Cardan Reduction

Cardan Reduction
Proposer(s): Matt DiPalma
Proposed: 2017
Alt Names: CR
Variants: CR+, Step 3-4 of Heise Method
No. Algs: 144 (72 with mirrors) for CR2 substep
Avg Moves: 24.90 (for CR1, CR2 and CR3 substep)
Previous state: F2L-1 + EO cube state
Next state: Solved cube state

F2L-1 + EO cube state -> Cardan Reduction step -> Solved cube state

The Cardan Reduction step is the step between the F2L-1 + EO cube state and the Solved cube state.

Cardan Reduction is a novel "LS/LL" approach developed by Matt DiPalma for methods that pre-orient edges before the last slot (ZZ, Petrus, Heise, CFOP with edge control). It features a particularly low case count and movecount, in comparison with conventional LS/LL approaches. "LS/LL" is in quotes because the solution is not discretized in that way. This variant leverages cancellations, statistically common cases, rotational symmetry, inverses, and reflections to efficiently reduce the cube to a commutator/conjugate.

Cardan Reduction has 3 steps after EOF2L-1 (F2L-1 + EO) is completed.


  1. (CR1) F2L-1 + EO cube state to F2L-1C + EO + 2x1x1 block cube state : Insert FR edge and create a U-layer 2x1x1 block.
    • the U-layer pair has a fairly high likelihood (32/75) of solving itself while the FR edge is inserted
    • if not, this can take an average of 8 moves to do manually
    • pairs can be preserved during F2L to drastically reduce this movecount (see examples)
  2. (CR2) F2L-1C + EO + 2x1x1 block cube state to F2L-1C + LL-2C cube state : Solve the 2x1x1 pair, all edges, and a corner (72 cases, and their mirrors).
    • AUF the 2x1x1 pair so it points over the FR edge
    • if the pair is a clockwise pair (UR edge and URF corner)
      • determine edge permutation (6 possibilities)
      • determine destination of UFL corner (12 possibilities)
      • apply alg from speadsheet
    • if the pair is an anticlockwise pair (UF edge and URF corner)
      • rotate y (so FR edge is in LF)
      • determine edge permutation (6 possibilities)
      • determine destination of UFR corner (12 possibilities)
      • apply alg that is mirrored from spreadsheet
    • Algorithms can be found here.
  3. (CR3) F2L-1C + LL-2C cube state to Solved cube state : Solve the remaining 3 corners using a commutator/conjugate.

External links