Combination puzzles - AbsoluteAstronomy.com

A combination puzzle, also known as a sequential move puzzle, is a puzzle

Puzzle

A puzzle is a problem or enigma that tests the ingenuity of the solver. In a basic puzzle, one is intended to put together pieces in a logical way in order to come up with the desired solution...

which consists of a set of pieces which can be manipulated into different combinations by a group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

of operation

Operation (mathematics)

The general operation as explained on this page should not be confused with the more specific operators on vector spaces. For a notion in elementary mathematics, see arithmetic operation....

s. The puzzle is solved by achieving a particular combination starting from a random (scrambled) combination

Combination

In mathematics a combination is a way of selecting several things out of a larger group, where order does not matter. In smaller cases it is possible to count the number of combinations...

. Often, the solution is required to be some recognisable pattern such as 'all like colours together' or 'all numbers in order'. The most famous of these puzzles is the original Rubik's Cube

Rubik's Cube

Rubik's Cube is a 3-D mechanical puzzle invented in 1974 by Hungarian sculptor and professor of architecture Ernő Rubik.Originally called the "Magic Cube", the puzzle was licensed by Rubik to be sold by Ideal Toy Corp. in 1980 and won the German Game of the Year special award for Best Puzzle that...

, a cubic puzzle in which each of the six faces can be independently rotated. Each of the six faces is a different colour, but each of the nine pieces on a face is identical in colour, in the solved condition. In the unsolved condition colours are distributed amongst the pieces of the cube.

The mechanical construction of the puzzle will usually define the rules by which the combination of pieces can be altered. This leads to some limitations on what combinations are possible. For instance, in the case of the Rubiks Cube, there are a large number of combinations that can be achieved by randomly placing the coloured stickers on the cube, but not all of these can be achieved by manipulating the cube rotations. Similarly, not all the combinations that are mechanically possible from a disassembled cube are possible by manipulation of the puzzle. Since neither unpeeling the stickers nor disassembling the cube is an allowed operation, the possible operations of rotating various faces limit what can be achieved.

Although a mechanical realization of the puzzle is usual, it is not actually necessary. It is only necessary that the rules for the operations are defined. The puzzle can be realized entirely in virtual space or as a set of mathematical statements. In fact, there are some puzzles that can only be realized in virtual space. An example is the 4-dimensional 3×3×3×3 tesseract

Tesseract

In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8...

puzzle, simulated by the MagicCube4D software.

Properties

There have been many different shapes of Rubik type puzzles constructed. As well as cubes, all of the regular polyhedra and many of the semi-regular and stellated

Stellation

Stellation is a process of constructing new polygons , new polyhedra in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again...

polyhedra have been made.

Regular cuboids

A cuboid

Cuboid

In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing definitions of a cuboid in mathematical literature...

is a rectilinear

Rectilinear polygon

A rectilinear polygon is a polygon all of whose edges meet at right angles. Thus the interior angle at each vertex is either 90° or 270°. Rectilinear polygons are a special case of isothetic polygons....

polyhedron

Polyhedron

In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

. That is, all its edges form right angles. Or in other words (in the majority of cases), a box shape. A regular cuboid, in the context of this article, is a cuboid puzzle where all the pieces are the same size in edge length.

Picture	Data	Comments
	Commercial name: Rubik's Cube Geometric shape: Cube Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... Piece configuration: 3×3×3	The original Rubik's Cube
	Commercial name: Rubik's Revenge Geometric shape: Cube Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... Piece configuration: 4×4×4	Solution is much the same as 3×3×3 cube except additional (and relatively simple) algorithm Algorithm In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning... (s) are required to unscramble the centre pieces and edges and additional parity not seen on the 3x3x3 Rubik's Cube.
	Commercial name: Professor's Cube Geometric shape: Cube Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... Piece configuration: 5×5×5	Solution is much the same as 3×3×3 cube except additional (and relatively simple) algorithm Algorithm In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning... (s) are required to unscramble the centre pieces and edges.
	Commercial name: Pocket Cube Geometric shape: Cube Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... Piece configuration: 2×2×2	Simpler to solve than the standard cube in that only the algorithm Algorithm In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning... s for the corner pieces are required. It is nevertheless surprisingly non-trivial to solve.
	Commercial name: V-CUBE Geometric shape: Cube Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... Piece configuration: 2×2×2 to 11×11×11 Main articles: V-Cube 6 V-Cube 6 The V-Cube 6 is the 6×6×6 version of Rubik's Cube. Unlike the original puzzle , it has no fixed facets: the center facets are free to move to different positions... , V-Cube 7 V-Cube 7 The V-Cube 7 is the 7×7×7 version of Rubik's Cube. It was invented by Panagiotis Verdes and is produced by his company, Verdes Innovations SA. Like the 5×5×5, the V-Cube 7 has both fixed and movable center facets.- Mechanics :...	Panagiotis Verdes holds a patent to a method which is said to be able to make cubes up to 11×11×11. He has fully working products for 2×2×2, 5×5×5, 6×6×6 and 7×7×7 cubes. Solution strategies are similar to those of the 4×4×4 and 5×5×5.
	4-Dimensional puzzle Geometric shape: Tesseract Tesseract In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8... Piece configuration: 3×3×3×3	This is the 4-dimensional analog of a cube and thus cannot actually be constructed. However, it can be drawn or represented by a computer. Seriously more difficult to solve than the standard cube, although the techniques follow much the same principles. There are many other sizes of virtual cuboid puzzles ranging from the trivial 3×3 to the unsolved 5-dimensional 7×7×7×7×7.
http://www.twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=24 http://www.twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=21 http://www.twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=1357 http://www.twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=1285	Non-uniform cuboids Geometric shape: Cuboid Cuboid In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing definitions of a cuboid in mathematical literature... Piece configuration (1st): 2×2×3 Piece configuration (2nd): 2×3×3 Piece configuration (3rd): 3×4×4 Piece configuration (4th): 2×2×6	Most of the puzzles in this class of puzzle are generally custom made in small numbers. Most of them start with the internal mechanism of a standard puzzle. Additional cubie pieces are then added, either modified from standard puzzles or made from scratch. The four shown here are only a sample from a very large number of examples. Those with two or three different numbers of even or odd rows also have the ability to change their shape.
http://www.twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=1271	Siamese cubes Geometric shape: Fused cubes Piece configuration: two 3×3×3 fused 1×1×3	Siamese cubes are two or more puzzles that are fused so that some pieces are common to both cubes. The picture here shows two 3×3×3 cubes that have been fused. The largest example known to exist is in The Puzzle Museum and consists of three 5×5×5 cubes that are siamese fused 2×2×5 in two places.
http://www.twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=439	Extended cubes Geometric shape: Box Piece configuration: 3×3×3 extended 3×3×5	These puzzles are made by bonding additional cubies to an existing puzzle. They therefore do not add to the complexity of the puzzle configuration, they just make it look more complex. Solution strategies remain the same, though a scrambled puzzle can have a strange appearance.
http://www.twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=13	Commercial name: Boob cube Geometric shape: Box Piece configuration: 1×1×2	Very possibly the simplest regular cuboid puzzle to solve. Completely trivial solution as the puzzle consists of only two cubies.
	Commercial name: Void cube Geometric shape: Menger Sponge Menger sponge In mathematics, the Menger sponge is a fractal curve. It is a universal curve, in that it has topological dimension one, and any other curve is homeomorphic to some subset of it. It is sometimes called the Menger-Sierpinski sponge or the Sierpinski sponge... with 1 iteration Piece configuration: 3x3x3 with a hole in the middle.	Solutions to this cube is similar to a regular 3x3x3. However, parity errors are possible in these solutions. This cube uses a special mechanism due to absence of a central core.

Pattern variations

There are many puzzles which are mechanically identical to the regular cuboids listed above but have variations in the pattern and colour of design. Some of these are custom made in very small numbers, sometimes for promotional events. The ones listed in the table below are included because the pattern in some way affects the difficulty of the solution or is notable in some other way.

Picture	Data	Comments
No Picture	Commercial name: Junior Cube Geometric shape: Cube Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... Piece configuration: 2×2×2	Mechanically identical to the Pocket Cube. However, much easier to solve as it only uses two colours.
http://www.twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=663	Commercial name: Fooler Cube Geometric shape: Cube Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... Piece configuration: 3×3×3	Mechanically identical to the standard 3×3×3 cube but not a real puzzle since all the faces are the same colour. There are also cubes which have only three colours, either one colour per pair of opposite faces or one colour per layer. Also known as the Dodo cube.
	Commercial name: Calendar Cube Geometric shape: Cube Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... Piece configuration: 3×3×3	Mechanically identical to the standard 3×3×3 cube, but with specially printed stickers for displaying the date. Much easier to solve since five of the six faces are ignored. Ideal produced a commercial version during the initial cube craze. Sticker sets are also available for converting a normal cube into a calendar.
http://www.twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=63	Rubik's Cube for the blind Geometric shape: Cube Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... Piece configuration: 3×3×3	Mechanically identical to the standard 3×3×3 cube. However the pieces are in some way tactile to allow operation by blind persons, or to solve blindfolded. The cube pictured is the original "Blind Man's Cube" made by Politechnika. This is coloured the same as the standard cube, but there is an embossed symbol on each square which corresponds to a colour.
	Commercial Name: Magic Cube Geometric shape: Cube Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... Piece configuration: 3×3×3	Mechanically identical to the standard 3×3×3 cube. However, the numbers on the centre pieces force the solver to become aware that each one can be in one of four orientations, thus hugely increasing the total number of combinations. The number of combinations of centre face orientations is 4⁶. However, odd combinations (overall odd number of rotations) of the centre faces cannot be achieved with legal operations. The increase is therefore x2¹¹ over the original making the total approximately 10²⁴ combinations. This adds to the difficulty of the puzzle but not astronomically; only one or two additional algorithm Algorithm In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning... s are required to effect a solution. Note that the puzzle can be treated as a number magic square Magic square In recreational mathematics, a magic square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n2... puzzle on each of the six faces with the magic constant Magic constant The magic constant or magic sum of a magic square is the sum of numbers in any row, column, and diagonal of the magic square. For example, the magic square shown below has a magic constant of 15.... being 15 in this case.
http://www.twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=237	Patterned cubes Geometric shape: Cube Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... Piece configuration: 3×3×3	Mechanically identical to the standard 3×3×3 cube. The pattern, which is often a promotional logo or pictures of performers, will usually have the effect of making the orientation of the centre pieces 'count' in the solution. The solution is therefore the same as the 'Magic Square' cube above.
	Commercial name: Sudoku Cube Geometric shape: Cube Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... Piece configuration: 3×3×3	Identical to the Rubik's Cube in mechanical function, it adds another layer of difficulty in that the numbers must all have the same orientation and there are no colors to follow. The name reflects its superficial resemblance to the two-dimensional Sudoku Sudoku is a logic-based, combinatorial number-placement puzzle. The objective is to fill a 9×9 grid with digits so that each column, each row, and each of the nine 3×3 sub-grids that compose the grid contains all of the digits from 1 to 9... number puzzle.
Over The Top	Commercial name: Over The Top Geometric shape: Cube Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... Piece configuration: 17x17x17 Inventor: Oskar van Deventer	A remarkable extension to the basic Rubik's Cube. Experimental; made by 3-D printing of plastic. Corners are much larger in proportion, and edge pieces match that larger dimension; they are narrow, and do not resemble cubes. The rest of the cubelets are 15x15 arrays on each side of the whole cube; as planned, they would be only 4 mm on a side. The mechanism is a 3x3x3 core, with thin "vanes" for the center edges; the rest of the cubelets fill in the gaps. The core has a sphere at its center. Surrounding the core are six concentric spherical shells (or more, depending on your definition). The scheme is quite different from that of Panagiotis Verdes, the inventor of the V Cubes. Mr. van Deventer is a noted inventor of puzzles.

Irregular cuboids

An irregular cuboid, in the context of this article, is a cuboid puzzle where not all the pieces are the same size in edge length. This category of puzzle is often made by taking a larger regular cuboid puzzle and fusing together some of the pieces to make larger pieces. In the formulae for piece configuration, the configuration of the fused pieces is given in brackets. Thus, (as a simple regular cuboid example) a 2(2,2)x2(2,2)x2(2,2) is a 2×2×2 puzzle, but it was made by fusing a 4×4×4 puzzle. Puzzles which are constructed in this way are often called "bandaged" cubes. However, there are many irregular cuboids that have not (and often could not) be made by bandaging.

Picture	Data	Comments
	Commercial name: Skewb Geometric shape: Cube Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... Piece configuration: -	Similar to the original Rubik's Cube, the Skewb differs in that its four axes of rotation pass through the corners of the cube rather than the centres of the faces. As a result, it is a deep-cut puzzle in which each twist scrambles all six faces.
http://www.twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=530	Bandaged Cubes Geometric shape: Cube Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... Piece configuration: various	The example shown in the link is a simple example of a large number of bandaged cubes that have been made.
	Commercial name: Square One Geometric shape: Cube Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...	A variation on the original Rubik's Cube where it can be turned in such a manner as to distort the cubical shape of the puzzle. The Square One consists of three layers. The upper and lower layers contain kite and triangular pieces. The middle layer contains two trapezoid pieces, which together may form an irregular hexagon or a square. Square One is an example of another very large class of puzzle — cuboid puzzles which have cubies that are not themselves all cuboid.

Other polyhedra

Picture	Data	Comments
	Commercial Name: Pyraminx Geometric shape: Tetrahedron Tetrahedron In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids... Piece configuration: 3×3×3	Pyramid shaped puzzle similar to Rubik's cube in operation and solution.
	Commercial Name: Pyramorphix Geometric shape: Tetrahedron Tetrahedron In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids... Piece configuration: 2×2×2	Pyramid shaped puzzle with a 2×2×2 cube mechanism.
	Commercial Name: Megaminx Geometric shape: Dodecahedron Piece configuration: Similar to 3×3×3	12-sided polyhedron puzzle similar to Rubik's cube in operation and solution.
	Commercial Name: Gigaminx, Teraminx, Petaminx Geometric shape: Dodecahedron	Megaminx variants with multiple layers per face. The Gigaminx has 2 layers per face, for a total of 5 layers per edge; the Teraminx has 3 layers per face, 7 layers per edge; and the Petaminx has 4 layers per face, 9 layers per edge.
	Commercial Name: Impossiball Geometric shape: Rounded icosahedron Icosahedron In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids.... Piece configuration: -	Rounded icosahedron Icosahedron In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids.... puzzle similar to Pocket Cube in operation and solution.
	Commercial Name: Alexander's Star Geometric shape: Great dodecahedron Piece configuration: -	12-sided Nonconvex uniform polyhedron Nonconvex uniform polyhedron In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting... puzzle similar to Rubik's cube in operation and solution.
	Commercial Name: BrainTwist Geometric shape: Tetrahedron Tetrahedron In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids... Piece configuration: -	The BrainTwist is a unique tetrahedral puzzle with an ability to "flip", showing only half of the puzzle at a time.
	Commercial Name: Dogic Geometric shape: Icosahedron Icosahedron In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids.... Piece configuration: -	The Dogic is an icosahedron cut into 60 triangular pieces around its 12 tips and 20 face centers.
	Commercial Name: Skewb Diamond Geometric shape: Octahedron Octahedron In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.... Piece configuration: 3×3×3	An octahedral variation on the Skewb, it is a deep-cut puzzle very similar to the Skewb and is a dual-polyhedron transformation.
	Commercial Name: Skewb Ultimate Geometric shape: Dodecahedron Piece configuration: -	While appearing more difficult than the Skewb Diamond, it is functionally the same as the Skewb and Skewb Diamond. The puzzle is cut in a different manner but the same solutions can be used to solve it by identifying what pieces are equivalent.
	Commercial Name: Barrel Cube Geometric shape: Octagonal Prism Octagonal prism In geometry, the octagonal prism is the sixth in an infinite set of prisms, formed by square sides and two regular octagon caps.If faces are all regular, it is a semiregular polyhedron.- Use :... Piece configuration: 3×3×3	Mechanically identical to the 3×3×3 cube. It does, however, have an interesting difference in its solution. The vertical corner columns are different colours to the faces and do not match the colours of the vertical face columns. The corner columns can therefore be placed in any corner. On the face of it, this makes the solution easier, however odd combinations of corner columns cannot be achieved by legal moves. The solver may unwittingly attempt an odd combination solution, but will not be aware of this until the last few pieces.
	Commercial Name: Diamond Cube Geometric shape: Rhombicuboctahedron Rhombicuboctahedron In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each. Note that six of the squares only share vertices with the triangles... Piece configuration: 3×3×3	Mechanically identical to the 3×3×3 cube although the example pictured is easier to solve due to the restricted colour scheme. This puzzle is a rhombicuboctahedron Rhombicuboctahedron In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each. Note that six of the squares only share vertices with the triangles... but not a uniform Uniform polyhedron A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive... one as the edge pieces are oblong rather than square. There is in existence a similar puzzle actually called Rhombicuboctahedron which is uniform.
	Commercial Name: Pyraminx Crystal Geometric shape: Dodecahedron Piece configuration: -	A dodecahedron cut into 20 corner pieces and 40 edge pieces. It is similar to the Megaminx, but is deeper cut, giving edges that behave differently from the Megaminx's edges when twisted.
	Commercial Name: Magic 120-cell Geometric shape: 120-cell Piece configuration: 3×3×3×3	Virtual 4-dimensional puzzle, the 4-D analogue of the Megaminx.

Other geometric shapes

Picture	Data	Comments
	Commercial Name: Magic Ball Geometric shape: Sphere Sphere A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point... Piece configuration: 3×3×3	Also known as Rubik's Sphere. Mechanically identical to the 3×3×3 cube in operation and solution. The only practical difference is that it is rather hard to grip. This accounts for the poor condition of this specimen, as the coloured stickers tend to get forced off in use.

Non-Rubik style three-dimensional

Picture	Data	Comments
	Commercial Name: Rubik's Clock Piece configuration: 3×3×2 12-position dials	Rubik's Clock is a two-sided puzzle, each side presenting nine clocks to the puzzler. There are four wheels, one at each corner of the puzzle, each allowing the corresponding corner clock to be rotated directly.
	Commercial Name: Rubik's Snake Piece configuration: 24 pieces	Some would not count this as a combinational puzzle despite it bearing the Rubik name. Also known as Rubik's Twist. There is no one solution to this puzzle but multiple different shapes can be made.

Two-dimensional

Picture	Data	Comments
	Sliding piece puzzle Piece configuration: 7×7	These ubiquitous puzzles come in many sizes and designs. The traditional design is with numbers and the solution forms a magic square Magic square In recreational mathematics, a magic square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n2... . There have been many different designs, the example shown here uses graphic symbols instead of numbers. The solution requires that there are no repeated symbols in any row column or diagonal. The picture shows the puzzle unsolved.
	Sliding piece puzzle with picture Piece configuration: 7×7	Mechanically, no different from the puzzle above. However, the picture on the pieces gives the puzzle something of the nature of a jigsaw in addition to being a combination puzzle. Note that the picture consists of multitude of a polyhedra which have been made into Rubik puzzles.
	Fifteen puzzle Piece configuration: 4×4-1	The original sliding piece puzzle.
	Rubik's Magic	Not entirely 2D. Involves flipping parts back onto itself.
	Rubik's Master Magic	The five ringed version of the Rubik's Magic
	Commercial name:2D Magic Cube Geometric shape:Square Square (geometry) In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles... Piece configuration: 3×3	Another virtual puzzle in the Rubik series, but this time a very simple one.
	Klotski Piece configuration: 4×5-2 with some fused pieces	A traditional sliding piece puzzle. There are now endless variations of this original puzzle implemented as computer games.

External links

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