (http://en.wikipedia.org/wiki/Megaminx#/media/File:Teraminx.jpg)
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