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From the outside, a Mixup Cube looks almost like a regular cube. The difference shows up when you turn the middle layer by 45 degrees instead of 90. This lets you make moves that are impossible on a classic cube, and the edges and centers start swapping places. After a few such moves, a piece that used to be a center can end up in an edge position. The first 3x3x3 Mixup cube was designed by Oskar van Deventer and is produced by WitEden. In this category you will find WitEden models as well as FangShi models from the LimCube series.
On a regular 3x3x3 cube, the centers stay in place and define the color of each face. A Mixup cube works differently, because the centers travel around the cube together with the edges. So first you have to restore the cube's normal shape and send every piece back to its proper place, and only then finish solving. Knowing a 3x3 method helps a lot, but it is not enough on its own. You will also need your own algorithms or a bit of experimenting. Mixup is not a WCA event. People usually buy it as a brain teaser and a collector's item rather than for practicing solve times.
If this is your first Mixup cube, choose the WitEden & Oskar 3x3x3 Mixup. Its mechanism is based on the familiar 3x3 cube, so it is easier to understand how the pieces swap places. If you are comfortable with the 4x4, you can go straight for the 4x4x4 Mixup. The Edge-rebuilt and Plus versions, the 30 degrees models and the FangShi LimCube puzzles are for those who already know the basic Mixup and are looking for a harder challenge or something new for their collection.