Notice that row 9, column 9 and box 9 all have 2 candidates for digit 3, which now have opposite colors.
In the example, green and blue colors have been applied. We can use colors to help us remember which candidates must be true or false simultaneously. In coloring terminology, these candidates form a conjugate pair. When a row, column or box contains only 2 candidates for a digit, one of them must be true and the other must be false. The following example demonstrates the principle:Īll the candidates for digit 3 are highlighted. 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.Simple Colors Simple Colors is a subtype of coloring which only uses 2 colors. This is not allowed, so the 8 can be removed from all the blue 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. The value 8 cannot be in some blue cells and some green cells. 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. 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'. Once the tree is complete look for two cells of the same color in the same unit. The two cells in row 4 tagged 'A' and 'B' start this tree. The colored chain of conjugate pairs based on the value 8 in Figure 2 is an example of Simple Coloring Type 2.Īs before, start with any conjugate pair and build up the colored tree. This strategy looks for two cells of the same color in a colored tree of conjugate pairs in the same unit. The cells tagged with a 'A', 'C', 'E' and 'D' would have been enough of a tree in this particular case. You might notice the whole tree was not needed to make this deduction. 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.Įither way, the value 4 cannot be placed in the grey cell so it can be removed from the candidates. 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. 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. 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. 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'.
Continue this process until no more pairs can be found.Īll 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. One of these cells is already blue so color the other cell tagged with a 'D' green. 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 green so color the other cell tagged with an 'E' blue. 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. To create the tree one cell of the first conjugate pair is colored blue and the other cell is colored green.Īssume the first conjugate pair found was the two cells tagged with an 'A' and a 'C' in column 6. In each case, the value 4 only exists in two cells of a unit. The Simple Coloring Type 1 example in Figure 1 shows a colored tree of conjugate pairs based on the value 4. 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.
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