# Generalized Sue de Coq

Later in the day of Sue de Coq’s historic post, the proposer posted a “generalized version” of the elimination rationale. Putting it in more familiar terms,

Here is an example from Andrew Stuart’s The Logic of Sudoku, immediately after line marking. The dj cells r1c56 hold intersection values 789, removing 235 from the  box remainder less dj cell r3c2.  This cell holds 23, removing 35789 from r1c79.

Why does this work?

It’s about what can be in the intersection. NWr1 can have only one of 789, because ds r1c56 has two. Likewise, NWr1 can have only one of dj 23 values.

These two facts force NWr1 to have the 5 value! Now a logical formula for NWr1 is  5(2+3)(7+8+9), because it must contain 2 or 3, and it must contain  7 or 8 or 9, as well as 5. And we know that r1c79 can’t have 7,8, or 9. Between them, the intersection and the 789 dj cells have each of these values.

Stating the rationale generally, the intersection can place only 2 or 3 of its values.  Each remainder requires (# ds values – # ds cells)  placements. When placements are available, extra remainder values in non-ds cells are removed.

The SdC rule above works with ALS ds cells just as well, since a remainder ALS of n values is n – 1 cells.

In The Logic of Sudoku, the reader exercise Problem 19 is in the Sue de Coq section, so it’s a good place to check your spotting for “generalized” Sue de Coq. You can reconstruct all the prior moves from the givens from this grid.

The r1 naked triple, 8-wing, and boxline are followed by

and the resulting r9 naked pair 56 above. Then r9 naked pair 13.

It’s when you remove 1r2c5 that you spot the second ds cell, and the generalized Sue de Coq.

With bv dj cells and only 4 intersect values,  the contents of the intersection is

Nr3 =

6(2+4)(3+7),

and the normal and generalized rule above converge. You could use the logical formula instead of the rule to make the removals.

Unless you suppress the  WXYZ option, Stuart’s solver  produces this one, with wing cells 34 and 37, and two removals, instead of the Sue de Coq.

This result is enough, but the WXYZ suggests we look for a BARN.

We do get the naked quad in the Nr3 bent region. But this quad is not “almost restricted”. Every member value is locked within its box or its line.

No, look back on Sysudoku Advanced. It’s a rare Bob Hanson BNS0, duplicating the generalized Sue de Coq. Who wooda thunk?