A General Sudoku Logic

Rank and Covering Sets

Points. Rank is the relative number of sets and linksets used to link a structure, where rank 0 indicates equal numbers. More generally, rank measures the number of unknowns in a region of logic and determines the number of linksets required to cause eliminations.

Sets, Linksets, and Rank

Rank is defined as the minimum number of linksets required to link all candidates in a group of sets minus the number of sets, or M - N. Many Sudoku methods like fish are rank 0 have numbers of sets and linksets. Rank 1 structures include most chains, empty rectangles, and other fish such as finned fish and Sashimi.

In Sudoku, chain-like structures are often drawn with terminating strong links where the end candidates are not connected to anything else, like the simple chain below.

To evaluate the rank of a structure, it must be fully connected by linksets. This can always be done because every candidate belongs to four sets, some of which are available as linksets. However, the choice of sets makes a difference. If two sets are required to contain the end candidates then the rank is 3 - 2 = 1, like the simple chain below.

When both ends can be contained in one linkset then this structure becomes a simple loop like the X-Wing below, which has a rank of 2 - 2 = 0.

Complex structures may be fully connected by different combinations of linksets, which may use different numbers of linksets. Each group of linksets may have its own rank and cause its own eliminations.