How to solve hard sudoku?

Sudoku is a logic-based number-placement puzzle that requires you to fill a 9x9 grid with digits from 1 to 9, such that each row, column, and 3x3 box contains each digit exactly once. Hard sudoku puzzles are more challenging than easy ones because they have fewer given numbers and require more advanced strategies to eliminate candidates and find the correct placements.

Some of the common techniques for solving hard sudoku puzzles are:

Naked singles**: These are cells that have only one possible candidate left after eliminating the numbers that already appear in the same row, column, and box. You can place the naked single in the cell and remove it from the candidates of other cells that share the same row, column, or box. ¹

Hidden singles**: These are cells that contain the only occurrence of a certain candidate in a row, column, or box, even though there may be other candidates in the same cell. You can place the hidden single in the cell and remove it from the candidates of other cells that share the same row, column, or box.

Naked pairs**: These are two cells in the same row, column, or box that have the same two candidates left after eliminating the numbers that already appear in the same row, column, and box. You can remove these two candidates from the other cells in the same row, column, or box, since they must go in the naked pair.

Hidden pairs**: These are two cells in the same row, column, or box that contain the only occurrence of two candidates in that row, column, or box, even though there may be other candidates in the same cells. You can remove the other candidates from these two cells, since they must contain the hidden pair.

X-wing**: This is a pattern that involves four cells that form the corners of a rectangle, such that each row and column of the rectangle contains exactly two occurrences of the same candidate. You can eliminate this candidate from the other cells in the same columns or rows as the corners of the rectangle, since it must go in one of the corners.

Swordfish**: This is a pattern that involves six cells that form the corners of a 2x3 or 3x2 rectangle, such that each row and column of the rectangle contains exactly two occurrences of the same candidate. You can eliminate this candidate from the other cells in the same columns or rows as the corners of the rectangle, since it must go in one of the corners.

XY-wing**: This is a pattern that involves three cells that form a Y shape, such that each cell has exactly two candidates, and the three cells share a common candidate. You can eliminate this common candidate from the cells that see all three cells of the Y shape, since it must go in one of them.

XYZ-wing**: This is a pattern that involves three cells that form a Y shape, such that each cell has exactly two or three candidates, and the three cells share a common candidate. You can eliminate this common candidate from the cells that see all three cells of the Y shape, except the cell that has three candidates, since it must go in one of them.

Unique rectangle**: This is a pattern that involves four cells that form the corners of a rectangle, such that two of the cells have exactly two candidates, and the other two cells have the same two candidates plus one more. You can eliminate the extra candidate from the cells that have three candidates, since otherwise the puzzle would have two possible solutions, which is not allowed in sudoku.