Sue de Coq is an elimination method named by its inventor in a forum post. Sue de Coq, or SdC, is among the methods to look for in a scan over every box of the line marked grid. Here we first describe the original form first posted, and several related forms rwell suited for the bv scan.
Original Sue de Coq
To find a Sue de Coq Scan the bent regions, the intersections of box and line, with four matching candidates in chute (wall) and the box and line remainders
You have an original SdC when the true contents of the chute that can be described by the logical expression N(a+b)(c+d), where N is a clue and the two remainders contain an ALS with value groups a and b, and another with value groups c and d.
Of the two chute cells available for candidates, one must contain a or b, and the other must contain c or d. The logical expression for the chute candidates has two alternates. An ALS can give up only one of its values, making those in the ALS remainders alternates in the chute. Extra candidates in each remainder, but outside its chute matching ALS, are removed. Extra chute candidates are removed.
Actually, instead of a clue, the logical expression is satisfied by a slink or aligned triple in one candidate. Just so the candidate N must be present in the chute solution.
This example of the original SdC is from Frank Longo’s The Nastiest Sudoku Book Ever, puzzle 640.
The Sc4 chute is marked. The sure number is 9 and there are choices for possible alternates. Where a chute has enough numbers, I just look at the remainders for matching ALS. Bv are available for
Sc4 = 9(3+4)(5+8).
Numbers 5 and 8 are shared between the chute and bv 58 in the box remainder. The extra 8 candidate is removed. In the line remainder, the bv 34 forces 3 and 4 to alternate in the chute, but there are no extra candidates to eliminate. It shows why the most effective ALS for Sue de Coq is the bv.
There are two single alternate forms of the original Sue de Coq. One occurs when two values must be present in the chute, usually two clues. Then only one ALS is required to force alternation in the single free cell.
A second single alternate form is the SASdC or Single Alternate SdC trial, with only one matching ALS in the remainders. The logical expression for chute contents then has a second term accounting for the possibility that one set of alternates is missing from the chute. The SASdC is described further in the Trials section of The Guide.
Later on the day of Sue de Coq’s historic post, he posted this “generalized version” of his elimination rationale:
Putting this in more familiar terms:
With two or three cells (c) to fill in the chute, and at least two more values (v >=4 or 5), the generalized SdC requires at least one cell in each remainder with candidates of these values only. Call these the dj cells. The values in the dj cells of the remainders must be disjoint.
Then the chute values not in the dj cells of each remainder can be eliminated from the non-dj cells of the other remainder.
In 640 above, V = 34589. VR = 34, the VB =85, and V/VR = 589 is removed from r9c6 and V/VB = 349, from r2c4 and r4c4.
Sue-de-Coq’s generalized version was accepted on the forum, and on many expert websites as the preferred way to include cells of more than two values in Sue de Coq. For example, Hobiger’s explanations of the examples linked above are based on the generalized version. The primary reason for the generalization is to allow more than 4 or 5 candidate values in the chute, when extra dj cells can be found.
Sysudoku doesn’t go along, because it seems that full employment by ALS beyond the bv has more focus and is practically as effective as the generalized SdC. Besides, being confined to the values of the chute, multiple dj cells in a remainder tend to be ALS.
The examples shown and linked above illustrate the ALS form satisfies tie subtle requirements of generalized ALS. The remainder ALS often contain additional values that become locked in the ALS cells as the ALS loses a matching value to the chute. All other candidates of these locked values in the remainder are eliminated.
As a closing illustration, from Andrew Stuart’s The Logic of Sudoku. Andrew uses this case to explain generalized Sue de Coq.
The chute has 6 values and no sure numbers. Generalized coverage of the three chute cells is afforded by three dj cells of 789, 89, and 23.
However, recognizing that the ALS 789 can release only one value, we can describe the contents of NW r1 as NWr1 = 5(2+3)(7+8+9). One chute position that can only be filled with 5. No matter if the matching value is 7, 8 or 9 the other two values are locked in the ALS, producing eliminations.