BrainTwist
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The Hoberman BrainTwist is a 3D mechanical puzzle designed and marketed by Chuck Hoberman's company Hoberman Designs. The puzzle is in the same family as the Rubik's Cube and other puzzles that involve manipulating and scrambling colored face elements with the goal of returning them to their original order from a randomized state, commonly called twisty puzzles. This puzzle is unique in that in addition to solving one set of tetrahedral faces the puzzle can be flipped inside-out through an intermediate stellated shape to reveal another (dual) tetrahedron with a set of 4 different colored faces. The puzzle also has an alternate solution in which the apices are each a uniform color.
Basic elements
The BrainTwist consists of 12 colored triangular elements each with a detent to aid in aligning the apices of the puzzle after rotating them. There are eight colors in the puzzle: red, orange, yellow, light and dark green, blue, violet and magenta. Each of the 12 basic units has a unique combination of two of the colors - one per side. In the tetrahedral configuration each apex consists of 3 such elements. Each face of the tetrahedron is also composed of 3 basic color elements.
Manipulation
When folded into the tetrahedral arrangement, each of the 4 apices (corners) can be rotated so that the elements are oriented toward one of 3 faces. The puzzle locks into position when an element is aligned with a face. 3 clockwise rotations or 3 counter clockwise rotations returns the apex to its original orientation.
By lifting the three elements of a face and bringing them towards each other, the entire puzzle opens up into the stellated form shown above. Continuing the "flip" move completely inverts the tetrahedron into its dual tetrahedron. The elements that were face elements now comprise the apices of the new tetrahedron and vice versa.
By performing arbitrary combinations of flips and rotations the pieces can be scrambled almost completely. The only restriction is that the mechanism prevents a 180 degree rotation of a piece so that the colors on the obverse and reverse of each piece never interchange. Consequently, the two tetrahedrons are restricted to show only 4 of the eight colors at a time. There is a red-orange-violet-magenta (ROVM) tetrahedron and a yellow-blue-light green-dark green (YBLD) tetrahedron.
Solutions
A clue to solving a shuffled BrainTwist puzzle is to examine the stellated configuration. It is possible to solve for the ROVM tetrahedron while leaving the YBLD tetrahedron shuffled and vice versa. The only way to make sure the puzzle is completely solved is to check that the adjacent colors make sense in the stellated configuration
The puzzle is chiral: if one tries to build the faces one at a time then one might end up putting colors clockwise that should have been placed counterclockwise. The puzzle can only be solved if, for example, facing the Red face in stellated form the Blue, LightGreen, and Yellow faces proceed clockwise in that order (BLY). Sometimes this may not work.