Beginner Method

The first step to solve a 4x4 is to solve the center pieces as the 4x4 does not
have center pieces.

Scrambled 4x4
Scrambled 4x4
Centers Completed
Centers Completed

Next, solve the edges by matching the seperated edge pieces.

Centers Completed
Centers Completed
Edges Completed
Center and Edge pieces completed

Finally, solve the 4x4 like a 3x3

Edges Completed
Center and Edge pieces completed
Completed 4x4
Completed 4x4

There is a weird thing about 4x4, and that is parity errors. These is due
to the absence of center pieces in the 4x4. There are two parity cases
and there are algorithms to solve each of them.

First Parity case
First Parity case
Second Parity case
Second Parity case

Now you have a completed 4x4!

Below is an in-depth tutorial on the 4x4 Beginner Method

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This method can be applied to bigger cubes such as 5x5. See video below