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Solve the Centers
A 5x5 has a total of 54 center pieces (nine on each face). Eight center pieces on each face can be moved; one cannot. Begin solving the centers by creating rows of three center pieces each and matching them up with the appropriate face. Once at least two adjacent centers have been solved, the remaining centers cannot be completed without disturbing the already completed centers. Continue by using the "moving out of the way" principle: Rotate a completed row into the place of another completed row, rotate the center so that the new row is in another location, and then replace the old row. Doing this allows you to move pieces into the appropriate place without disrupting the rest of the cube.
Solve the Edges
A 5x5 has a total of 36 different edge pieces. Each edge is composed of three different pieces. To solve an edge, locate two pieces that match. Rotate them so that when facing a single side, one is on the top, and one is on the bottom. Align them and rotate the edge out of the way so that a new "broken" edge takes its place. Re-align the center and repeat this process for the third piece on the same edge. Continue on all 12 edges until they are all complete.
Complete the 5x5 Just Like a 3x3
After solving the centers and edges, the 54 center pieces have become six large "center squares." The 36 edge pieces have become 12 large "edges." The eight corners still remain. With 12 edges, six centers and eight corners, the 5x5 now has the same dimensions as a 3x3. The pieces are still scrambled, but you can orient them properly using any favored method to solve a 3x3.
The most basic approach to solving a 3x3 is the "layer by later" method, in which each layer is solved one at a time. Other methods include the Petrus method, invented by Lars Petrus, and the Fridrich method, invented by Jessica Fridrich. Any method for the 3x3 will work to complete the 5x5 at this point.
Parity
Because a 5x5 has so many permutations, patterns can arise once all of the centers and edges are solved that would be impossible in a regular 3x3. This is known as parity. On a 5x5, this happens when two edges are "flipped." They appear in a different orientation than they would were the cube a 3x3. If a parity situation occurs, it is impossible to solve the cube using the same method as a 3x3, even after completing the edges and centers. But there is a simple way to correct a parity situation on a 5x5 should it occur. Rotate the two flipped edges so that when facing one side, one is on the top, and one is on the bottom. Then rotate the two axes on the right clockwise one-quarter turn. Rotate the top axis one-half turn. Repeat this four more times. This will scramble four edges, including the flipped edges, allowing you to solve them normally.