Greater Than Sudoku – Hard

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"Greater Than Sudoku," also known as "Compdoku," is a variant of the classic Sudoku puzzle. It incorporates inequality signs (">" or "<") between adjacent squares to introduce an additional layer of complexity and challenge to the traditional Sudoku format.

Basic Rules

1. Grid: The puzzle is typically presented in a 9x9 grid, divided into 3x3 subgrids (also known as regions or boxes).
2. Numbers: You need to fill each square with a number, usually from 1 to 9, ensuring that each number appears exactly once in each row, column, and 3x3 subgrid.
3. Inequality Signs: Pay attention to the greater-than (">") or less-than ("<") signs placed between some squares. The numbers you place in those squares must satisfy these inequalities.

Additional Rules

• Inequalities: The inequalities between squares must be strictly adhered to; if a square has a ">" sign pointing to its neighbor, the number in that square must be strictly greater than the number in the neighboring square.
• No Repetition: Even with the inequalities, the standard Sudoku rule that a number cannot be repeated within a given row, column, or 3x3 subgrid still applies.

Basic Solving Tips

• Start with Certainties: Identify squares where a number is definitively determined by the inequalities and standard Sudoku rules.
• Use Inequalities: Utilize the inequalities to narrow down the possible numbers that can go into a particular square.
• Avoid Violations: Ensure that placing a number doesn’t violate either the Sudoku rules or the inequality constraints.
• Pencil Marks: Use pencil marks for possible numbers in a square and update them as you gather more information from solving neighboring squares.