Helicopter Cube

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. 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

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, 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.

The ~71° turn in preparation for a jumbling move

The start of a jumbling move

Thoroughly jumbled helicopter cube

Solutions

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. 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

A scrambled Helicopter Cube that isn't jumbled (mixed with only 180 degree twists) has ${\frac {7!\cdot 3^{6}\cdot 6!^{4}}{2}}$ combinations. The expanded number is 493,694,233,804,800,000 (approximately four hundred ninety three quadrillion).

A Helicopter Cube that is scrambled with jumbling moves but still retains its cube shape has ${\frac {7!\cdot 3^{6}\cdot 24!}{4!^{6}}}$ combinations. The expanded number is 11,928,787,020,628,077,600,000 (approximately 12 sextillion)

There are a total of 1,305,133 possible shapes when jumbled. Multiplying this with the previous result gives 15,568,653,590,593,384,802,320,800,000 (approximately 15 octillion) jumbled positions altogether.

See also

• Square One (puzzle), another shape-changing cubical puzzle.