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Sudoku Algorithms

Singles Chains - Twice in a Unit
The Singles Chains sudoku-solving strategy is an advanced and powerful technique, sometimes also referred to as Single Coloring. This method involves tracing a chain of single, interrelated candidates across different cells and coloring mutually exclusive candidates in two distinct colors. Mutually exclusive are for example two candidates in the same row or the same column. If one of them is ON then another should be OFF and vice versa. If a set of candidates colored in one color contradicts the sudoku rules (for example, if they appear in the same column or row), we can exclude this set and retain the candidates marked by the second color. The image below illustrates an example of the Singles Chains technique.
Singles Chains Example There are two sets of single candidates for the number 3: E6 and D7, marked in yellow, and B5, B7, D4, and E8, marked in red. The two red candidates in cells B6 and B7 are in the same row, which contradicts the sudoku rule. Therefore, all candidates marked in red in cells B5, B7, D4, and E8 can be removed, and the value '3' can be assigned to the cells E6 and D7, marked in yellow.
Singles Chains Example The image on the left presents another example of the Singles Chains sudoku-solving technique. There are two sets of mutually exclusive single candidates with a value of 7, colored yellow (C4, D6, and E8) and red (B6, B8, C9, D9, and E4). The two red candidates in cells C9 and D9 are in the same column, the red candidates in cells B8 and C9 are in the same square, and red candidates B6 and B8 are in the same row which contradicts the sudoku rules. Therefore, all candidates marked in red can be removed, and the value '7' can be assigned to the cells marked in yellow.

Singles Chains - Two colors 'elsewhere'
Singles Chains Example Another type of contradiction occurs when one chain candidate is linked to two chain candidates with different colors. In the following example, candidate '6', marked red in cell D1, is in the same row as candidate '2', colored orange in D8, and in the same column as candidate '6', colored yellow in cell G1. Coloring candidate 6 in cell D1 yellow or orange will break the Sudoku rule. Therefore, in this example, we should remove candidate 6 from cell D1.