Sudoku HowTo

# Simple Coloring

Simple Coloring is also referred to as Singles Chains.

Colors are used to indicate the opposing sides of conjugate pairs to allow potential candidate removal to be identified.

A chain of conjugate pairs exists when multiple conjugate pairs are linked together. For this technique, conjugate pairs, all on the same value, are linked together and the cells in the tree are colored with two alternating colors. This means cells from a single conjugate pair are never colored the same.

Once a conjugate pair tree using a single value is formed, all the cells of one color and only one color have to contain the considered value. The value cannot exist in some cells of each color. It has to be in all the cells of one color and no cells of the other color.

The simple coloring process does not define which set of cells the considered value will end up in but we can use the simple coloring chain to do candidate removal on other cells
Simple Coloring Type 1. This strategy looks for a cell in the puzzle that is not in the conjugate pair tree but is in the same unit as a cell of each color in the colored tree of conjugate pairs.

The Simple Coloring Type 1 example in Figure 1 shows a colored tree of conjugate pairs based on the value 4. In each case, the value 4 only exists in two cells of a unit. To create the tree one cell of the first conjugate pair is colored blue and the other cell is colored green.

Assume the first conjugate pair found was the two cells tagged with an 'A' and a 'C' in column 6. One is blue and one green. Using the green cell tagged with a 'C', another conjugate pair, the cells tagged with a 'C' and an 'E' can be found in box 8. One of these cells is already green so color the other cell tagged with an 'E' blue. Using the blue cell tagged with an 'E', another conjugate pair, the cells tagged with an 'E' and a 'D' can be found in row 9. One of these cells is already blue so color the other cell tagged with a 'D' green.

You get the idea. Continue this process until no more pairs can be found.

All other cells that can contain the value 4 can now be checked for a cell from the tree of each color in the same unit. The grey cell is in the same row as the blue cell tagged with an 'A', and in the same column as the green cell tagged with an 'D'.

The conjugate pair chain properties mean all the green cells need to contain the value 4 or all the blue cells need to contain the value 4. This strategy does not determine if the value 4 will be in the blue or the green cells but it does mean the value 4 cannot be in the grey cell.

If the value 4 is in the blue cells then there will be a 4 in row 5 in the cell tagged with an 'A' so the value 4 cannot be in the grey cell in row 5. If the value 4 is in the green cells there will be a 4 in column 2 in the cell tagged with a 'D' so the value 4 cannot be in the grey cell in column 2.

Either way, the value 4 cannot be placed in the grey cell so it can be removed from the candidates.

You might notice the whole tree was not needed to make this deduction. The cells tagged with a 'A', 'C', 'E' and 'D' would have been enough of a tree in this particular case.
Simple Coloring Type 2. This strategy looks for two cells of the same color in a colored tree of conjugate pairs in the same unit.

The colored chain of conjugate pairs based on the value 8 in Figure 2 is an example of Simple Coloring Type 2.

As before, start with any conjugate pair and build up the colored tree. The two cells in row 4 tagged 'A' and 'B' start this tree.

Once the tree is complete look for two cells of the same color in the same unit. In this example there are two blue cells in column 4 tagged 'A' and 'E', two blue cells in box 8 tagged 'E' and 'G' and two blue cells in row 9 tagged 'G' and 'H'.

The properties of conjugate pair chains mean that all the blue cells or all the green cells have to contain the considered value, the value 8 in this case. The value 8 cannot be in some blue cells and some green cells. If all the blue cells contain the value 8 then there will be two value 8s in column 4, box 8 and row 9.

This is not allowed, so the 8 can be removed from all the blue cells.

If the value 8 cannot be in the blue cells then it has to be in the green cells so the value 8 can be inserted as the solution in all the green cells.

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