Combination puzzles
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Combination puzzles, also called sequential move puzzles, consist of a set of pieces which can be manipulated into different combinations by a group of operations. The puzzle consists of achieving a particular combination starting from a random (scrambled) 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 randomly 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.
Typically, these puzzles are simple in concept but can be fiendishly difficult to solve. Their popularity lies in this very thing; everyone can understand what is required and the operations needed to achieve it, but few can actually do it. Also, a brightly coloured 'toy' is inherently more interesting than dry mathematical equations, even though they are equivalent.
Although a mechanical realisation 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 realised entirely in virtual space or as a set of mathematical statements. In fact, there are some puzzles that can only be realised in virtual space. An example is the 4-dimensional 3x3x3x3 tesseract puzzle.
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[edit] Puzzle 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 polyhedra have been made. It seems, in fact, that solid geometry is hard pressed to come up with a shape that cannot be made into a combination puzzle.
[edit] Regular Cuboids
A cuboid is a rectilinear polyhedron. 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 |
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Commercial name: Rubik's Cube |
The original Rubik's Cube |
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Commercial name: Rubik's Revenge |
Solution is much the same as 3x3x3 cube except additional (and relatively simple) algorithm(s) are required to unscramble the centre pieces and also to solve parity problems not seen with the 3x3x3. |
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Commercial name: Professor's Cube |
Solution is much the same as 3x3x3 cube except additional (and relatively simple) algorithm(s) are required to unscramble the centre pieces and also to solve parity problems not seen with the 3x3x3. |
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Commercial name: Pocket Cube |
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. |
| [2] |
Commercial name: V-CUBE |
Panagiotis Verdes holds a patent to a method which is said to be able to make cubes up to 11x11x11. He has fully working products for 5x5x5, 6x6x6 and 7x7x7 cubes. |
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4-Dimensional puzzle |
This is the 4-dimensional analog of a cube and cannot, of course, 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 3x3 to the unsolved 5-dimensional 7x7x7x7x7. |
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Non-uniform box shapes |
This class of puzzle is 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 three shown here are only a sample from a very large number of examples. | |
| [6] |
Siamese cubes |
Siamese cubes are two or more puzzles that are fused so that some pieces are common to both cubes. The picture here shows two 3x3x3 cubes that have been fused. The largest example known to exist is in The Puzzle Museum [1] and consists of three 5x5x5 cubes that are siamese fused 2x2x5 in two places. |
| [7] |
Extended cubes |
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. They can however, add slightly to the difficulty of the solution as they can obstruct some moves. This will prevent certain algorithms being executed. Again, there is a very large variety of these around, all custom made. |
| [8] |
Commercial name: Boob cube |
Very possibly the simplest regular cuboid puzzle to solve. Completely trivial solution as the puzzle consists of only two cubies. |
[edit] 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. It would be impossible, and probably pointless, to list every model that has ever been. 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 |
Mechanically identical to the Pocket Cube. However, much easier to solve as it only uses two colours. |
| [9] |
Commercial name: Fooler Cube |
Mechanically identical to the standard 3x3x3 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. These are easier to solve than the standard cube but definitely non-trivial. |
| [10] |
Rubik's Cube for the blind |
Mechanically identical to the standard 3x3x3 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. This cube is no longer on sale and is something of a collectors item commanding a high price. However, there are several other "blind" cubes available. One of the more presentable for the home (rather than something to throw in the toy box is this one[11] made of a bronze like metal. |
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Commercial Name: Magic Cube |
Mechanically identical to the standard 3x3x3 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 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. |
| [12] |
Patterned cubes |
Mechanically identical to the standard 3x3x3 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. In practice, the cube may be harder to solve because the nature of the pattern causes confusion. The tartan cube pictured here, for instance, could cause eye strain from the Fresnel interference patterning. |
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Commercial name: Sudokube |
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, instead the cuber must solve the Sudoku puzzle. |
[edit] 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 2x2x2 puzzle, but it was made by fusing a 4x4x4 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 |
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Commercial name: Skewb |
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. |
| [13] |
Bandaged Cubes |
The example shown here is a simple example of a large number of bandaged cubes that have been made. |
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Commercial name: Square One |
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. |
[edit] Other Polyhedra
| Picture | Data | Comments |
|---|---|---|
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Commercial Name: Pyraminx |
Pyramid shaped puzzle similar to Rubik's cube in operation and solution. |
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Commercial Name: Pyramorphix |
Pyramid shaped puzzle which functions out to be identical to a 2x2x2 cube. |
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Commercial Name: Megaminx |
12-sided polyhedron puzzle similar to Rubik's cube in operation and solution. |
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Commercial Name: Alexander's Star |
12-sided Nonconvex uniform polyhedron puzzle similar to Rubik's cube in operation and solution. |
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Commercial Name: Dogic |
The Dogic is an icosahedron cut into 60 triangular pieces around its 12 tips and 20 face centers. |
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Commercial Name: Skewb Diamond |
An octahedral variation on the Rubik's Cube, it is a deep-cut puzzle very similar to the Skewb. |
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Commercial Name: Skewb Ultimate |
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. |
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Commercial Name: |
Mechanically identical to the 3x3x3 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. |
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Commercial Name: Diamond Cube |
Mechanically identical to the 3x3x3 cube although the example pictured is easier to solve due to the restricted colour scheme. This puzzle is a rhombicuboctahedron but not a uniform one as the edge pieces are oblong rather than square. There is in existence a similar puzzle actually called Rhombicuboctahedron which is uniform. |
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Commercial Name: Magic 120-cell |
Virtual 4-dimensional puzzle, the 4-D analogue of the Megaminx. |
[edit] Other Geometric Shapes
| Picture | Data | Comments |
|---|---|---|
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Commercial Name: Magic Ball |
Also known as Rubik's Sphere. Mechanically identical to the 3x3x3 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. |
[edit] 3D Puzzles distinct from the 'Rubik' series
| Picture | Data | Comments |
|---|---|---|
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Commercial Name: Rubik's Clock |
Unofficially called Rubik's gas meter by some due to its resemblance to the little dials on old style utility company meters. Despite its obvious physical differences, the moves and algorithms of solutions of the Clock have many similarities to the Cube. |
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Commercial Name: Rubik's Snake |
Some would not count this as a combinational puzzle despite it bearing the Rubik name. |
[edit] 2D Puzzles
| Picture | Data | Comments |
|---|---|---|
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Sliding piece puzzle |
These ubiquitous puzzles come in many sizes and designs. The traditional design is with numbers and the solution forms a magic square. 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. |
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Sliding piece puzzle with picture |
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. |
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Rubik's Magic |
Not entirely 2D |
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Commercial name:2D Magic Cube |
Another virtual puzzle in the Rubik series, but this time a very simple one. |
[edit] See also
[edit] Notes
[edit] External links
- A large database of Rubik type puzzles.
- 4-D/5-D Rubik's Cube Simulator
- The Puzzle Museum
- Rubik's official site
- Short Cube history and Erno Rubik biography
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