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If you solve Sudoku puzzles, you almost certainly know the basic technique: find a cell where only one number is possible and fill it in. This is called a naked single, and it is enough to solve most Easy and Medium puzzles. But when you move to Hard, you hit a wall. You scan every row, column, and box, and there are no naked singles anywhere. The puzzle stares back at you with three or four candidates in every cell.

The answer is not to guess. The answer is to learn three techniques that eliminate candidates without placing a number, eventually creating the naked singles that let you finish the puzzle.

1. Hidden pairs (and triples)

A hidden pair occurs when two numbers appear as candidates in exactly two cells within a row, column, or box — and those same two cells also contain other candidates. Even though the cells have extra options, those two numbers must go in those two cells (because they have nowhere else to go). This means every other candidate in those two cells can be eliminated.

For example: in a box, suppose the number 3 appears as a candidate only in cells A and B, and the number 7 also appears only in cells A and B. Cells A and B might also list 1, 5, 9 as candidates. But since 3 and 7 have to go somewhere, and their only options are A and B, those two cells must contain 3 and 7 (in some order). You can erase 1, 5, and 9 from both cells.

Hidden triples work the same way with three numbers in three cells. They are harder to spot but follow the same logic: if N numbers can only go in N cells, those cells must contain those numbers and nothing else.

2. Pointing pairs

A pointing pair occurs when a number's candidates within a box are all confined to a single row or column. Since the number must appear in that box, and its only options within the box are in that row or column, it cannot appear in that row or column outside the box. All candidates for that number in the same row or column but in different boxes can be eliminated.

For example: in the top-left box, the number 5 only appears as a candidate in row 2. That means wherever 5 goes in this box, it will occupy a cell in row 2. So 5 cannot appear in row 2 in the top-middle or top-right boxes — you can eliminate those candidates.

Pointing pairs are often the key that breaks a Hard puzzle open. They create cascading eliminations: removing a candidate from one cell might reveal a hidden pair in another unit, which eliminates more candidates, which creates a naked single somewhere else.

3. Box-line reduction

Box-line reduction is the reverse of a pointing pair. Instead of starting from a box and looking at a line, you start from a line and look at a box. If a number's candidates within a row or column are all confined to a single box, the number can be eliminated from all other cells in that box (in different rows or columns).

For example: in row 5, the number 8 only appears as a candidate in the middle box. That means 8 in row 5 will be placed in the middle box. So 8 cannot appear anywhere else in the middle box — you can eliminate it from all cells in the middle box that are not in row 5.

The solving loop

These techniques do not place numbers directly — they eliminate candidates. The solving loop for a Hard puzzle looks like this:

Step 1: Scan for naked singles. If you find one, fill it in and scan again.

Step 2: When there are no naked singles, scan for hidden pairs, pointing pairs, and box-line reductions. Eliminate candidates.

Step 3: After eliminating candidates, go back to Step 1. The eliminations often create new naked singles.

Step 4: Repeat until the puzzle is solved.

Most Hard puzzles on PlayVersusAI can be solved with just these three techniques plus naked singles. You almost never need to guess.

How to practise

Open a Hard Sudoku puzzle and turn on pencil marks (the small candidate numbers). Instead of looking for cells to fill, look for candidates to eliminate. Find a pointing pair, cross off the invalid candidates, and see what opens up. Within a few puzzles, you will start seeing these patterns automatically — and Hard puzzles will feel like Medium ones.