Twisty Puzzles‎ > ‎

F 8 : Octahedron

An "Octahedron" twisty-puzzle is, overall, a regular octahedron (no surprise here!) which has the following key properties:
  • 8 faces (each an identical equilateral triangle, except for color) -- an equilateral triangle is a regular polygon of 3 sides
  • 12 edges (each a line where two faces meet)
  • 6 vertices (each a point where four edges meet)
Like all twisty puzzles discussed on my site, each face is divided into multiple facets which are (usually 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 puzzles discussed on this page have eight unique colors (one for each face), but there are some which duplicate colors in pairs and thus have only four unique colors (the reason is explained below).

I know of 3 variants of this puzzle, each described in some detail further below.  But a quick summary is:
  • Skewb Diamond [n=2] (22=4 triangles per face * 8 faces = 32 triangle facets; 6 octahedrons + 8 tetrahedrons = 14 surface blocks)
  • Face-Turning (F.T.) Octahedron [n=3] (32=9 triangles per face * 8 faces = 72 triangle facets; 6+3*2+6=18 octahedrons + 3*8=24 tetrahedrons = 42 surface blocks)
  • Corner-Turning (C.T.) Octahedron [n=3] (32=9 triangles per face * 8 faces = 72 triangle facets; 6 pentahedrons + 6 decahedrons + 18 tetrahedrons = 30 surface blocks)
The octahedron puzzles can be divided into two classes:
  • "True" octahedrons puzzles (the Face-Turning variety, including the Diamond Skewb)
  • "Fake" octahedron puzzles (the Corner-Turning variety)
An important thing to note about the "true" Octahedron 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 "true" (face-turning) Octahedron twisty puzzles is built of the following two types of 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)
Tetrahedron: pyramid with a 3-sided base
Tetrahedron
Note that both of these building blocks are Platonic solids.  Two interesting things about any Platonic solid: all sides 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 "true" (face-turning) Octahedron family appears to be composed of identical blocks ("cubies" in the popular twisty-puzzle lingo), each puzzle is actually composed of two different types of blocks: octahedrons (same shape as the overall puzzle) and tetrahedrons (very different shape: a pyramid with a 3-sided base).  Critically, 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).

Octahedron: two pyramids (each with a 4-sided base) fused together at their bases (thus, only triangular faces are visible).
Octahedron

I would love to give you a true and full description of the octahedron shape, but words fail me.  You really need to hold one in your hand (and examine from multiple angles) to appreciate this Platonic Solid.  But I will try to describe it for you.. so each face is triangular, but from certain ("side") views, the shape may also appear as either a parallelogram or a hexagon.  If you stand it on a corner, it looks like a "diamond".  When viewed as a whole (and resting on a face), it looks like a mad scientist genetically merged a Rubik's Cube with a Pyraminx!  (I warned you I can't describe it with words...).  The only other way I can think to describe the 3D shape is like two equilateral triangles (each 60 degrees "out of phase") and with both having 3 triangular sides (so 2*3 sides + 1 top + 1 bottom = 8 faces).

Trivia:
  1. the octahedron is the mathematical "dual" of the cube (octahedron has 6 vertices and 8 faces, while cube has 6 faces and 8 vertices)
  2. "true" octahedron puzzles are built of the same two block types (octahedron and tetrahedron) as the "Pyraminx" puzzles

Important Note: Although the corner blocks of the "true" Octahedron puzzles look like the "trivial" tips of the Pyramix family, the corner blocks of these puzzles can not be rotated freely!  The reason is because the corner blocks are not tetrahedrons (contrary to the corners of the Pyraminx family)... in fact, the corner blocks are Octahedrons.  The corner octahedrons are firmly held in-place (can't rotate) by 4 tetrahedrons... this is analogous to the way that corner-blocks of the Rubik's Cube [family] are held in-place by 3 edge-blocks.

So though it may "look" like the corners should rotate, they do not, and can not, due to the geometry of octahedra in a "true" face-turning puzzle.

Some non-obvious facts about the "true" (F.T.) octahedron puzzles:

  • The corner blocks have only 2 orientations each (as opposed to 4 orientations in "false" [C.T.] puzzles)
  • The edge blocks have only 1 orientation each (as opposed to 2 orientations in "false" puzzles)
  • Each middle/wing block of the same color can only show up on 4 of the 8 faces (as opposed to 7 faces in "false" puzzles)

Hopefully the last fact makes it clear that the colored facets can not fully intermix.  Interestingly, the first two facts above also limit the way colors can mix in the same way!  So it might not be obvious, but any block type (corner, edge, or middle/wing) will have each color on its visible facet(s) limited to appearing on only 4 of the 8 faces of the puzzle.  I call this the "no mixing rule": all the colored facets that belong to the same face (when the puzzle is solved) can only ever appear on that face (the "home" face) or the three triangular faces which touch the "home" face at a vertex.  In other words, the colors of a face may never appear on the face opposite of "home", nor on any of the three triangular faces which share an edge with the "home" face.

Because of the "no mixing rule" of colors, some variants of the Face-Turning Octahedron have only four colors instead of eight.  In these, each color is used on two different faces, but the duplicate colors appear on faces which do not mix colors.  I've never played one, but it sounds like a nasty beast that would often confound the player with "parity" errors (you would have to pay very close attention to which of two sets a piece belongs because you couldn't rely on color).

There is also a "false" (Corner-Turning) variety of Octahedron twisty-puzzles.  I call them "false" octahedron puzzles because none of their blocks ("cubies" in popular twisty-puzzle lingo) are octahedrons, and because their two middle layers do not twist relative to each other (instead they form a "double-thick" layer).  These look exactly like a "true" (Face-Turning) puzzle, but they are geometrically different, and thus playing with them is also different.  These puzzles are extremely similar to the Pyraminx family.  If you can solve a Pyraminx, then it should take little additional effort to solve a "fake" (C.T.) octahedron.

The "false" (C.T.) octahedron puzzles do not have a strict color mixing rule.  So every version produced (AFAIK) has 8 colors.  The "edge" blocks can show their colors on any face (no restrictions).  The color that appears on a corner or middle/wing facet can appear on any face except the opposite face (of its solved face).  In other words, the non-edge blocks have a minor mixing rule of "only 7 of 8" (and not Seven of Nine to the dismay of some Voyager fans).

Although the number of combinations is limited by the "no mixing rule" in a "true" (face-turning) octahedron puzzle, many (most?) players find the "true" octahedron puzzle to be more difficult than the "false" (center-turning) puzzle.  One reason is the corners pieces may be simply turned in the "false" octahedron (it requires at least 3 twists on a "true" puzzle and will "flip" at least one other corner).  The main reason, however, is the "false" octahedron puzzle has far fewer combinations than the "true" octahedron puzzle.  Now this may sound like a contradiction, so let me spell it out for you.  The "true" (F.T.) octahedron has a greatly reduced number of combinations (because of the "no mixing rule"), but nonetheless, the "false" (C.T.) octahedron has a many fewer combinations than the "true" octahedron (despite more color mixing).

It probably sounds illogical to think that a puzzle with notably reduced color combinations (the "true" octahedron) has more combinations than a puzzle which allows almost complete color mixing (the "false" octahedron).  The real reason is the "true" octahedron has its 24 middle/wing facets on 24 different blocks (tetrahedron in shape) which can move about with almost complete freedom (only restricted by the "no mixing rule"), while the "false" octahedron has its 24 middle/wing facets on only 6 different blocks (decahedron in shape) which can not change position at all (these "axial" blocks can only rotate in-place).


 
Skewb Diamond  

This face-turning (n=2) puzzle is simple.  It has 4 colored triangles (facets) per face.  Like all octahedra, it has 8 faces.  Thus it has 4*8=32 colored polygons (facets) in total.

The Skewb Diamond is composed of only 2 block types:
  • 6 octahedron corners (with 3 visible facets [colors] each)
  • 8 tetrahedron centers (with 1 visible facet [color] each)
Interestingly there is no "core" block... the center of this puzzle is a single (zero-dimensional) point!  Anyway, this puzzle has 14 surface blocks (6+8).  It has an average of (approximately) 2.285714 visible facets per surface block (32/14 = 16/7).

The Skewb Diamond has 138,240 combinations: 6!/2*25 [corners] * 4!/2 [centers].
A scrambled Diamond Skweb

 
Face-Turning (F.T.) "true" Octahedron
 

This face-turning (n=3) puzzle has no commonly accepted name (so if you're buying one, be sure to find out if it is a face-turning or corner-turning octahedron).  It is composed of 5 different block types:
  • 6 octahedron "corner" blocks (which do not twist), each with 4 visible facets (colored triangles) and 2 orientations
  • 12 octahedron "edge" blocks, each with 2 visible facets but only 1 orientation
  • 12 tetrahedron "A-center/wing" blocks, each with 1 visible facet and only a single, discernible orientation
  • 12 tetrahedron "B-center/wing" blocks, each with 1 visible facet and only a single, discernible orientation
  • 9 hidden "core" blocks: 8 tetrahedrons and 1 octahedron -- their orientation is irrelevant because they are invisible
The second (very tall) image on the right is a diagram (exploded view) of the blocks which make up a face-turning octahedron (click it to enlarge).  The transparent blocks are not visible to the player; these are the "core" blocks.  Because the exploded view makes it hard to imagine how the "core" blocks combine into the central core of this puzzle, I've added another diagram [below] showing only the "core" (which consists of 9 blocks = 1 octahedron + 8 tetrahedra).

So this puzzle has 42 visible "surface" blocks (6+12+12+12).  It also has 9*8=72 visible facets (colored triangles).  Thus it has about 1.714285 visible facets per surface block (72/42 = 12/7).  Interestingly, this puzzle has (in geometric theory) an impressive 9 hidden/core blocks (8 tetrahedrons + 1 octahedron)!  But if you disassemble one [see photo below], the "core" is quite different in reality (almost spherical).

More importantly, the "center/wing" blocks (24 total) are divided into two sets of 12 each.  These two sets can not intermix.  For a real life example, the first photo on the right shows 4 faces (and four unique colors).  The edge colors (purple, green, yellow) and the hidden bottom (blue) colors can mix with each other, but the other 4 colors (the top [red] and 3 hidden from view) can never mix with any of the first four.  This is the "no mixing" rule of "true" (face-turning) octahedron puzzles.As a (partial) consequence of the "no mixing" rule, a few versions of this puzzle have only 4 (instead of 8) unique colors.

Anyway, this puzzle has 31,408,133,379,194,880,000,000 (about 31.4 sextillion [1021]) combinations = 6!/2*25 (corners) * 12! (edges) * [12!]/[3!4] (A-centers) * [12!]/[3!4] (B-centers) / 24 (no puzzle orientation).

A concept that may help players is knowing that the 12 "edge" pieces are attached to the core of an octahedron (see the photograph [below] of a disassembled octahedron).  This is similar to the 6 "center" pieces that attach to the core of a Rubik's Cube.  The two differences are: the "edge" blocks of this puzzle rotate in groups of 3 (unlike the Cube where each center rotates independently) and each "edge" block exposes 2 facets (unlike the Cube centers which each expose 1 facet).  Also, you can see in the photograph that "core" isn't truly a fusion of 8 tetrahedra and 1 octahedron (as suggested by geometry and my diagrams).  For mechanical reasons, the core more closely resembles a sphere.

Of particular note, the edge pieces of a "true" (F.T.) octahedron rotate in groups of 3 and their orientation is fixed (for example, a purple-white edge-block can never be "flipped" into a white-purple edge-block). [In contrast, a "false" (C.T.) octahedron rotates its edges in groups of 4 and any edge-block may be "flipped" in orientation.]

Diagram of the core blocks in a face-turning (n=3) octahedron puzzle
F.T. Octahedron Core
Photograph of the core and edge blocks in a face-turning (n=3) octahedron puzzle
F.T. Octahedron Core with 12 edges

Photo of an (n=3) octahedron puzzle
Diagram of the blocks in a face-turning (n=3) octahedron puzzle

 
Corner-Turning (C.T.) "false" Octahedron  

This corner-turning (n=3) puzzle has no commonly accepted name (occasionally I've heard it called a "Magic Octahedron" or a "Crystal Octahedron", but most often simply "Octahedron"), so if you're buying one, be sure to find out if it is a face-turning or corner-turning octahedron.  Refer to the photograph at the right to understand my explanation of twisting this puzzle (it plays very differently than a "true" [F.T.] puzzle).  So any corner block (blue/white, white/yellow, yellow/red, red/blue [which are literally at the corners in the photo] or the blue/white/yellow/red block [at the center of the photo]) may be rotated by 90 degrees (clockwise or counter-clockwise).  The layer directly "below" a corner may also be rotated by 90 degrees; for example, the 3 blue triangles and 3 white triangles immediately "below" the blue/white corner may be twisted together as a group (a "layer").  Importantly, the layer below that (to continue the example, the 5 blue triangles and 5 white triangles [which border the red and yellow triangles]) may not be twisted.  Hopefully you can see that if that second layer (with 5 blue and 5 white triangles) could rotate then it would split the blue/white/yellow/red corner block (middle of photograph).  In other words, the two middle layers (5 blue + 5 white, and 5 red + 5 yellow) are fused and do not twist relative to each other.

The "false" (C.T.) Octahedron consists of 25 blocks.  Refer to "exploded view" diagram at right (hidden facets are dark gray, the "core" block is transparent [white], and the 6 axes are dark cyan).  The 25 blocks, grouped by shape, are:
  • 6 pentahedron "corner" blocks (which do twist) -- these are shaped like Egyptian pyramids [see diagram below] and each has 4 exposed (colored) equilateral triangle facets and 1 normally hidden square facet (partially visible when you rotate a corner)
  • 6 decahedron "axial" blocks [see diagram below] -- these have 4 exposed (colored) equilateral triangle facets, 4 normally hidden "wide" isosceles triangle facets (you can see part of one when rotating a layer), and 2 square facets (one is partially visible when you rotate a corner and the other [larger] one is always hidden [attached to the core]).  The hidden square facets are parallel to each other and rotated by 45 degrees to each other.
  • 12 irregular tetrahedron "edge" blocks [see diagram below] -- these have 2 exposed (colored) equilateral triangle facets and 2 normally hidden "wide" isosceles triangle facets (part of these facets are visible when rotating a layer).
  • 1 cube "core" block -- this has 6 totally hidden square facets (each facet attaches to an "axial" block).
So this puzzle has 6+6+12=24 visible blocks (of 3 different shapes).  It also has 9*8=72 visible facets (everyone is an equilateral triangle).  Thus it has an average of (exactly) 3 visible facets per visible block (72/24).  Did you notice the deceptive statistics?  If not, I'll spell it out for you: statistically, each visible block has exactly 3 visible facets (colored triangles), but in reality no block has 3 visible facets!!  So in reality, a visible block will have either 2 or 4 visible facets (never 3).  It just goes to show you can't trust statistics with common sense.

This "false" Corner-Turning Octahedron has 8,229,184,826,926,694,400 (about 8.23 quintillion [1018]) combinations = 46 (corners) * 46 (axials) * 12!/2*211 (edges).  In summary, this corner-turning puzzle has about 4000-times-fewer combinations than a "true" (face-turning) puzzle of the same size ([n=3] 9 triangles per face).

In case my diagram of blocks (or my description in the lengthy intro to this page) isn't clear about the fixed/axial nature of the "false" (C.T.) Octahedron puzzle, I've included a photograph of a partially disassembled "false" Octahedron [see below].  The "edge" blocks have been removed.  What remains is the "core" (hidden very well), the six "axial" blocks, and the trivial "corner" blocks.  Hopefully you can see the remaining blocks lie on fixed axes (there are 6 axes, but you will have to trust me because the photo isn't clear on that point).  Hopefully you can also see that these "fixed" pieces comprise the majority of the puzzle (only the "edge" pieces were removed).  [So maybe now you can appreciate how this "false" Octahedron is easier to solve than a "true" Octahedron, despite its lack of a strict "no mixing rule"].

Although this "false" octahedron has no commonly accepted name, a more well-known variant called "Christoph's Jewel" exists.  The only real difference is that the "trivial" corner blocks have been removed from Christoph's Jewel.  Both versions play the same way, but Christoph's Jewel has "only" 2,009,078,326,886,400 (about 2 quadrillion [1015]) combinations -- exactly 4096 fewer combinations than a "false" (C.T.) Octahedron puzzle.

Also, some people consider this puzzle to be a disguised form of the Cube family of puzzles.  Presumably because it has a cube "core", six axes, and no blocks are octahedron in shape.

Photo of an (n=3) octahedron puzzle
Diagram of the blocks and axes of a "false" (center-turning) octahedron puzzle
Diagram of a pentahedron (pyramid with a square base)
Pentahedron Block
Diagram of an irregular decahedron (4 equilateral triangles + 4 isosceles triangles + 2 squares)
Decahedron Block
Diagram of an irregular tetrahedron (2 equilateral triangles + 2
Irregular
Tetrahedron Block
Photo of a partially disassembled "false" (C.T.) Octahedron which shows the "axial" and "corner" blocks (but no "edge" blocks)
C.T. ("false") Octahedron
with "edge" blocks removed
Photo of Christoph's Jewel (a corner-turning Octahedron puzzle missing the 'trivial tips')
Christoph's Jewel
(C.T. Octahedron w/o "corners")

© 2017, H2Obsession
Image of "Diamond Skewb" derived from an image at www.thecubicle.us provided under copyright fair use.
Image of "Christoph's Jewel" from Jaap's Puzzle Page: www.jaapsch.net/puzzles/octahed.htm
Comments