Notation
Let the faces be denoted by the letters L, R, F, B, U and D (Left, Right Front, Back, Up and Down). Clockwise quarter turns of a face are denoted by the appropriate letter, anti-clockwise quarter turns by the letter with an apostrophe (i.e. L', R', F', B', U' or D'). Half turns are denoted by the letter followed by a 2 (i.e. L2, R2, F2, B2, U2 or D2).
Note that on this cube the moves LR' is simply a rotation of the whole cube. Similarly FB' or UD'. Therefore moving only three adjacent faces is enough to solve the cube. If we only use the F, R and D faces to solve it, then the UBL corner is never moved from its place, and the other pieces are simply placed so they line up with it. The solution below will use all the faces because it is based on a solution for the normal Rubik's Cube.
Solution
Phase 2: Place the bottom layer pieces, ignoring orientation.
Phase 3: Twist the bottom layer pieces correctly.
| 1. To twist DFL-, DBL+ | do R'D'R F' DR'DR D2F2 |
| 2. To twist DFL+, DBL- | do F2D2 R'D'RD' F R'DR |
| 3. To twist DFL-, DBR+ | do R2D'R D2R'D2R D'R2D |
| 4. To twist DFR-, DRB-, DBL- | do R'D'RD' R'D2RD2 |
| 5. To twist DFR+, DRB+, DBL+ | do D2R'D2R DR'DR |
| 6. To twist DFR-, DRB+, DBL-, DLF+ | do R2D2 R D2R2 D |
| 7. To twist DFR+, DRB-, DBL-, DLF+ | do RDF R2D2F2 DF'DR2 |
This solution takes at most 24+8+10=42 moves. The other solution included in the Solving section is loosely based on the "thistlewaite" algorithm adapted for the 2 cube. It uses fewer moves than this solution, but is far more complicated since it uses a large table of sequences.