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Maltese Gear Cube
Solution ideas from zergosaur: https://www.youtube.com/watch?v=CMM1YoUvfaU
1) Solve the corners, centers and edges (ie all pieces except for the gears) using a Mixup Gear Cube method:
a) Solve the corners as a 2×2, using: http://www.rubiksplace.com/cubes/2x2/
b) 3-Cycle the edge + centers using: (R U R U R) (U' R' U' R' U') - this midpoint might be worth investigating further as the 2×2 face is unchanged but the back 3x3s are modified at this point. Follow with: (L' U' L' U' L') (U L U L U)
2) Permute the gears
a) 2-Cycle across the right edge:
b) 3-Cycle across the left edge and left upper edge: (R U R U R) (U'R'U'R'U') -match- (R U R U R) (U'R'U'R'U') -match- (U'L'U'L'U') (L U L U L) -match- (U'L'U'L'U') (L U L U L) -match- (DLD') (L'U'L'U'L') (U L U L U) (L'U'L'U'L') (U L U L U) (U R U R U) (R'U'R'U'R') (U R U R U) (R'U'R'U'R') (DL'D') using https://youtu.be/uBU_0Qb-AGI or https://youtu.be/wn4itiyq2pU
3) To rotate 2 gears by 90°:
(R'D'RD)*4 D (RURUR U'R'U'R'U')*2 (U'L'U'L'U' LULUL)*2 D' (D'R'DR)*4 D (L'U'L'U'L' ULULU)*2 (URURU R'U'R'U'R')*2 D'
This rotates the front left gear +90°, and the front right gear -90°.
Alternatively, a small variation of this alg
(R'D'RD)*4 LD (RURUR U'R'U'R'U')*2 (U'L'U'L'U' LULUL)*2 D'L' (D'R'DR)*4 LD (L'U'L'U'L' ULULU)*2 (URURU R'U'R'U'R')*2 D'L'
rotates 4 gears:
front right -90° top left: +90° left top: -90° left bottom: +90°
Do the above algs until all gears are either solved or need to be flipped 180° (ie only need to fix gears which need to be rotated 90° or 270°).
4) Doing https://youtu.be/uBU_0Qb-AGI twice flips all 4 gears which were part of the cycle by 180°. Use this to orient all of the gears correctly.
Additional Algs (not used in above method):
a) The following 1000+ move alg (!) rotates the two gears located on the front and back faces next to the L face by 90°.
(R' D' R D)*132 D (R' D' R D)*132 D'
b) Use https://youtu.be/L3Sd1mS5DTI to rotate pairs of gears of the same color by 90°.