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### F 4 : Tetrahedron

A "Tetrahedron" twisty-puzzle is, overall, a regular tetrahedron (no surprise there) which has the following key properties:
• 4 equilateral triangle faces (identical, except for color) -- an equilateral triangle is a regular polygon of 3 sides
• 6 edges (each a line where two faces meet)
• 4 vertices (each a point where three edges meet)
Note that a regular tetrahedron is a pyramid with a triangular base (and 4 faces).  This contrasts with the Great Pyramids of Egypt which each have a square base (and 5 faces).

Like all twisty puzzles discussed on my site, each face is divided into multiple facets which are (for this class) colored triangles each identical in shape and size.  The goal is to arrange the facets (by twisting layers) such that every face shows a single color.  Most tetrahedron puzzles (all discussed here) have four unique colors (one for each face).

There are several variations of this puzzle, each described in some detail further below.  But a quick summary is:
• Mini-minx / Junior-Pyraminx [n=2] (22=4 triangles per face * 4 faces = 16 triangles; 4 tetrahedrons + 1 octahedron = 5 surface blocks)
• Pyraminx [n=3] (32=9 triangles per face * 4 faces = 36 triangles; 4+6=10 tetrahedrons + 4 octahedrons = 14 surface blocks)
• Master Pyraminx [n=4] (42=16 triangles per face * 4 faces = 64 triangles; 4+12+4=20 tetrahedrons + 4+6=10 octahedrons = 30 surface blocks)
• Professor's Pyraminx [n=5] (52=25 triangles per face * 4 faces = 100 triangles; 4+18+12=34 tetrahedrons + 4+12+4=20 octahedrons = 54 surface blocks)
An important thing to note about the Tetrahedron twisty-puzzles is that, conceptually, each puzzle is built out of two (2) types of blocks!  Compare this with the Rubik's Cube puzzles where each is built out of only one (1) type of block.  In other words, any size Rubik's cube is built (theoretically) out of a large number of identical (yet smaller) cubes.  These building blocks are called "cubies" in popular twisty-puzzle literature, but the term really only applies to the Rubik's Cube family (and perhaps a few others).  Most important, the Tetrahedron ("Pyraminx") family of twisty puzzles is built of two different types of building blocks:
• Regular Tetrahedron (pyramid with a 3-sided base; 4 triangular facets total)
• Regular Octahedron (two pyramids, each with a 4-sided base, fused together at their bases; 8 triangular facets total)

Note that both of these building blocks are a Platonic solid.  Two interesting things about any Platonic solid: all faces have the same shape and each vertex (corner) has the same number of faces.  In particular for this puzzle class:

• both Tetrahedrons and Octahedrons have facets which are equilateral triangles
• Tetrahedrons have 3 triangles converging at each vertex and a total of 4 triangular facets (see 1st image on right)
• Octahedrons have 4 triangles converging at each vertex and a total of 8 triangular facets (see 2nd image on right)

In summary, although each puzzle of the Tetrahedron ("Pyraminx") family appears to be composed of identical blocks ("cubies" in the popular twisty-puzzle lingo), each Tetrahedron puzzle is actually composed of two different types of blocks: tetrahedrons (same shape as the overall puzzle) and octahedrons (very different shape, but each facet is still an equilateral triangle).  In other words, every exposed facet of the different blocks is an equilateral triangle, and because they are all the same shape (and size) the whole puzzle appears to be built from identical "cubies" (but it is an illusion).

The irony, in my opinion, is that the tetrahedron puzzles are conceptually more complex (2 block types) than the cube puzzles (1 block type), and yet for the same number of stickers per face (triangles or squares, depending on puzzle type), the cube puzzles are far more difficult in terms of possible combinations.  I'm guessing this is because a cube has more faces and more orientations than a tetrahedron..

Uniquely, the tetrahedron ("Pyraminx") puzzles are the only ones which are fully Dual-Turning.  That is, you can rotate both Faces and Corners (or any layer in-between).  All other known twisty puzzles are either Face-Turning or else Corner-Turning, but not both (note there are also rare Edge-Turning puzzles).  This is because, for a tetrahedron shape, each corner has a face on its "opposite" side (and vice versa).  All other regular polyhedron shapes have a face opposite a face, and a corner opposite a corner!

Finally note that I said the tetrahedron puzzles are "fully" Dual-Turning (all layers operate the same).  It is conceivable that another Platonic Solid (like an octahedron) could be built as a Dual-Turning twisty puzzle.  I do not know of any which actually exist, but if one were to be constructed, it would have to built from non-Platonic (irregular) blocks.  I think of such a construct as impure (not built from blocks of Platonic Solids).  An actual example is the Icosaix (which has an overall icosahedron shape).  It allows twisting of both corners and faces, but the corners can not turn in layers (only the "trivial" tips) and thus I consider that puzzle to be mainly Face-Turning (or "very limited" Dual-Turning).

All of the tetrahedron twisty puzzles, except for the Mini-Minx, lack fixed center blocks.  As far as the number of combinations go, you might expect these puzzles to have values divided by 12 (4*3) to account for no fixed orientation like the way a "center-less" cube puzzle has its combinations divided by 24 (6*4).  No "orientation" divisor is needed for any tetrahedron puzzle because the orientation is fixed by the four corners!  You might think that turning the face of one of these puzzles would move 3 of the corners (which it does), but mathematically this is equivalent to rotating all the layers "below" a face in the opposite direction.  The result is the 3 corners on the face remain fixed, and the fourth corner simply rotates in place.

 Mini-Minx
 This dual-turning [n=2] puzzle (also known as a "Junior Pyraminx") is a joke... only the corners need to be rotated (which destroy nothing).  Entertaining for pre-school children, boring for everybody else! This puzzle has only 4 "truly manipulable" pieces (the "tips").  The "tips" only rotate in-place (they do not move).  It also has 4 center facets, but these are all part of a single ("core") block.  Trivia: this is the only twisty-puzzle that I know which has a visible "core" block.  Challenge: modify a Void Cube to display the removed "core" block! After searching the web, I have never found an "official" number of combinations, but with a degree in engineering and real-world experience with statistics, I believe the number of combinations is only 81 (34).  You can prove this to yourself (if you have one to play with), by treating the puzzle like an abacus... each corner can represent a single digit of a 4-digit number encoded in ternary.  You can "count" up to 81 by twisting the corners!  Not impressed?  I already told you it was child's play... On the right is a diagram that shows the blocks which comprise a Mini-minx. There are two block types (as described above); the "core" is an octahedron and each "tip" is a tetrahedron. The blocks (often called "cubies") of this puzzle are: four (4) tetrahedron "tips", each with 3 visible facets (colored triangles) and 1 hidden facet one (1) octahedron "core" with 4 visible facets and 4 hidden facets Thus, the Mini-Minx has 3*4+4 = 16 visible facets and 4+1 blocks.  This gives an average of exactly 3.2 visible facets per block (16/5). I'm getting ahead of myself, but in "bigger" Tetrahedron puzzles (described below), there are references to "axial" blocks. All of the bigger puzzles have four of them. In this puzzle, I like to think of the "core" block (the single octahedron) as an "overlay" of 4 axial blocks, all in the same location (this is mathematically called a "degenerate case").

 Pyraminx
 This dual-turning [n=3] puzzle is easy to solve.  First solve the 4 "tips", which is super-easy (just twist each to align with its partner, "axial" piece).  Next, choose your favorite vertex (like the top one) as a point of reference, and rotate (in place) the three remaining axial+tip pairs so their exposed facets match the colors on your favorite tip. Last, there are only 6 "edge" pieces which need to be "fixed" (half the number of edges in Rubik's Cube).  The last step involves a small amount of logic, but most people can get it with practice (trial and error).  Of course you can always cheat and lookup algorithms on the web. Note there is no "center" triangle, so you (the player) may experience "parity" errors, but these are easy to avoid in the Pyraminx (because they aren't true "parity" errors).  Avoid the errors by noting which colors are on the "tips", and rotating the "axial" pieces to match the "tips"; this gives a reference "color frame" for the final "edge" pieces. The "official" number of combinations is 75,582,720 (about 75 million) = 34 ["tips"] * 34 ["axials"] * 6!/2*25 ["edges"].  Note the edges have only 25 orientations (and not 26), because the final [6th] edge is fixed by the prior 5 edges.  The count of combinations includes the trivial "tips".  If you omit the "trivial tips", there are just under a million combinations (933,120 = 75,582,720 / 81) which is very simple (compared to Rubik's Cube which has billions of billions of combinations)!  In fact, once the axial pieces are rotated so their colors match (very easy), and neglecting the trivial tips, there only 11,520 combinations!!  Thus the puzzle is extremely easy and can be solved by almost anyone with some patience and dexterity. The diagram on the right shows how a Pyraminx has four (4) fixed axes.  Each axis consists of two blocks: an "axial" and a "tip".  The Pyraminx has 4 types of blocks: 4 "tip" blocks, each a tetrahedron with 3 visible colors (a fourth facet attaches to an "axial") -- these simply rotate in-place 4 "axial" blocks, each an octahedron with 3 visible colors (three other facets attach to an "edge", one hidden facet attaches to a "tip", and one hidden facet attaches to the "core") -- To many players, it is not immediately obvious that the 3 colored facets of an "axial" are fixed relative to each other.  Like the "tips", these can not move but simply rotate in-place. 6 "edge" blocks, each a tetrahedron with 2 visible colors (two hidden facets each "glide over" a hidden facet of an adjacent "axial" block) 1 "core" block, a tetrahedron with all facets hidden (each is attached to a hidden facet of an "axial" block) The transparent block at the center of the diagram is, theoretically, a tetrahedron "core".  This contrasts with the (smaller) Mini-minx [with an octahedron core] and the (larger) Master Pyraminx [with a "core" of 4 tetrahedrons joined at a single point]... in theory!  This obscure fact is due to the way tetrahedrons and octahedrons tessellate 3D space (form a honeycomb).  An important practical note is that if you disassemble a Pyraminx puzzle, you will see that the "core" is actually spherical in shape!  You will also see that the hidden/inside edges of the "axial" and "edge" pieces are also curved.  This allows the layers of the puzzle to rotate more smoothly. The "edge" blocks use every possible combination of two different colors (from the set of four colors).  Because there are 4*3=12 different-color combinations and half of them are identical (e.g., red & green = green & red), it should be no surprise there are 12/2=6 "edge" pieces in a Pyraminx.  Each edge-block can be in one of two possible orientations (this may seem trivial now, but becomes important in the bigger Master Pyraminx and Professor's Pyraminx).  Another piece of trivia: the Pyraminx is the only puzzle in the Tetrahedron family which doesn't have a center triangle on each face!

 Master Pyraminx