Combination puzzle

A combination puzzle, also known as a sequential move puzzle, is a puzzle which consists of a set of pieces which can be manipulated into different combinations by a group of operations. Many such puzzles are mechanical puzzles of polyhedral shape, consisting of multiple layers of pieces along each axis which can rotate independently of each other. Collectively known as twisty puzzles, the archetype of this kind of puzzle is the Rubik's Cube. Each rotating side is usually marked with different colours, intended to be scrambled, then solved by a sequence of moves that sort the facets by colour. Generally, combination puzzles also include mathematically defined examples that have not been, or are impossible to, physically construct.

Description

A combination puzzle is solved by achieving a particular combination starting from a random combination. 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, 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. Puzzles like the Rubik's Cube which are manipulated by rotating a section of pieces are popularly called twisty puzzles. They are often face-turning, but commonly exist in corner-turning and edge-turning varieties.
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 Rubik's 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 puzzle, simulated by the MagicCube4D software.

Types

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 polyhedra have been made.

Regular cuboids

A cuboid is a rectilinear polyhedron. That is, all its edges form right angles. Or in other words, 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. Pieces are often referred to as "cubies".

Picture	Data	Geometric shape	Piece configuration	WCA event	Comments
Image:Pocket cube solved.jpg\|2×2×2 Cube Puzzle\|frameless\|upright=0.5	Commercial name: Pocket Cube	Cube	2×2×2		Simpler to solve than the standard cube in that only the algorithms for the corner pieces are required. It is nevertheless surprisingly non-trivial to solve.
Image:Rubiks cube solved.jpg\|3×3×3 Cube Puzzle\|frameless\|upright=0.5	Commercial name: Rubik's Cube	Cube	3×3×3		The original Rubik's Cube
Image:Rubiks revenge solved.jpg\|4×4×4 Cube Puzzle\|frameless\|upright=0.5	Commercial name: Rubik's Revenge	Cube	4×4×4		Solution is much the same as 3×3×3 cube except additional algorithm are required to unscramble the centre pieces and edges and additional parity not seen on the 3x3x3 Rubik's Cube.
Image:Professor's cube solved.jpg\|5×5×5 Cube Puzzle\|frameless\|upright=0.5	Commercial name: Professor's Cube	Cube	5×5×5		Solution is much the same as 3×3×3 cube except additional algorithm are required to unscramble the centre pieces and edges.
Image:V-Cube 6 small.jpg\|6×6×6 Cube Puzzle\|frameless\|upright=0.5	Commercial name: V-CUBE	Cube	2×2×2 to 11×11×11	6×6×6 and 7×7×7 8×8×8 and higher	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 - 9×9×9 cubes.
Image:4-cube solved.png\|3×3×3×3 Hypercube Puzzle\|frameless\|upright=0.5	4-Dimensional puzzle	Tesseract	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. Significantly 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 5-dimensional 7×7×7×7×7 which has only been solved twice so far. However, the 6×6×6×6×6 has only been solved once, since its parity does not remain constant
Image:Cuboid collage.png\|Four different cuboid-shaped combination puzzles\|frameless\|upright=0.5	Non-uniform cuboids	Cuboid	A: 2×2×4 B: 3×3×2 C: 2×2×3 D: 3×3×4		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. The 2×2×4 is sold by Rubik's as the Rubik's Tower and is able to be configured into irregular shapes. The 2×2×3 was manufactured by Chronos and distributed by Japanese company Gentosha Education; it is the third "Okamoto Cube". It does not change form, and the top and bottom colours do not mix with the colours on the sides.
	Siamese cubes	Fused cubes	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. there is also a "2 3x3x3 fused 2x2x2" version called the fused cube. The first Siamese cube was made by Tony Fisher in 1981. This has been credited as the first example of a "handmade modified rotational puzzle".
	Commercial name: Void cube	Menger Sponge with 1 iteration	3x3x3-7.		Solutions to this cube is similar to a regular 3x3x3 except that odd-parity combinations are possible with this puzzle. This cube uses a special mechanism due to absence of a central core.
	Commercial name: Crazy cube type I Crazy cube type II	Cube	4x4x4.		The inner circles of a Crazy cube 4x4x4 move with the second layer of each face. On a crazy cube type I, they are internally connected in such a way that they essentially move as 8 distinct pieces, not 24. To solve such a cube, think of it as a 2x2x2 trapped inside a 4x4x4. Solve the 2x2x2 first, then solve the 4x4x4 by making exchanges only. Solving the type II is much more difficult.
	Commercial name: Over The Top	Cube	17x17x17		Experimental cube made by 3-D printing of plastic invented by Oskar van Deventer. 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 original 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. As of 2023, it is being mass produced by the Chinese companies YuXin and ShengShou.

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
Image:Rubik's calendar cubes.jpg\|3×3×3 calendar cube\|frameless\|upright=0.5	Commercial name: Calendar Cube Geometric shape: Cube 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.
Image:Combination Puzzle Rubik Cube with numbers.jpg\|3×3×3 Cube puzzle with numbers\|frameless\|upright=0.5	Commercial Name: Magic Cube Geometric shape: Cube 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 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 algorithms are required to affect a solution. Note that the puzzle can be treated as a number magic square puzzle on each of the six faces with the magic constant being 15 in this case.