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A 13x13x13 cube has thirteen layers on every axis, 866 moving pieces and 1014 stickers or tiles on its faces. People usually buy a cube like this for the challenge itself rather than for a time on the timer. The method is the same as on a 4x4 or 7x7: centers first, then edge pairing, and finally solving it like a 3x3 with parities. With this many pieces, though, your first solve can take several hours. In this category you'll find 13x13 models from MoYu, ShengShou, YuXin and DianSheng, both magnetic and non-magnetic.
The most important decision is whether the cube should have magnets. On a cube this big, magnets help the layers line up. When a dozen or so layers aren't perfectly aligned, lockups happen easily. That's why magnetic versions are noticeably more pleasant for longer solving sessions, but they are heavier and more expensive. Non-magnetic versions are lighter and are a good choice if you just want to try a 13x13 or you're buying a cube for your collection.
Sizes also vary a lot. A smaller cube sits better in your hand and weighs less, but its pieces are tiny. A larger one gives your fingers more room on each individual layer.
This is not a good first cube. It's best to pick one up once you can comfortably solve the 5x5 and 7x7, because by then you know how to deal with parities and pairing long edges. A note for parents buying a gift: a 13x13 is a great fit for a child who already solves big cubes, not for someone who is just learning the 3x3. Big cubes should also be protected from drops. With so many small pieces, a fall from table height can make the cube pop apart.