How to Solve the Crazy 3x3 Plus Cube/Mercury
This page explains how to solve the Mercury version of the Crazy 3x3 Plus Cube. It will be assumed that you already know how to solve the Rubik's Cube.
This puzzle has one side with a circle that will turn with the rest of the layer, usually the white side. The other sides turn as shown in the picture below and to the right. It will be assumed for the rest of this page that the 1 side (side whose circle turns with the rest of the layer) is the white side, like it shows on the box that probably came with your cube. It will also be assumed that the yellow side is opposite the white side.
Notice a few things about your cube. The inner edges that are around the white side do not move relative to each other. Same with the inner edges that are around the side opposite the white side. It is impossible to move them off of their layer.
If your puzzle is still unscambled, try this: scramble it without turning the white side, then solve the outside in the same fashion as a Rubik's Cube without turning the white side either. If you do this right, you should notice that all of the inner corners are solved. This is because there is no way of moving the inner corners relative to the edge behind them without turning the white side.
Now let's learn how to solve the cube. We will solve the inner edges first, then the outside pieces, then the inner corners.
Solve all the inner edges edit
This is the easiest step. It is done intuitively. After this step, you will have a + shape on each side.
Start with the side opposite the white side (yellow side). Place all the inner edges correctly in this side, using the white side to turn them into position if needed. Then look for pieces on the white side that don't belong there and put them into their proper spots too. Solve all the inner edges in this way.
Solve the first two layers, ignoring the inner corners edit
In this step, you solve the "outside shell" of the first two layers of the cube, with white on the bottom of the first two layers. Use whatever method you are familiar with, but be sure to place the pieces around their proper inner edge pieces. Otherwise, you've done the first step for nought.
Solve the outside pieces of the last layer edit
If you did the last step properly, the yellow side will be the only side left unsolved. Place this layer on top. In this step, in order to facilitate placing the edges on top of their proper inner edges, we will use four steps, in order:
 Orient the edges
 Place the edges on top of their corresponding inner edges
 Orient the corners
 Permute the corners
If you are algorithmsavvy, you may be able to combine some of these steps into one.
Orient the edges (F R U R' U' F', etc.) then place them on top of the right inner edges using the "Sune" (R U R' U R U2 R' or R U2 R' U' R U' R'). Think about what this does to the edges. If the pieces are not solvable using either of these sequences, do (M' E2 M) (R U' R U R U R U' R' U' R2) U (M' E2 M) then try again. This last algorithm migrates the yellow pieces to the white side, then does a Uperm or 3edge cycle, then moves them back.
Then, for the remaining corners, use algorithms that don't affect the position of the solved edges. It takes some experimentation to see which ones work, but here are a few that do:

F (RUR'U') (RUR'U') (RUR'U') F'

(The "chameleon")
L F R' F' L' F R F' or
(L D' L' D L D' L') U' (L D L' D' L D L') U 
(The "bowtie")
F R' F' L F R F' L' (the inverse of the one before) or (L D' L' D L D' L') U2 (L D L' D' L D L') U2 
R U' L' U R' U' L U

(L D' L' D L D' L') U (L D L' D' L D L') U'

Lw' U R' D2 R U' R' D2 R2

F R' F' L F R F' Rw2 U' R U L U' R' U (essentially a "bowtie" and a "chameleon")
Some of these algorithms above are commutators and conjugates. Stay tuned....
Solve the inner corners edit
For this step, it is important to know what a commutator is. The concept is described here for standard Rubik's cubes. For the Crazy Cube Mercury, the same concepts apply but the execution is slightly more convoluted. You can solve the rest of the cube using only this concept.
In the prologue of this page, you were asked to perform an experiment in which, with a solved cube, you scrambled the puzzle without turning white then solved the outside of the cube without turning white. This goes to show that it is impossible to move the inner corners without using the white side.
To solve the inner corners, the first part of the algorithm (call it operation X) will put a piece from the white side onto the outside of the yellow layer (i.e. the inner corners of the green, blue, red and orange sides on the white side) without disturbing the rest of the yellow layer but allowing the rest of the cube to be "taken apart". Then, we will rotate the yellow layer such that another piece occupies the position that we inserted the piece into, then we will undo operation X. Since the yellow layer was untouched by X except for two locations, this should restore the rest of the cube and leave only a yellow layer that is turned.
This is the algorithm: Put white on U then do: (R U' R' F R' F' R) D'/D/D2 (R' F R F' R U R') D/D'/D2. This moves U(LF) to F(RD) to R(BD)/L(FD)/B(LD) back to U(LF). (Explanation of notation is here). Its mirror, which is also useful, is (L' U L F' L F L') D/D'/D2 (L F' L' F L' U' L) D'/D/D2. Understanding commutators, try to understand how this algorithm works.
Quite often, the pieces that you want to cycle are not in these positions. In that case, you will need to move them so that they are. This is a simple concept known as conjugation. Also remember that there are four inner corners of every colour, and that it doesn't matter which ones go where as long as they are all together on the right side in the end. That is to say, if you do a 3cycle of inner corners of which two inner corners are the same colour, it will appear as if you have swapped only two instead of cycled three.