Sudoku – Hard

Sudoku

Sudoku is one of the most well-known logic puzzles. The classic puzzle consists of a 9x9 grid divided into nine 3x3 blocks. Some numbers are already given, and the remaining cells must be filled with numbers from 1 to 9.

The goal is to place each number exactly once in each row, each column, and each 3x3 block. It does not involve calculations: the numbers serve merely as nine different symbols. The puzzle is solved through logical elimination and combination of the three areas to which each cell belongs.

Sudoku is also known under the older name "Number Place." Besides the classic 9x9 Sudoku, there are numerous variants and other grid sizes. The following rules and strategies relate to the classic Sudoku with 3x3 blocks.

Basic Rules

• The playing field consists of 9 rows and 9 columns.
• The grid is also divided into nine 3x3 blocks.
• Exactly one number from 1 to 9 is entered in each empty cell.
• Each row must contain each number from 1 to 9 exactly once.
• Each column must contain each number from 1 to 9 exactly once.
• Each 3x3 block must contain each number from 1 to 9 exactly once.
• The given numbers may not be changed.
• The puzzle is solved when all 81 cells are filled and all three area rules are met.

Solving Strategies

1. Start with nearly complete rows

A good starting point is rows, columns, or blocks where many numbers are already present. For example, in the seventh row of the following example, only one cell is missing:

The seventh row already contains 1, 3, 4, 5, 6, 7, 8, and 9. Only the 2 is missing. The empty cell between 5 and 7 must therefore contain a 2.

This conclusion does not require a candidate list. The row itself fully determines the missing value.

2. Naked Single Candidate: Only one number remains in a cell

In the case of a naked single candidate, row, column, and block of a cell are checked together. In the following example, we examine the top-left cell.

In the first row, the numbers 1, 4, 6, and 9 are missing.

• The first column already contains 3, 5, 6, 7, and 9. So 6 and 9 are excluded.
• The top-left 3x3 block already contains a 4. So 4 is excluded.
• The only possible number remaining is 1.

Therefore, the top-left cell must be a 1.

The 1 is not guessed. All other missing numbers in the row are excluded by the column or block.

3. Hidden Single Candidate in a Row

Sometimes a cell has multiple candidates. Still, a particular number can only be in one place within a row.

In the first row, the numbers 1, 4, 6, and 9 are missing. We specifically look for the 4.

• The first and third cells are in the left upper 3x3 block, where a 4 is already present.
• The seventh cell excludes the 4 based on the seventh column.
• Only the last cell of the first row remains possible for 4.

The last cell of the first row must contain a 4.

The cell previously had multiple candidates. The position of the 4 became clear only by asking: "Where can the 4 possibly be in this row?"

4. Hidden Single Candidate in a 3x3 Block

The same logic applies within a block. We consider the upper middle 3x3 block and look for the number 9 there.

The considered block consists of these nine cells:

583
2..
..6

• In the second row, a 9 is already present. Therefore, 9 cannot be in the two empty cells of this row within the block.
• The middle empty cell in the third row lies in a column that already contains a 9.
• Only the first empty cell of the third row remains in the block.

There, the 9 must be entered.

Again, the cell itself isn't immediately obvious. The position of the desired number within the block is clear.

5. Systematically Note Candidates

When no immediate entry is visible, possible numbers are noted for empty cells. These candidates arise from the elimination of numbers already in the row, column, or block.

In the eighth row, the missing numbers are 2, 6, and 8:

For the middle empty cell in this row, only the 6 remains. All other numbers are already present in this 3x3 block.

Next, only 2 and 8 are missing in the eighth row. These remaining cells form a simple pair of numbers: one contains 2, the other 8. The exact order is determined later by the columns.

6. Track a Specific Number Systematically Through the Grid

Instead of checking each empty cell individually, you can track a specific number through all blocks. Existing numbers block their rows and columns for the same number elsewhere.

For example, consider the number 9:

• The possible position of 9 in the first row is highly restricted.
• In the upper middle block, 9 can only be in the third row.
• Every newly placed 9 excludes its entire row, column, and block for more 9s.

This scanning method is especially helpful at the beginning because you focus on only one number at a time.

7. Block-Row Interaction: Remove Candidates Outside a Block

Sometimes a number in a 3x3 block is not yet assigned to a single cell. But if all possible positions lie in the same row, the number can be removed from outside the block in that row.

In the upper middle block, the 4 can only be in the two empty cells of the second row.

The two possible positions for 4 within the upper middle block imply that one of these must contain the 4 of the block. Therefore, no 4 can be in the same row within the right 3x3 block.

This step does not add a number but removes two candidates and can later create a single candidate.

8. Understand Candidate Pairs Correctly

A candidate pair arises when in a row, column, or block, two numbers are limited to exactly two cells. These two numbers must occupy these two cells, even if their order is still unknown.

After placing 6 in the eighth row, only the numbers 2 and 8 remain for the two open cells:

This ensures:

• The first and third cells of the eighth row contain 2 and 8.
• No other number can be in these two cells.
• The final order is determined by the columns.

For more difficult Sudokus, such pairs can help eliminate candidates from other cells in the same area. It is important that the pair is strictly limited to two numbers and two cells.

9. When is a Entry Truly Certain

A cell should only be filled when there is a clear, logical reason. A number is certain if at least one of the following statements is true:

• After checking the row, column, and block, only this number remains for the cell.
• The number has only this possible position in a row.
• The number has only this possible position in a column.
• The number has only this possible position in a 3x3 block.
• An correctly applied candidate pattern excludes all other options.

If two or more possibilities remain, no entry is allowed yet. Candidates can be noted, but the decision must wait.

Typical Solution Sequence

1. First check almost complete rows, columns, and 3x3 blocks.
2. Look for cells where only one number is possible.
3. Then look for the only possible position for each missing number within a region.
4. Track individual numbers block-wise across the grid.
5. Note candidates when no direct entries are visible.
6. Use block-row and block-column interactions to remove candidates.
7. Search for pairs or similar candidate patterns if simple singles are insufficient;
8. Update each affected row, column, and block after each entry.
9. Regularly check that no number is duplicated.
10. Do not guess. Wait until a possibility is logically certain.

Common Mistakes

• Checking only the row and forgetting the column or 3x3 block.
• Entering a number when multiple candidates are still possible.
• Changing given numbers.
• Not updating candidates after a new entry.
• Overlooking hidden single candidates by only searching for cells with one candidate.
• Assuming a candidate pair when numbers might still appear in other cells of the area.
• Eliminating a number from a block without fully checking its row or column.
• Guessing too early and noticing an error only many steps later.

Tips for Beginners

• Start with areas that already contain the most numbers.
• Always check each cell in the order row, column, then block.
• Regularly switch perspective: rows, columns, blocks, and individual numbers.
• Write candidates small and clearly.
• Mark a candidate only if it truly meets all three area rules.
• Ask yourself before each entry: "Why can no other number be here?"
• Use safe eliminations as consistently as direct entries.
• After each new value, review the three affected areas again.

Sudoku is purely a logic puzzle. Numbers are not calculated but positionally determined by elimination. Systematically alternating between cells, numbers, and areas enables larger solution chains to be traced confidently, without guessing."