A General Logic for Sudoku

Rank 5 Elimination

Example Rank 5 Elimination

This 34 set, rank 5 elimination comes from a solution (by ttt) to monster #224 from the 1465 list, which was presented here. The elimination in Eureka notation is given by:

(step 4): (9)r3c3=(9-4)r3c2=(hp34)r78c2-(4=hp19)r9c13-(19)r9c89=(hp19)r7c89-(19=hp34)r7c26-(134=5)r7c5-[(5)r4c5 & (6)r7c8]={(6)r1c1=(6)r2c1-(6)r2c8=(6-5)r4c8=(5)r4c1}-(5)r1c1=(ht126)r12c1/r1c2
=> r3c23<>12

In terms of sets (below), the logic has 14 sets, 19 linksets, and a rather high rank of 5. The logic also contains several loops, as shown in the 3D diagram.

In terms of set logic, each of four candidates (12)r3c23 is contained in two linksets, one box (b11 or b12) and one cell (n32 or n33). The candidates need 4 more constraints for elimination because the rank is 5. Linkset triplets B and C both point in the direction (brown arrows) of cell r9c3 where they join. This branch then flows (brown arrows) through triplet A and into row set r39 thus providing 3 extra constraints. Node 6r2c1 in box 1 has one triplet D pointing towards the box 1 linksets. This provides one more constraint and allows the overlap linksets to eliminate the candidates.