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Shape mods change their shape as you turn them. After a few moves they no longer look like a cube. Most of them use the mechanism of a classic 3x3x3 cube, but the cuts are rotated or shifted relative to the faces. That is why the pieces come in different shapes and sizes. The best-known examples of this type are the YJ cubes: Fisher (YiLeng), Windmill (FengHuoLun) and Axis (KingKong).
If you can solve a 3x3x3, you can solve most of these puzzles with the same method. You just have to apply it to an unusual shape. The hardest part is recognizing which piece is a corner and which is an edge when the puzzle is scrambled. On the Axis, Fisher and Windmill you also need to get the center orientation right, which you usually don't check on a regular cube. Shape mods are not a WCA event. People buy them for variety and a new challenge, not to break records.
Besides the models based on the 3x3x3, this category also includes puzzles with a different mechanism and look. For example, the MoYu Puppet Cube has an offset center and from some sides looks like a 2x2. You will also find DianSheng corner-only puzzles and the UFO cube, the MoYu Four Leaf Clover and Pandora Cube, and from Z-Cube the Twisted 3x3x3, the Pentacle Cube and the festive Christmas Tree Cube.
The Fisher, Windmill and Axis are available in versions from several manufacturers, including YJ, QiYi, MoYu and ShengShou. The versions differ in build quality and how the mechanism turns. If you can't decide between two versions of the same shape mod, check the product descriptions to see whether they have magnets and what finish they have.