Coloring: What Can Same-Color Conflicts and Two-Color Traps Remove?

Two colors do not decide the answer

Coloring uses Simple Coloring to follow whether candidates for one number are true or false. Choose one number. When it has exactly two possible cells in a row, column or box, join those cells with a solid line. Exactly one of the two is true, so give the endpoints different marks.

This article uses an accent-colored ● and a secondary-colored ○. Even without distinguishing the colors, you can follow the same logic using symbols and coordinates. Within one connected network, if all ● candidates are true, all ○ candidates are false, and vice versa. You have not yet decided which group is true.

Conditions — one number and connected strong links

1. Write every candidate without omissions, then choose just one number.
2. Find rows, columns or boxes with only two possible positions for that number and join them with solid lines. These are strong links.
3. Alternate ● and ○ along the solid lines. Keep each cell's symbol the same at branches too.
4. Use the two symbols in the same network to find a conflict or a trap as described below.

Two unconnected groups can swap their ● and ○ symbols independently. Do not combine ● and ○ from different groups to remove candidates.

Example ① — a trap between two colors

c1c2c3c4c5c6c7c8c9

r1r2r3r4r5r6r7r8r9

8

4

7

9

3

6

1

6

8

3

5

2

7

9

9

7

3

6

1

2

8

4

5

3

8

4

6

5

9

7

7

124

5

9

3

12

8

9

8

7

3

24

9

5

8

12

3

9

3

2

4

8

6

3

8

24

7

6

1

9

5

24

●○○○●

The number is 2. Follow r5c8(●)—r6c9(○)—r9c9(●)—r9c3(○). Each link has exactly two positions for 2 in box 6, column 9 and row 9, respectively, so they are joined by solid lines. A strong link in column 8 also connects r5c8 to r7c8(○).

The uncolored cell r5c3 sees r5c8(●) through row 5 and r9c3(○) through column 3. Whichever symbol's group is true, one of these two cells is 2, so remove 2 from r5c3. Its candidates shrink from 1, 2 and 4 to 1 and 4. When a candidate outside the network sees both symbols like this, it is called a trap.

Example ② — a wrap caused by a same-color conflict

c1c2c3c4c5c6c7c8c9

r1r2r3r4r5r6r7r8r9

2

6

3

9

8

4

7

1

5

1

78

4

6

5

3

28

9

78

5

9

3

1

4

6

28

4

2

5

3

1

78

6

9

78

9

78

6

5

2

78

1

3

4

78

3

1

6

4

9

28

5

3

9

2

8

7

6

5

4

1

5

4

7

1

9

3

28

28

6

6

1

8

4

5

2

9

7

3

●○○●○●○●●●○●

This time the number is 8. After alternating colors along the solid lines, r3c9(●) and r4c9(●) are both in column 9. If the ● group were true, column 9 would contain two 8s, a contradiction. Therefore, all ● candidates for 8 in this network are false. This is a wrap.

Remove 8 from seven cells: r2c2, r3c9, r4c9, r5c6, r6c1, r6c7, r8c8. They leave only 7, 2, 7, 7, 7, 2, 2, respectively. You remove the candidate from every cell with the same symbol in the network, not just the two conflicting cells, because they share the same truth value.

When you cannot use it, and common mistakes

• The chosen number has three candidate cells in a row or column. You cannot pick two and join them with a solid line. Both might be false while the third is true.
• The colors match, but the networks are different. Using the same symbol does not mean they have the same truth value.
• A cell sees only one symbol in a trap. The other group could still be true, so you cannot remove the candidate.
• You remove other numbers in a wrap. This argument applies only to the one chosen number.
• You think ○ means false. The empty circle only distinguishes the two groups. In this article's wrap, it is the ● group that is false.

Practice puzzle

c1c2c3c4c5c6c7c8c9

r1r2r3r4r5r6r7r8r9

1

5

8

2

357

6

1457

9

3

4

1

4

7

8

3

5

9

2

6

9

6

2

4

1

7

3

5

8

5

8

3

2

7

2

5

9

7

357

9

8

4

1

456

7

2456

8

9

●○●○

Look at the number 5. What can you remove from r2c7? Is this a wrap or a trap?

Show answer

Remove 5 using a trap. r2c2(●)—r8c2(○)—r9c3(●)—r9c7(○) follows strong links in column 2, box 7 and row 9, respectively. r2c7 sees r2c2(●) through row 2 and r9c7(○) through column 7. So remove 5 from candidates 1, 4, 5 and 7, leaving 1, 4 and 7.

Further reading

The prerequisites are How to Use Sudoku Candidate Notes and Do You Need Trial and Error to Solve Sudoku?. Next, read BUG+1: Why Must You Assume a Unique Solution?.

The example boards were found using the sudoku-core generator and logical solver, rather than copied from external puzzles. The seeds, difficulty and first target step numbers are recorded in the article metadata. Technique definition reference: HoDoKu Coloring.