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.
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 | Comments |
Commercial name: Pocket Cube Geometric shape: Cube Piece configuration: 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. | |
Commercial name: Rubik's Cube Geometric shape: Cube Piece configuration: 3×3×3 | The original Rubik's Cube | |
Commercial name: Rubik's Revenge Geometric shape: Cube Piece configuration: 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. | |
Commercial name: Professor's Cube Geometric shape: Cube Piece configuration: 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. | |
Commercial name: V-CUBE Geometric shape: Cube Piece configuration: 2×2×2 to 11×11×11 | 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. | |
4-Dimensional puzzle Geometric shape: Tesseract 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. 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. | |
| Non-uniform cuboids Geometric shape: Cuboid Piece configuration : 2×2×3 Piece configuration : 2×3×3 Piece configuration : 3×4×4 Piece configuration : 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. The Tower Cube 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 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. 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”. | |
Extended cubes Geometric shape: Box Piece configuration: 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. | |
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 with 1 iteration Piece configuration: 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: Over The Top Geometric shape: Cube Piece configuration: 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 2017, it is being mass produced by the Chinese company YuXin. |
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 |
Commercial name: Junior Cube Geometric shape: Cube Piece configuration: 2×2×2 | Mechanically identical to the Pocket Cube. However, much easier to solve as it only uses two colours. | |
Commercial name: Fooler Cube Geometric shape: Cube 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 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. | |
Rubik's Cube for the blind Geometric shape: Cube 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 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 effect 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. | |
Patterned cubes Geometric shape: Cube 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 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 colours to follow. The name reflects its superficial resemblance to the two-dimensional Sudoku number puzzle. |
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, a 2x2x2 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 be made by bandaging.| Picture | Data | Comments |
Commercial name: Skewb Geometric shape: Cube Piece configuration: 3x3x3 | 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. | |
Bandaged Cubes Geometric shape: Cube Piece configuration: various | The example shown in the link is a simple example of a large number of bandaged cubes that have been made. A bandaged cube is a cube where some of the pieces are stuck together. | |
Commercial name: Square One Geometric shape: Cube | 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. | |
Commercial name: Tony Fisher's Golden Cube Geometric shape: Cube | First rotational puzzle created that has just one colour, requiring the solver to restore the puzzle to its original cube form without colour aids. | |
Commercial name: Lan Lan Rex Cube Geometric shape: Cube |
Other polyhedra
Other geometric shapes
| Picture | Data | Comments |
Commercial Name: Magic Ball Geometric shape: Sphere 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. |