After the inimitable

*glork (sense 4)*, one of the most enduring contributions made to the English language by

Douglas Hofstadter's

*Metamagical Themas* column must be the delightful

portmanteau word

*Skewb*. The

Rubik-style toy to which it refers, invented by

British journalist Tony Durham, had been marketed by

Uwe Mèffert under the clunky and confusing name “Pyraminx Cube” (to emphasise the connection with Mèffert's earlier tetrahedral invention

Pyraminx.) Subsequently, however, he adopted Hofstadter's suggested name, and went on to sell other related puzzles (

*vide infra*)

branded with the Skewb name instead.

The Skewb is a cube-shaped puzzle, but unlike in Rubik's Cube, where each of the six faces rotates about an axis through its centre, the axes of the Skewb go through the eight corners. However, the plane along which the cube is cut for each move bisects it into two congruent pieces. Consequently there are effectively only four choices of axis of rotation, since rotating either half is equivalent to rotating the other half in the opposite direction. (The same argument would apply to the 2x2x2 Rubik's Cube, but not the usual 3x3x3 one.)

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Each face of the cube has four cuts through it, corresponding to the four different axes of rotation. Moreover, if you

visualise the shape of the plane along which the cube is sliced, you'll find it's a regular

hexagon, with each side corresponding to the intersection with a different face of the cube. It therefore follows that

**every twist affects every face**. It is this property, together with the

inherent counterintuitiveness of the “skew” twist, that make the Skewb so

deceptively difficult to solve, even though

combinatorially speaking it is much simpler than Rubik's Cube.

Note that although the dividing plane is hexagonal, each half of the divided cube has only threefold rotational symmetry and therefore you can only rotate each piece through integer multiples of 120°. Thus in the most natural description of the Skewb group there are eight primitive moves: for each of four possible axes you can rotate some canonically chosen half either 120° clockwise or 120° the other way. Or to put it weirdly, there are eight possible halves, one for each corner, and you can rotate any of them 120° clockwise.

## Skewb internals

One of the first observations one makes in a combinatorial

analysis of the Skewb is that it can never be possible to interchange two

*adjacent* corner pieces: in fact the eight corners divide

alternately into two sets of four. The corners in each set can be

permuted among themselves, but never shall the

twain meet. Indeed, this is obvious by

induction since each

twist cycles three corners from the same set.

It turns out that this is fundamental to the construction of the Skewb, for the two sets of corners are radically different in construction and their similarities are purely superficial. Four corners are fixed on rotating axes which meet in the centre. The face centres are hooked under them so that they are held in but can rotate freely, and the other four corners are hooked under the face centres. In this respect the fixed corners, faces and free corners play the respective rôles of the faces, edges and corners in the construction of Rubik's cube.

## Variants of the Skewb

It's possible to build different shapes of

mechanical puzzle using the same four-axis construction of the Skewb. For example, you can build out the cube to its

dual polyhedron the

octahedron by putting a corner point directly above each face centre (at twice the distance from the centre of the cube as the face was, for those of you

trying this at home) and taking the

convex hull. (In theory, you could instead make a

*smaller* octahedron by

sanding away everything that's not in the convex hull of the original face centres, but this won't work with a real Skewb because you'll destroy the

mechanism before you succeed.) Cutting the octahedron along the planes defined by the original puzzle and painting each face a different colour gives you another Rubik-style solid, the

*Skewb Diamond*. This puzzle has the same number of pieces arranged in the same way as the Skewb, and the same moves to

permute them—after all, you built one just now by changing the shapes of a few pieces—so it's a precisely equivalent puzzle and if you can solve one you can solve the other, right?

Well, not quite. If you scramble a Skewb Diamond and try to solve it by “translating” a solution for the ordinary Skewb, something goes wrong. Some of the corner pieces will (probably) be in the wrong orientation, and your Skewb solution simply does not mention anything corresponding to this problem. Why not? Well, the corner pieces correspond to face centres of the Skewb, and it doesn't matter if the face centres are the wrong way round, because they're monochromatic squares and they look the same either way. There's a dual of this observation as well: the eight corners of the Skewb have become the face centres of the Diamond, so for a similar reason you no longer have to worry about their orientation.

So somehow, each of these puzzles is failing to make use of the whole group generated by the underlying mechanism, because in each case the colouring causes different internal states to appear identical—in particular, it allows multiple “solutions” which are visually identical to the starting position but internally distinct. One way to address this is to print pictures on each face instead of plain colours, as has been done by many people to Rubik's cube (which too becomes harder thereby, for the same reason). But a very elegant and perhaps surprising solution was discovered by Durham, the original inventor of the Skewb. If you construct a circumscribed dodecahedron on the cube, extend the cuts and colour the faces as before, the result is the *Ultimate Skewb*, previously marketed as *Pyraminx Ball*.

##
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# *| **..........**:|*:#
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Key: # edges of dodecahedron
|- edges of underlying Skewb (not visible on real puzzle)
* cuts

This puzzle has a striking

asymmetry as each

pentagonal face is cut along two of its five

diagonals, corresponding to an

arbitrary choice of one of

the five cubes inscribed in the dodecahedron. (The correspondence is that the two cut diagonals meet in the

midpoint of the

unique edge of the chosen cube which lies inside that face.)

Nevertheless, each face piece of the Skewb becomes an edge of the

dodecahedron, and each corner piece of the Skewb remains a corner piece. Consequently the

orientation of both types of piece is visible and thus this is the hardest possible Skewb-structured puzzle.

It has 100,776,960 possible positions, which—let's not lose sight of this—is still **less than the square root** of the number of positions of Rubik's Cube (4.3×10^{19}). If you're prepared to suppose that the difficulty of a puzzle is roughly proportional to the logarithm of its number of states, then Rubik's Cube is still 2.45347 times harder.

## Easier versions

On the other hand, it is possible to make easier variants of the Skewb by building

polyhedra with more

indistinguishable states.

Mèffert's original

Pyraminx was one such. This is a tetrahedral structure in which each face is divided, at least visually, into a “3x3”

grid of smaller

triangles. It turns out that this can be regarded as a Skewb in which four corners (obviously the “fixed” ones in the mechanical description) have remained corners, the face centres have become the six

edges, and the “free” corners don't exist at all! (The remaining triangles on each face, which appear as though they might play the rôle of face centres, are in fact each attached to the axle for the nearest corner.) Consequently, one does not merely not have to worry about

orientating these “

phantom corners” correctly, but nor even about interchanging one with another. (A subsequently discovered variant of this called the

Halpern-Meier pyramid restores these phantom corners as face centres, so their orientation doesn't matter but their position does.)

We can put all this information together and construct a poset showing relations between four-axis puzzles:

Skewb Ultimate -----> Halpern-Meier pyramid -----> Pyraminx
| |
| |
V V
Skewb Skewb Diamond

Here the arrows mean “

*is harder than*” in the strict sense, meaning that if you have a

method for solving the first then it will also solve the second.

**References:** *There are many references for this material on the Web. One which I used for some names and numbers, and which has some much better pictures of the puzzles than mine here, is* __http://www.geocities.com/jaapsch/puzzles/__. There is a very comprehensive listing of Rubik-style puzzles at __http://twistypuzzles.com/__.

#### Not to be confused with ...

The intended scope of this writeup is *puzzles with four axes*. This is very far from exhaustive of the possibilities for making even the Platonic solids into mechanical puzzles. In particular, the Skewb Ultimate should not be confused with either the Megaminx or the Pyraminx Crystal, which are both dodecahedral puzzles which rotate around face centres and therefore have six axes (and are different from each other); similarly the Skewb Diamond (dual of the Skewb) differs from the three-axis octahedral puzzle obtained by dualizing Rubik's Cube in the same manner. This latter has been called Trajber's Octahedron, although the writer of that node seems to believe that it is the same as the Halpern-Meier pyramid (which, although the way the triangular faces are divided is similar, isn't even octahedral!).