# Cardan Reduction

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

## Steps

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)