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 . The puzzle is solved
by 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
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
Rubik's Cube which
are manipulated by rotating a layer of pieces are popularly called
TWISTY PUZZLES.
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.
CONTENTS
* 1 Properties
* 1.1 Regular cuboids
* 1.1.1 Pattern variations
* 1.2 Irregular cuboids
* 1.3 Other polyhedra
* 1.4 Other geometric shapes
* 1.5 Non-Rubik style three-dimensional
* 1.6 Two-dimensional
* 1.7 One-dimensional
* 1.8 Geared puzzles
* 2 See also
* 3 References
* 4 External links
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.
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. Pieces
are often referred to as "cubies".
PICTURE
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Commercial name: Pocket Cube
Geometric shape:
Cube
Cube
Piece configuration: 2×2×2
Main article: Pocket
Cube
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.
Commercial name: Rubik's Cube
Geometric shape:
Cube
Cube
Piece configuration: 3×3×3
Main article: Rubik\'s
Cube
Cube The original Rubik's Cube
Commercial name: Rubik's Revenge
Geometric shape:
Cube
Cube
Piece configuration: 4×4×4
Main article: Rubik\'s Revenge Solution is much the same as
3×3×3 cube except additional (and relatively simple) algorithm (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
Piece configuration: 5×5×5
Main article: Professor\'s
Cube
Cube Solution is much the same as
3×3×3 cube except additional (and relatively simple) algorithm (s)
are required to unscramble the centre pieces and edges.
Commercial name: V-CUBE
Geometric shape:
Cube
Cube
Piece configuration: 2×2×2 to 11×11×11
Main articles: V-
Cube
Cube 6 , V-
Cube
Cube 7 , V-
Cube
Cube 8 " Panagiotis Verdes
holds an (expired) 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
Tesseract
Piece configuration: 3×3×3×3
Main article:
N-dimensional sequential move puzzles 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.
* Slim Tower or Tower
Cube
Cube
* Rubik\'s Tower
* 3×4×4
* 2×2×6
Non-uniform cuboids
Geometric shape:
Cuboid
Cuboid
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. The Tower
Cube
Cube was manufactured by
Chronos
Chronos and
distributed by Japanese company Gentosha Education ; it is the third
"Okamoto Cube" (invented by
Katsuhiko Okamoto ). It does not change
form, and the top and bottom colors do not mix with the colors 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
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 accredited 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.
Main article: Void
Cube
Cube 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.
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
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Commercial name: Junior Cube
Geometric shape:
Cube
Cube
Piece configuration: 2×2×2
Main article: Pocket
Cube
Cube § Variants Mechanically identical to
the Pocket Cube. However, much easier to solve as it only uses two
colours.
Commercial name: Fooler Cube
Geometric shape:
Cube
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
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
Rubik's Cube for the blind
Geometric shape:
Cube
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
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 (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.
Patterned cubes
Geometric shape:
Cube
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
Sudoku Cube
Geometric shape:
Cube
Cube
Piece configuration: 3×3×3
Main article:
Sudoku
Sudoku
Cube
Cube Identical to the
Rubik's Cube
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 number puzzle.
Over The Top
Commercial name: Over The Top
Geometric shape:
Cube
Cube
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 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.
Surrounding the core were 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. Once built,
however, the mechanism had excessive friction, and Mr. van Deventer
redesigned it for a much simpler structure. 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
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Commercial name: Skewb
Geometric shape:
Cube
Cube
Piece configuration: 3x3x3
Main article:
Skewb
Skewb Similar to the original Rubik's Cube, the
Skewb
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
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
Cube
Main article:
Square One (puzzle)
Square One (puzzle) A variation on the original
Rubik's Cube
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.
Golden Cube
Commercial name: Tony Fisher\'s Golden Cube
Geometric shape:
Cube
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
Cube (Flower Box)
Geometric shape:
Cube
Cube
OTHER POLYHEDRA
PICTURE
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Commercial Name: Pyraminx
Geometric shape:
Tetrahedron
Tetrahedron
Piece configuration: 3×3×3
Main article:
Pyraminx
Pyraminx Tetrahedral-shaped puzzle with axes on the
corners and trivial tips. It was invented in 1970 by
Uwe Mèffert .
Commercial Name: Pyramorphix
Geometric shape:
Tetrahedron
Tetrahedron
Piece configuration: 2×2×2
Main article:
Pyramorphix
Pyramorphix Edge turning tetrahedron shaped puzzle
with a 2×2×2 cube mechanism.
Commercial Name: Megaminx
Geometric shape:
Dodecahedron
Dodecahedron
Piece configuration: 3×3×3
Main article:
Megaminx
Megaminx 12-sided polyhedron puzzle similar to
Rubik's Cube
Rubik's Cube in operation and solution.
Commercial Name: Gigaminx, Teraminx, Petaminx
Geometric shape:
Dodecahedron
Dodecahedron
Piece configuration: gigaminx is 5x5x5, teraminx is 7x7x7, petaminx
is 9x9x9
Megaminx
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
Piece configuration: 2x2x2
Main article:
Impossiball
Impossiball Rounded icosahedron puzzle similar to
Pocket
Cube
Cube in operation and solution.
Commercial Name: Alexander's Star
Geometric shape:
Great dodecahedron
Great dodecahedron
Piece configuration: 3x3x3
Main article: Alexander\'s Star 12-sided Nonconvex uniform
polyhedron puzzle similar to
Rubik's Cube
Rubik's Cube in operation and solution.
Commercial Name: BrainTwist
Geometric shape:
Tetrahedron
Tetrahedron
Piece configuration: 2x2x2
Main article:
BrainTwist 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
Piece configuration: 4x4x4
Main article:
Dogic The
Dogic is an icosahedron cut into 60
triangular pieces around its 12 tips and 20 face centers.
Commercial Name:
Skewb
Skewb Diamond
Geometric shape:
Octahedron
Octahedron
Piece configuration: 3x3x3
Main article:
Skewb
Skewb Diamond An octahedral variation on the Skewb,
it is a deep-cut puzzle very similar to the
Skewb
Skewb and is a
dual-polyhedron transformation.
Commercial Name:
Skewb
Skewb Ultimate
Geometric shape:
Dodecahedron
Dodecahedron
Piece configuration: 3x3x3
Main article:
Skewb
Skewb Ultimate While appearing more difficult than
the
Skewb
Skewb Diamond, it is functionally very similar to the
Skewb
Skewb and
Skewb
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. Because faces of the
Skewb
Skewb Diamond correspond to corners
of the
Skewb
Skewb Ultimate, an additional constraint on the orientation of
these pieces appears. Any
Skewb
Skewb Diamond solution thus requires a few
additions in order to solve the
Skewb
Skewb Ultimate.
Commercial Name: Barrel Cube
Geometric shape:
Octagonal Prism
Octagonal Prism
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
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 but not a uniform one as the edge
pieces are oblong rather than square. There is in existence a similar
puzzle actually called
Rhombicuboctahedron
Rhombicuboctahedron which is uniform.
Commercial Name:
Pyraminx
Pyraminx Crystal
Geometric shape:
Dodecahedron
Dodecahedron
Piece configuration: 3x3x3
Main article:
Pyraminx
Pyraminx Crystal 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
Main article:
N-dimensional sequential move puzzles § Magic 120-cell
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
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
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Commercial Name: Rubik's Clock
Piece configuration: 3×3×2 12-position dials
Main article: Rubik\'s Clock
Rubik's Clock
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: 1x1x24
Main article: Rubik\'s Snake 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
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Sliding piece puzzle
Piece configuration: 7×7
Main article:
Sliding 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.
Sliding piece puzzle with picture
Piece configuration: 7×7
Main article:
Sliding puzzle Mechanically, no different from the
puzzle above. However, the picture on the pieces gives it something of
the nature of a jigsaw puzzle, in addition to being a combination
puzzle. Note that the picture consists of a multitude of polyhedra
which have been made into Rubik puzzles.
Fifteen puzzle
Piece configuration: 4×4-1
Main article:
Fifteen puzzle The original sliding piece puzzle.
Rubik's Magic
Main article: Rubik\'s Magic Not entirely 2D. Involves flipping
parts back onto itself.
Rubik's Master Magic
Main article: Rubik\'s Magic: Master Edition The five ringed
version of the Rubik's Magic
Commercial name:2D Magic Cube
Geometric shape:Square
Piece configuration: 3×3
Main article:
N-dimensional sequential move puzzles § 3x3 2D square
Another virtual puzzle in the Rubik series, but this time a very
simple one.
Klotski
Piece configuration: 4×5-2 with some fused pieces
Main article:
Klotski A traditional sliding piece puzzle. There
are now endless variations of this original puzzle implemented as
computer games.
Geranium
Piece configuration: 5 intersecting circular rotational groups of
oddly shaped pieces
A rotating piece puzzle. Some rank its difficulty very high
compared to complex 3D puzzles. There are other versions of this
puzzle type including "Mini", "Pocket" and "Super", which have 2, 3
and 10 intersecting circles. There is an "Upgrade" mod which splits
some of the large pieces into smaller ones. This puzzle's current
production status is unknown.
ONE-DIMENSIONAL
Tricky Animals. The puzzle consists of a permutation of animals. The
user has three buttons to solve the puzzle: A: permutes the first two
animals. X: permutes the animals in the middle. B: permutes the last
two animals. There are over 1500 puzzles in increasing difficulty.
GEARED PUZZLES
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PICTURE
DATA
COMMENTS
Gear cube
This twisty puzzle was invented by
Oskar van Deventer .
Gear cube extreme
Gear mixup
Gear 5x5
gear cube ultimate
David Gear cube
Gear shift
SEE ALSO
*
N-dimensional sequential move puzzles
*
Puck puzzle
REFERENCES
* ^ "MagicCube5D Hall of Insanity".
* ^ "2×2×3 (aka: Slim Tower)". TwistyPuzzles.com.
* ^ "Tower Cube" (in Japanese). Gentosha Education.
* ^ "2×3×3". TwistyPuzzles.com.
* ^ "Rubik’s Tower 2×2×4".
* ^ "Specter Cube". TwistyPuzzles.com.
* ^ "2×2×6". TwistyPuzzles.com.
* ^ "Collection of cube puzzles". The
Puzzle
Puzzle Museum. January 2003.
* ^ Slocum, Jerry (2009), The Cube. The Ultimate Guide to the
World’s Best Selling Puzzles Published by Black Dog & Leventhal
Publishers, Inc (ISBN 978-1-57912-805-0 )
* ^ Slocum, Jerry (2009), The Cube. The Ultimate Guide to the
World’s Best Selling Puzzles Published by Black Dog & Leventhal
Publishers, Inc (ISBN 978-1-57912-805-0 )
* ^ Slocum, Jerry (2009), The Cube. The Ultimate Guide to the
World’s Best Selling Puzzles Published by Black Dog & Leventhal
Publishers, Inc (ISBN 978-1-57912-805-0 )
* ^ Tony Durham, New Scientist, page 209, 9 September 1982
* ^ "top 5 hardest massproduced puzzles". TwistyPuzzles.com Forum.
EXTERNAL LINKS
* A large database of twisty puzzles
* The
Puzzle
Puzzle Museum
* The Magic Polyhedra Patent Page
* v
* t
* e