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Crazy cubes and the DaYan Bermuda series are puzzles built on the mechanism of a regular cube, usually a 3x3. Extra cuts or bandaged pieces change the way they are solved. If you can already solve a 3x3 cube, most of the algorithms you know will still be useful, but you will have to combine them with extra setups and plan the order of your moves differently. This category is a good direction if you have mastered the 3x3 and want a new challenge rather than a bigger cube. These puzzles are made more for figuring out than for breaking time records.
Each face of a Crazy cube has a circular cut that divides it into an inner circle and a ring of pieces around it. The design determines whether the circle turns together with the layer or stays in place. In the MF8 Crazy 3x3 planet series (including Mercury, Venus, Earth, Jupiter, Uranus and Neptune), each version has a different arrangement of these faces. Mercury has one face where the circle turns together with the layer, while Venus has two: white and yellow. So each planet is a separate puzzle, not just a different colour scheme.
ShengShou (SengSo) makes Crazy cubes in 2x2, 3x3 and 4x4 sizes, including the newer v2 and V3 versions. In the SengSo Crazy 3x3 v2, the rings do not turn together with the faces. In the Calvins Puzzle Full Function Crazy 3x3x3 (Center-Locking), the circles are locked in place, yet the cube can still be fully scrambled. The MF8 Double Crazy Cube has two rings on each face. On one pair of faces both rings turn, on the second pair both stay in place, and on the third pair only the inner circle stays in place. It is a good pick if you have already solved a regular Crazy 3x3. The category also includes unusual formats such as the DianSheng Crazy 2x3x3 and the WitEden 2x2x4 Crazy.
The Bermuda series was released in 2011 and is based on the 3x3. Some of its faces have triangular cuts (a small or a large triangle) or a layout known from the Fisher Cube. The triangles join neighbouring pieces together (bandaging), so some moves cannot be made until you have aligned other layers correctly. The eight basic models are named after planets and form a cube shape when solved. They differ in their set of faces: Earth has three regular faces, two with large triangles and one Fisher-type face, while Neptune has triangles on three adjacent faces.
The Bermuda Sunflower (a star-shaped prism), the Bermuda Cylindric (an octagonal cylinder) and the red-house and green-house versions are shape mods, i.e. puzzles that change shape as you solve them. In these, bandaging affects only one face. That makes them a good first contact with the series. Planets with several triangular faces require more thinking on the last layer.