Sudoku Rules

Sudoku is a logic-based puzzle game. The puzzle is about digit placements on a 9x9 grid. The goal is to fill the entire 9x9 grid with digits according to the Sudoku rules. The rules are very simple. But it needs your focus and logical thinking to play with the puzzles.

Now let's introduce the Sudoku rules so that you know what you need to follow when playing the Sudoku game.

Rules

Fill all 81 cells on a 9x9 Sudoku grid of nine rows, nine columns, and nine 3x3 blocks with digits from 1 to 9, so that

  1. Each row must contain each of the nine digits from 1 to 9 once, and only once.
  2. Each column must contain each of the nine digits from 1 to 9 once, and only once.
  3. Each 3x3 block must contain each of the nine digits from 1 to 9 once, and only once.

Each row, each column, or each 3x3 block has exactly 9 cells, and it must contain each of the nine digits from 1 to 9. As a result, a 9-cell unit (a row, a column, or a 3x3 block) cannot have the same digit more than once. That means there are no repeated digits in a row, in a column, or in a 3x3 block. This is a very important fact about Sudoku. We should always keep this in mind when playing Sudoku.

There are three rules for Sudoku. The first one is the rule for rows in the Sudoku grid. The second one is the rule for columns. The last one is the rule for 3x3 blocks. We will demonstrate each of these three rules with examples.

R1
R2
R3
R4
R5
R6
R7
R8
R9
1
2
3
4
5
6
7
8
9
3
8
5
7
2
5
9
1
4
Figure 1: The Rule for Rows in Sudoku

Figure 1 above is an example to demonstrate what digit placements in a row will or will not violate the Sudoku rule for rows.

  1. In row R3, all nine digits from 1 to 9 appear once each, and there are no repeated digits. Therefore, the digit placements in this row do not violate the Sudoku rule for rows.
  2. In row R7, the digit 6 is missing, and the digit 5 appears twice. Therefore, the digit placements in this row violate the Sudoku rule for rows.
C1
C2
C3
C4
C5
C6
C7
C8
C9
9
8
7
4
2
9
6
5
4
5
2
7
3
2
1
1
8
6
Figure 2: The Rule for Columns in Sudoku

Figure 2 above is an example to demonstrate what digit placements in a column will or will not violate the Sudoku rule for columns.

  1. In column C2, all nine digits from 1 to 9 appear once each, and there are no repeated digits. Therefore, the digit placements in this column do not violate the Sudoku rule for columns.
  2. In column C6, the digit 3 is missing, and the digit 2 appears twice. Therefore, the digit placements in this column violate the Sudoku rule for columns.
3
2
1
6
5
4
9
8
7
4
1
8
5
2
7
8
3
6
Figure 3: The Rule for 3x3 Blocks in Sudoku

Figure 3 above is an example to demonstrate what digit placements in a 3x3 block will or will not violate the Sudoku rule for blocks.

  1. In block B2, all nine digits from 1 to 9 appear once each, and there are no repeated digits. Therefore, the digit placements in this block do not violate the Sudoku rule for blocks.
  2. In block B4, the digit 9 is missing, and the digit 8 appears twice. Therefore, the digit placements in this block violate the Sudoku rule for blocks.

The above examples are for a row, a column, or a 3x3 block. As for the entire Sudoku grid, we need to check all nine rows, all nine columns, and all nine 3x3 blocks, and each of them cannot violate the Sudoku rules.

C1
C2
C3
C4
C5
C6
C7
C8
C9
R1
R2
R3
R4
R5
R6
R7
R8
R9
9
7
4
5
6
3
2
8
1
1
3
8
4
9
2
5
7
6
2
5
6
7
8
1
3
9
4
6
3
9
4
2
5
8
1
7
8
1
4
7
6
9
2
5
3
5
7
2
1
3
8
4
6
9
1
5
8
3
9
2
7
4
6
9
7
2
6
4
5
3
8
1
6
4
3
8
1
7
9
2
5
Figure 4: Two Violations in Columns C5 and C6

In Figure 4 above, all 81 cells in the Sudoku grid have been filled up with digits from 1 to 9. By checking each row, each column, and each 3x3 block of the Sudoku grid, we see that all rows and all 3x3 blocks satisfy the Sudoku rules. However, two of the nine columns violate the rules:

  1. The column C5 does not contain the digit 2, and the digit 7 appears twice in this column.
  2. The column C6 does not contain the digit 7, and the digit 2 appears twice in this column.

Because of these two violations, the entire Sudoku grid does not meet the Sudoku rules.

Actually, if we exchange the digit values in two cells (R7,C5) and (R7,C6) of row R7, then the entire Sudoku grid will satisfy the Sudoku rules.

C1
C2
C3
C4
C5
C6
C7
C8
C9
R1
R2
R3
R4
R5
R6
R7
R8
R9
9
7
4
5
6
3
2
8
1
1
3
8
4
9
2
5
7
6
2
5
6
7
8
1
3
9
4
6
3
9
4
2
5
8
1
7
8
1
4
7
6
9
2
5
3
5
7
2
1
3
8
4
6
9
1
5
8
3
9
2
7
4
6
9
2
7
6
4
5
3
8
1
6
4
3
8
1
7
9
2
5
Figure 5: No Violstions

Figure 5 shows the result after exchanging the digit values in two cells (R7,C5) and (R7,C6) of row R7. We can verify that

  1. All nine rows satisfy the Sudoku rule for rows.
  2. All nine columns satisfy the Sudoku rule for columns.
  3. All nine 3x3 blocks satisfy the Sudoku rule for blocks.

There are no violations. As a result, the entire Sudoku grid satisfies all the Sudoku rules.

After learning the rules of Sudoku, you may want to play the game now. Here are some tips for beginners. That will give you a quick start to the game.

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