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### F 6 : Hexahedron (Cube)

A "Cube" twisty-puzzle is, overall, a regular hexahedron which has the following key properties:
• 6 square faces (identical, except for color) -- a square is a regular polygon of 4 sides
• 8 corner points (vertices)
• 12 edges (each a line where two faces meet)
Trivia: the geometric dual of a hexahedron (cube) is an octahederon.  An octahedron has 8 faces and 6 vertices (the reverse of a cube).  Both shapes have 12 edges.

Like all twisty puzzles discussed on my site, each face is divided into multiple facets which are (usually for a Cube) colored squares 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 cube puzzles (all discussed here) have six 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:
• Dino Cube [n=2] (22=4 triangles per face * 6 faces = 24 triangles; 12 edges = 12 surface blocks)
• Helicopter Cube [n=2] (22*2=8 triangles per face * 6 faces = 48 triangles; 8 corners + 4*6 centers = 32 surface blocks)
• Pocket Cube [n=2] (22=4 squares per face * 6 faces = 24 squares; 8 corners = 8 surface blocks)
• Rubik's Cube [n=3] (32=9 squares per face * 6 faces = 54 squares; 8 corners + 12 edges + 1*6 centers = 26 surface blocks)
• Rubik's Revenge [n=4] (42=16 squares / face * 6 faces = 96 squares; 8 corners + 2*12 wings + 4*6 centers = 56 surface blocks)
• Professor's Cube [n=5] (52=25 squares / face * 6 faces = 150 squares; 8 corners + 12 edges + 2*12 wings + 9*6 centers = 98 surface blocks)
• V6 Cube [n=6] (62=36 squares / face * 6 faces = 216 squares; 8 corners + 2*12 edges + 2*12 wings + 16*6 centers = 152 surface blocks)
An important thing to note about the "Rubik's" twisty-puzzles (those with square facets) is that, conceptually, each puzzle is built from only one (1) block type!  Compare this with the Pyraminx, Octahedron, Dodecahedron, and Icosahedron puzzles (each built from various block types).  I can't say exactly why this is true, but I suspect this is due to the fact that the "cube" shape has 8 corners, and 8 can be "cube-rooted" into an integer (i.e., ∛8 = 2) which is not true of any other Platonic solid.  (If you are wondering why "cube-root" is important, it is because we inhabit a 3-D space.)  Thus, conceptually, the Rubik's Cube family of puzzles is the simplest.  Yet in terms of the number of combinations ("difficulty"), it is arguably the most challenging class of twisty-puzzles...

All but two of the puzzles on this page are from the "Rubik's" cube family.  The other "odd balls" are the Dino Cube and Helicopter Cube.  There are many variations of the Cube puzzle, and I don't have the time or patience to explain them all (would you have the time to read about them all?).  I include the Dino Cube because all of its visible blocks are identical, and (hopefully no surprise) all of its visible facets are also identical.  In other words, the Dino Cube is extremely regular, like the Rubik's family (and I hope you find it educational).  I also include the Helicopter Cube because it is an example of a rare form of twisty puzzle (with a familiar shape).  In summary, all the hexahedron puzzles from the Rubik's family are Face-Turning, while the Dino Cube is Corner-Turning and Helicopter Cube is Edge-Turning.

Another important aspect of the hexadedron ("Cube") puzzles is that some of them have no definite orientation.  In detail

• Pocket Cube (face-turning, n=2)
• Dino Cube (corner-turning, n=2)
• Helicopter Cube (edge-turning, n=2)
• Void Cube (face-turning, n=3)
• Rubik's Revenge (face-turning, n=4)
• V6 Cube (face-turning, n=6)
These are easy to visually identify because there is no center square (or other polygon) on any face.  All of them, except the Void Cube, are the so-called "even-n" puzzles.  These tend to give many players problems with "parity errors" (most notoriously the Rubik's Revenge).  The Void Cube is technically an "odd-n" puzzle, but because the centers and core have been removed, it may also cause "parity errors".  On the other hand, those puzzles which do have a center square on each face (n=3 for Rubik's Cube and n=5 for Professor's Cube), the so-called "odd-n" puzzles, are considered by many players to be easier... I believe this is because the centers give a "fixed" frame of reference for solving the puzzle.  However, I must note that the Professor's Cube (n=5) will often challenge players with "parity errors" even though it has fixed center blocks (because it also has "wing edges" and "non-middle centers" like the [n=4] Rubik's Revenge).

More trivia: an "official" Rubik's Cube (or higher/lower order variant) has a corner with the colors of red, white, blue (in clockwise order).  That is 3 of 6 colors.  The others are orange (opposite red), yellow (opposite white), and green (opposite blue).  It seems that some "knock-off" cubes (which were otherwise identical to an official Rubik's Cube) managed to skirt the patent laws by using a different color scheme.  I'm not a lawyer so I can't verify this is true, but the fact remains you can find a variety of "Cubes" with different color schemes.

 Dino Cube
 This corner-turning [n=2] puzzle is one (of the two) on this page which is not a member of the Rubik's Cube family.  Hopefully this is obvious from the photograph [on the right]: each face is divided into isosceles triangles (and not squares).  Perhaps not obvious, but extremely relevant for playing with this puzzle, is the fact that you can only twist the corners (all of Rubiks' Cubes are face-turning).  In geometric theory, the Dino Cube can be (de)constructed as: 12 visible "edge" blocks (irregular tetrahedrons) -- each has 4 facets: 2 visible (colored) "wide" isosceles triangles and 2 normally hidden equilateral triangles (partially visible when twisting a corner) 8 hidden "axial" blocks (regular tetrahedrons) -- each has 4 hidden facets (all equilateral triangles): 3 of them are partially visible when twisting a corner, the other is always hidden (attached to the "core") 1 hidden "core" block (a regular octahedron) -- it has 8 facets (each equilateral triangles) which are always hidden by the "axial" blocks The "tall" image on the right is an "exploded" diagram of the various blocks (listed above).  Normally hidden facets appear as dark gray.  The completely hidden "core" is shown as transparent (white).  I did not include axial lines to avoid clutter and confusion in the image, but I will describe them.  There are 8 axes which originate from the center of each face of the "core" block and terminate at a corner (the point where three surface blocks meet).  Trivia: none of this puzzle's blocks are cubes, contrary to every other ("Rubik's") puzzle on this page (which are built from only cubes)! I must point out that a "corner" is a group of 3 "edge" blocks (and a hidden "axial" block) which all share a common vertex (in the top photograph: the green/red, red/yellow, and yellow/green "edge" blocks comprise a "corner" [at the red/yellow/green "center" of the image]).  Since this puzzle can only twist "corners", it follows that each twist will move 3 "edge" blocks.  Trivia: each twist will also rotate a hidden "axial" block in-place (it turns but doesn't change position). So this puzzle has 12 visible (surface) blocks.  Because each face consists of four "wide" isosceles triangles (colored facets), the puzzle has a total of 4*6=24 visible facets.  Thus it has an average of (exactly) two (24/12) visible facets per surface block. The Dino Cube has 19,985,400 (about 20 million) combinations.  This is calculated as 12! ("edges") / 24 ("orientation").  Oh yeah, I should mention that because this puzzle has no center facets, the orientation is a bit ambiguous.  Since the standard for solving a twisty-puzzle is "every face has the same color", regardless of the orientation, we have to "factor-out" the various orientations (hence the final division term in the mathematical expression above). What I find fascinating is the Dino Cube [n=2] has more combinations than the Pocket Cube [also n=2].  I'm guessing this is mainly due to the greater number of visible blocks (12 for the Dino Cube but only 8 for the Pocket Cube). The photograph (below) shows a partially disassembled Dino Cube.  As you can see, the "axial" blocks are not true tetrahedrons because the base of each is not flat but curved (concave).  You can also see that the "core" is not really an octahedron but a sphere.  These "reality" facts contradict the "geometric" theory given at the start of this section.  Presumably these manufacturing choices were made to improve the twisty "feel" of this puzzle.

 Helicopter Cube
This edge-turning [n=2] puzzle is the other one on this page which is not a member of the Rubik's Cube family.  Hopefully the difference is obvious from the photograph [on the right]: each face is divided into "wide" isosceles triangles (and not squares).  Or, if you prefer, each square is cut diagonally in half.  Extremely relevant for playing this puzzle is the fact that you can only twist the edges (in contrast, all of Rubiks' Cubes are face-turning).  The second photograph [on the right] shows this puzzle with its top-left edge twisted clockwise by 60°.

In geometric theory, the Helicopter Cube can be (de)constructed as:

• 8 "corner" blocks (irregular hexahedrons) -- each has 6 facets (all "wide" isosceles triangle shape): 3 visible (colored) and 3 normally hidden (partially visible when twisting)
• 24 "central" blocks (irregular tetrahedrons) -- each has 4 facets: 1 visible (colored, "wide" isosceles triangle) and 3 hidden (partially visible when twisting -- two of them are identical triangles but irregular, and the third is an isosceles triangle which differs from the visible facet)
• 12 hidden "axial" blocks (irregular pentahedrons) -- each has 5 hidden facets: 4 identical but irregular triangles (partially visible when twisting) and 1 rhombus (attached to the "core" and always hidden)
• 1 hidden "core" block (a rhombic dodecahedron) -- it has 12 identical, rhombus-shaped facets which are always hidden by the "axial" blocks -- the "core" may be considered to built of a cube and six square-based pyramids

The tall image on the right (below the two photos) is an "exploded" diagram of the blocks listed above.  In the image, the "core" and "axial" blocks are shown transparent (these are the non-surface blocks).  There are 12 axes (not shown in the diagram to reduce clutter) which originate from the center of each face of the "core" block and terminate at a center edge of the full puzzle.  Trivia: none of this puzzle's blocks are cubes, contrary to every other ("Rubik's") puzzle on this page (which are built from only cubes)!

I must point out that a "center edge" is a group of 4 "central" blocks from two different faces (and a hidden "axial" block).  For example [see photograph on right], the intersection of the two yellow "central" blocks and the two red "central" blocks at a point along the top, left edge.  Since this puzzle can only twist "edges", each twist will move 6 blocks (4 "centrals" and 2 "corners").  Trivia: each twist will also rotate a hidden "axial" block in-place (it turns but doesn't change position).

So this puzzle has 32 (8+24) visible (surface) blocks.  Because each face has 8 colored triangles (visible facets), this puzzle has a total of 8*6=48 visible facets.  Thus it has an average of (exactly) 1.5 visible facets per surface block (48/32 = 3/2).

The Helicopter Cube has 493,694,233,804,800,000 (about 493 quadrillion) combinations.  This is calculated as 8!*37 ("corners") * [6!]4/2 ("centrals") / 24 ("orientation").  Oh yeah, I should mention that because this puzzle has no center facets, the orientation is a bit ambiguous.  Since the standard for solving a twisty-puzzle is "every face has the same color", regardless of the orientation, we have to "factor-out" the various orientations (hence the final division term in the mathematical expression above).

The photograph (below) shows a partially disassembled Helicopter Cube.  As you can see, the "axial" blocks are not true pentahedrons because the base of each is not flat but curved (concave).  Three of the "axial" blocks are fully exposed, and four more are easy to see, although partially obscured (two more are barely visible near the red/green "corner" block).  The non-identical hidden face of a "central" block can be seen twice in the photo (two different blocks), but the identical hidden faces of a "central" block are still obscured by "axial" blocks.  You can also see that the "core" is not really a rhombic dodecahedron but an intersection of the 12 axes.  These "reality" facts (curved pentahedrons and missing dodecahedron) contradict the "geometric" theory given at the start of this section.  Presumably these manufacturing choices were made to improve the twisty "feel" of this puzzle.

 "Corner" block: unequal bi-pyramid with regular triangle base (visible facets form the larger half) "Central" block: tetrahedron with two identical, irregular triangles and two different "wide" isosceles triangles "Axial" block: pyramid with rhombus base "Core" block: rhombic dodecahedron

 Pocket Cube
 The face-turning [n=2] Pocket Cube looks like a joke, but it is surprisingly challenging!  This puzzle has only 8 "fully manipulable" corners.  That is to say, the position and orientation of each "corner" block is virtually unconstrained (however, the orientation of the last [8th] block is not unique... it depends on the other 7 blocks). This puzzle has only one block type: a "corner" block with 3 visible facets (colored squares) and 3 hidden facets (partially visible when twisting).  This puzzle has eight "corner" blocks for a total of 24 (8*3) visible facets.  Because a cube has six faces, you would be correct to surmise that each face has 4 visible facets (24/6).  Of course you can look at the photograph if you don't trust my math!  Anyway, this puzzle has an average of (exactly) 3 visible facets per block (24/8). The Pocket Cube has 3,674,160 combinations = 8!*38-1 (corners) / 24 (orientation).  The divisor in that equation (24) is there because the puzzle can be held in 4*6 different orientations, but none of them can be distinguished because there are no "center" blocks to use for a frame of reference. On the right is a diagram that shows the blocks which comprise a Pocket Cube (in geometric theory).  Interestingly, there is no "core" block... the center is a zero-dimensional point!

 Rubik's Cube
 The "original" face-turning Cube [n=3] is moderately difficult to solve.  The most popular method is to solve this puzzle "layer-by-layer".  First solve the 4 "top" corners and then the 4 "top" edges... this solves the first of 3 layers.  Next solve the middle layer which is easy once you learn a simple algorithm for side/edge blocks.  Finally, solve the bottom layer.  This involves solving all 4 corners and all 4 edges (similar to the first/top layer, but you must avoid destroying all your prior work).  I won't give you algorithms for solving this puzzle, so give it your best shot (or you can always cheat and lookup algorithms on the web).   This puzzle has 4 different types of blocks: 8 "corner" blocks, each with 3 colors -- these do not twist, but each has 3 different orientations 12 "edge" blocks, each with 2 colors -- each has 2 different orientations 6 "center" blocks, each with 1 color -- each has 4 different orientations but are considered identical due to appearance 1 hidden "core" block with no visible color -- it has 24 orientations, but is irrelevant because it is invisible So there are 26 (8+12+6) surface blocks and one (1) core block.  This gives a total of 27 blocks, exactly as should be expected from a three-cubed (33) puzzle. Importantly, the six "center" blocks are fixed in position, relative to each other.  These provide a great way to orient the puzzle (which helps solving it), but have no effect on the "difficulty" (number of combinations). Since each face has 9 colored polygons (32) and a cube has 6 faces, this puzzle has 54 visible facets (colored squares).  This puzzle has an average of (exactly) 2 visible facets per surface block. The Rubik's Cube has 43,252,003,274,489,856,000 combinations, which is about 43.252 quintillion (1018) or 43.252 billion billion combinations.  This is calculated as 8!*37 (corners) times 12!/2*211 (edges).

 Rubik's Revenge
 This face-turning (n=4) puzzle is more complicated.  You can solve the corners just like the original "Rubik's Cube", but the edges and centers are very different!  It should be obvious that the "centers" are a cluster of 4 squares (not a single square), and more importantly, none of the "center" squares are actually in the middle!  In other words, there are no "true" middle pieces... this can make solving the puzzle difficult (especially for beginners). Equally important: each edge now has two (instead of one) blocks.  Among the 6 sides, there are a total of 24 "edge" blocks.  Although pairs of "edge" blocks look identical, they are not!  If you find a red&blue "edge" block which appears to be in the correct place, but the colors are reversed, then you actually have an "edge" block in the wrong place.  In other words, the "edge" blocks appear (in this puzzle) as pairs.  You can not swap their positions without also reversing their colors (flipping their orientation). Another way to say it: there are two (seemingly identical) types of "edge" blocks.  However these pairs may not be swapped without "flipping" their colors.  Really there are 2 different types of "edge" blocks which look identical.  This causes "parity errors" for both novice and experienced players.  A few people call these "wing" blocks, to distinguish them from the traditional "edge" blocks (which can be "flipped" anyway and suffer no "parity error").  In summary, the "edge/wing" blocks (which look identical) are actually segregated into two sets of "wing" blocks.  The two sets of "wing blocks" can not fully intermix...if you move a "wing" from one set to another, its colors will get swapped. I'm sorry if the above is confusing, but it goes to the heart of "parity errors" which are both difficult to describe and annoying to solve (if you don't know about them).  Like G.I. Joe says,  "Knowing is half the battle."  So I hope that warning you about the two "identical" block types (which are truly different) will help with solving this puzzle. This puzzle has 42=16 visible facets (colored squares) per face and six faces.  Thus it has 96 visible facets. Rubik's Revenge has four types of visible blocks: 8 "corner" blocks (three colors each) 24 "wing/edge" blocks (two colors each) 24 "central" blocks (one color each) 8 (=23) hidden "core" blocks (the transparent blocks in the diagram on the right) So the puzzle has a total of 8+24+24=56 visible "surface" blocks + 8 hidden "core" blocks = 64 total blocks.  This is exactly what one should expect from a 43 (four cubed) puzzle!  This puzzle has about 1.714285 visible facets per surface block (96/56 = 12/7).  Note the number of "core" blocks is just geometric theory.  If you disassemble this puzzle you will typically find a sphere inside!  This seems like it would be more difficult to manufacture than a set of mini-cubes, but presumably this is done for smooth and reliable twisting. The Rubik's Revenge has 7,401,196,841,564,901,869,874,093,974,498,574,336,000,000,000 (about 7.4 quattuordecillion [1045]) combinations =  8!*37 "corners" * 24! "wing/edges" * 24!/(4!6) "centrals" / 24 (no center orientation).  Note that the expression for the "wings" (confusingly symmetrical edges) doesn't have a multiplication factor for orientation (like the "edges" of a Rubik's Cube) which is due to the fact that each "wing" (in a given position) can only have one orientation (read above).  The divisor (24) in the above expression is present because no orientation is preferred due to lack of true "center" blocks.

 Professor's Cube
 This face-turning (n=5) is the most complex "flat and symmetric" cube that is possible in human reality (three dimensions of space)!  It has 52=25 visible facets (colored squares) per face and has six faces.  Thus it has 150 visible facets. This puzzle has 7 different types of blocks ("cubies"): 8 "corner" blocks (with 3 colors each) 12 "[middle] edge" blocks (with 2 colors each) 24 "wing [edge]" blocks (with 2 colors each) 24 "central corner" blocks (with 1 color) 24 "central edge" blocks (with 1 color) 6 "[true] center" blocks (with 1 color) 27 "core" blocks (completely hidden) That totals to 98 "visible" surface blocks (8+12+24+24+24+6).  This puzzle also has a "core" of 27 blocks (33).  Thus a Professor's Cube has a total of 98+27=125 blocks, exactly as should be expected from a five-cubed [53] puzzle.  Approximately, it has an average of 1.530612245 visible facets per surface block (150/98 = 75/49).  The number of "core" blocks is geometric theory.  If you disassemble one of these puzzles, will likely find a sphere or perhaps an array of axial bars (depending on manufacturer). I will note that like the edge blocks, which can be divided into two groups ("edges" and "wings"), the center blocks can also be divided into groups.  However, there are three (not just two) groups of "center" blocks: "True" centers -- there is only 1 on each face... it is in the middle of the center blocks (think ·) "Edge" centers -- there are 4 on each face... these share a side with the "true/middle" center (think +) "Corner" centers -- there are 4 on each face... these only touch the "true/middle" center at a corner (think X) Each of three types of "center" blocks listed above, although they look identical, can not interact with each other (although they may mix with themselves).  Similarly, the two kinds of "edge" blocks (the "edges" and "wings"), which appear identical, can not mix with each other (but may mix with themselves).  In other words, this puzzle has multiple classes of blocks which seem identical, but which in fact are segregated.  Without knowing this, a player may become hopelessly lost in a "parity error".  By telling this to you, I can't prevent a "parity error", but hopefully you can now recognize it.  So now, at least, you should know... and knowing is half the battle: G.I. Joe!! The "true" center blocks are fixed in position, relative to each other.  These make the player's job easier (helps with orientation and color-matching), and don't increase the "difficulty" (number of combinations) of the puzzle. The Professor's Cube has an incomprehensible 282,870,942,277,741,856,536,180,333,107,150,328,293,127,731,985,672,134,721,536,000,000,000,000,000 combinations (about 283 trevigintillion or 2.8287*1074).  This number comes from 8!*37 ("corners") * 12!/2*211 ("edges") * 24! ("wings") * 24!/[4!6] ("central corners") * 24!/[4!6] ("central edges").  This is almost (smaller by a factor of about 500,000 [if the universe only had hydrogen]) the number of atoms in the observable universe...  Think about it:: the count of universal atoms held in the palm of your hand!

 V6 Cube
 This face-turning (n=6) twisty-puzzle (also known as a "6x6[x6] Cube") is "impossible" to build with identically-sized blocks and flat faces (at least in normal/human 3D space).  Manufacturers have devised two ways to deal with this geometric problem.  One way is to curve all the faces of the puzzle, resulting in a "pillowed" version [see first photo at the right].  Another way is to vary the size of blocks of the puzzle [see second photo at the right].  This second version, which I will call the ShengShou version, looks like all blocks are equal in size at first glance.  But if you look very carefully, you can see that although the colored stickers are the same size, the blocks along the edges ["corners" and "edges"] are slightly larger than all the other blocks (a great optical illusion). Each face of this puzzle is divided into six types of "identical" facets: 4 "corner" facets 8 "wing edge" facets 8 "middle edge" facets 4 "[outer] central corner" facets 8 "[outer] central edge" facets -- there are actually two different types (read below) 4 "[inner] center" facets Thus this puzzle has 36 (62) colored squares (facets) per face (4+8+8+4+8+4), or a total of 36*6=216 (visible) facets.  The V6 Cube is composed of 8 different types of blocks: 8 "corner" blocks (each with 3 visible facets of different colors) 24 "wing edge" blocks (each with 2 visible facets of different colors) 24 "middle edge" blocks (each with 2 visible facets of different colors) 24 "[outer] central corner" blocks (each with 1 visible facet [colored square]) 24 "[outer] central A-edge" blocks (each with 1 visible facet [colored square]) 24 "[outer] central B-edge" blocks (each with 1 visible facet [colored square]) 24 "[inner] center" blocks (each with 1 visible facet [colored square]) 64 hidden "core" blocks (each without any visible facet) Thus the V6 Cube has 152 surface blocks (8+24+24+24+48+24).  Adding those with the 64 core blocks (43) results in 216 total blocks, exactly as you should expect from a 63 puzzle.  Anyway, this puzzle has about 1.421052631578947368 visible facets per surface block (216/152 = 27/19).  The number of "core" blocks is just geometric theory; expect to find something else inside (like a sphere) if you disassemble one of these puzzles.  The Wikipedia article says the interior of the "official" V6 is composed of 60 pieces (but there is no photo nor description so their shapes are unknown). I must point out that the "central edge" blocks (the central squares adjacent to the middle edges) which appear identical (on a flat V6 Cube) or at least symmetrical (on a pillowed V6 Cube) are actually segregated into two different sets which can not mix.  This is unlike the similar "middle edge" blocks, where two adjacent blocks may swap position if they flip their orientation.  Specifically, two "central edge" blocks which are adjacent may never swap their positions!  So really there are two different types of "central edge" blocks which I call (for lack of better terms) the "central A-edge" and "central B-edge" blocks.  Each set appears 4 times per face (or 24 times in the whole puzzle).  Not surprisingly, because there are two sets, there is a total of 8 "central edge" blocks per face (or 48 blocks in the full puzzle). Another interesting fact is that the V6 has 4 types of "center" blocks ("corners", "A-edges", "B-edges", and "inner").  In contrast (but similarly), the "next lower-order" version of this puzzle, the Professor's Cube, has 3 types of center blocks.  All other "lower-order" Cube puzzles have only one (or no) type of center block. This puzzle has an incomprehensible number of combinations. A good approximation is 157.152858401024 * 10114 or about 15.7 googol quadrillion.  It is exactly equal to 8!*37 ("corners") times 24! ("middle edges") times 24! ("wing edges") times 24!/[4!6] ("central corners") times 24!/[4!6] ("central A-edges") times 24!/[4!6] ("central B-edges") times 24!/[4!6] ("centers") divided by 24 (no puzzle orientation). Pillowed V6 Cube Flat V6 Cube