en>Headbomb: /* References */Various citation cleanup and WP:AWB general fixes using AWB

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References: Various citation cleanup and WP:AWB general fixes using AWB

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{{Notability|August 2010|date=September 2010}}
[[Image:Helicube-black.jpg|thumb|right|Mèffert's Helicopter Cube, black body]]
The '''Helicopter Cube''' is a [[Rubik's Cube]]-like puzzle invented by Adam G. Cowan in 2005 and built in 2006.<ref>{{cite web | url = http://www.mefferts.com/products/details.php?lang=en&category=13&id=327 | title=Helicopter Cubes Black body | publisher = Mèffert's | accessdate = 2010-09-01 | quote = The Helicopter Cube was conceived by Adam G. Cowan in 2005, but wasn’t built until 2006, when Adam discovered that 3D printing could be used to realize the parts. }}</ref><ref>{{cite web | url = http://www.puzzlemaster.ca/browse/other/3256-helicopter-cube-white-body | title = Helicopter Cube - White Body | publisher = Puzzle Master Inc. | accessdate = 2010-09-01 }}</ref><ref>{{cite web | url = http://puzzles.schwandtner.info/list.html | title = Extremely Puzzling | author = Goetz Schwandtner | accessdate = 2010-09-01 | quote = Helicopter Cube black ... By Adam Cowan }}</ref><ref name="zanden">{{cite web | url = http://www.tomvanderzanden.nl/puzzle.php?puz=Curvy%20Copter | title = Curvy Copter | accessdate = 2010-09-01 | author = Tom van der Zanden | quote = The Curvy Copter is my most popular puzzle yet. It is a variation on Adam G. Cowan's Helicopter Cube. }}</ref><ref>{{cite web | title = System of twisty puzzles | url = http://deep-cut.net/system/single_cut.htm | quote = Helicopter Cube was designed and built by Adam G. Cowan (Puzzlemaster42) and Katsuhiko Okamoto (Katsuhiko) in 2007 | accessdate = 2010-09-01 }}</ref><ref name="fan2cube">{{cite web | url = http://fan2cube.free.fr/helicopter1.php | title = L'Helicopter Cube (French) | accessdate = 2010-09-01 | publisher = fan2cube }}</ref><ref>{{cite web | url = http://www.puzzleforge.com/main/index.php?option=com_content&view=article&id=49:hcannounce&catid=1:latest-news&Itemid=50 | title = The Helicopter Cube | accessdate = 2010-09-01 | publisher = Puzzle Forge | author = Jason Smith }}</ref> It is also in the shape of a [[cube]], but cut differently, and twists around cube edges rather than cube faces. The purpose of the puzzle is to scramble the colors, and then restore them back to their original state of a single color per face.

==Description==

[[Image:Helicube-scrambled.jpg|thumb|right|Helicopter cube, scrambled]]
The Helicopter Cube is made in the shape of a cube, cut into 8 corner pieces and 24 face center pieces. Each corner piece has 3 colors, and each face center piece has only a single color. Unlike the Rubik's Cube, its faces do not rotate; rather, the pieces are scrambled by rotating around a cube edge.

When twisting the puzzle, a 180° turn exchanges two corner pieces and swaps two pairs of face center pieces, but preserves the cube shape. The entire puzzle can be scrambled in this way.

However, it is also possible to twist an edge by ~71°, such that the base of two groups of a corner piece and a face center piece each is aligned with the rotational plane of a different edge. The second edge can then be rotated, thus intermixing the corner pieces and the face center pieces and leaving the puzzle in a non-cubical shape. This kind of intermixing is known as a ''jumbling move''. Due to the differing shapes of the intermixed pieces, some rotations possible in the cubical shape may no longer be possible in the jumbled shape. By using a combination of such "jumbling" moves, it is possible to return to cubical shape but with some face center pieces in the wrong orientation, thus jutting outwards like spikes rather than lie flat on the face of the cube. More subtle changes may also be introduced, which are described later.

There are four variants of the Helicopter Cube:
* the original Helicopter Cube, manufactured by The Twisty Store (sold also by [[Uwe Mèffert]]), consisting only of 8 corner pieces and 24 face center pieces;
* the "Curvy Copter" by Tom van der Zanden,<ref name="zanden"/> which has an additional 12 edge pieces with 2 colors each;
* the "Curvy Copter Plus", also created by Tom van der Zanden, with additional cuts in the middle of the face center pieces, allowing the puzzle to jumble even more;
* the "Helicopter Skewb", also by Tom Van Der Zanden, which looks exactly the same as the original Helicopter Cube but it can also twist like the [[Skewb]].

:{|
|[[Image:Helicube-jumbling-1.jpg|thumb|The ~71° turn in preparation for a jumbling move]]
|[[Image:Helicube-jumbling-2.jpg|thumb|The start of a jumbling move]]
|[[Image:Helicube-4.jpg|thumb|Thoroughly jumbled helicopter cube]]
|}

==Solutions==

[[Image:Helicube-white.jpg|thumb|right|Mèffert's Helicopter Cube, white body, solved]]
If the puzzle is only scrambled using 180° twists, then it is obviously solvable using only 180° twists. However, if some jumbling moves were made, even if the puzzle was subsequently returned to cube shape, it may not be possible to solve it using only 180° twists. The reason for this is that using only 180° twists, each face center piece can only be permuted within a 6-member cycle, often referred to as its ''[[Orbit (group theory)|orbit]]''.<ref name="fan2cube"/> Face center pieces in different orbits cannot be interchanged using only 180° twists. However, jumbling moves are able to permute face center pieces between different orbits, thus leaving the puzzle in a state that cannot be solved by 180° twists alone.

==Number of combinations==

{{Expand section|date=May 2010}}
The Helicopter Cube has <math>\frac{7!\cdot 3^6 \cdot 6!^4}{2}</math> combinations.<ref name="fan2cube"/>

The expanded number is 493 694 233 804 800 000 (approximately four hundred ninety three quadrillion).

==See also==

* [[Square One (puzzle)]], another shape-changing cubical puzzle.

==References==
{{Reflist}}

{{Rubik's Cube}}

[[Category:Rubik's Cube]]
[[Category:Combination puzzles]]
[[Category:Mechanical puzzles]]
[[Category:Puzzles]]

Bochner measurable function - Revision history

en>Headbomb: /* References */Various citation cleanup and WP:AWB general fixes using AWB