Sudoku – Hard

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Sudoku, also known as "Number Place," is a logic-based placement puzzle that has gained immense popularity around the world. The classic version of this puzzle is presented in a 9x9 grid, subdivided into 3x3 subgrids, referred to as "regions." Some cells within the grid are pre-filled with numbers, which are referred to as "givens." The objective is to fill the empty cells with numbers, adhering to specific rules.

Basic Rules

1. Grid Structure: The puzzle consists of a 9x9 grid, divided into 3x3 subgrids or regions.
2. Givens: Some cells are pre-filled with numbers, known as "givens," which cannot be changed.
3. Objective: The goal is to fill the empty cells with numbers 1 through 9.

Additional Rules

• Row Constraint: Each row must contain all of the numbers 1 through 9, without repetition.
• Column Constraint: Each column must also contain all the numbers 1 through 9, ensuring no repetition.
• Region Constraint: Each of the 3x3 regions must contain all the numbers 1 through 9, without repetition.

Basic Solving Tips

1. Single Possibility: If a cell is the only one in its row, column, or region that can contain a particular number, then it must contain that number.
2. Exclusion: If a number is already present in a row or column, other cells in that row or column cannot contain that number.
3. Double Pairs: If two cells in a row, column, or region can only be two numbers (like 3 and 6), then exclude these two numbers from other cells in that area.
4. Look for Grid Symmetry: Often, Sudoku puzzles have symmetrical solutions. If you solve a number on one side of the grid, look for a similar solution on the opposite side.
5. Focus on a Region: Sometimes focusing on a single region, row, or column can help identify numbers that can or cannot be placed in certain cells.