Diagonal Sudoku: Rules and Strategy

Diagonal sudoku follows every classic rule, plus one more: the two long diagonals, corner to corner, must each contain the digits 1 to 9 exactly once. That gives the puzzle two extra units to check, and the best strategy is to treat them exactly like a row, column or box when you scan and take notes.

You may also see it called Sudoku X or X-Sudoku. If the standard rules are new to you, read how to play sudoku first. Everything below builds on them.

The rules

• Each row contains 1 to 9 once.
• Each column contains 1 to 9 once.
• Each 3×3 box contains 1 to 9 once.
• The main diagonal (top-left to bottom-right) contains 1 to 9 once.
• The anti-diagonal (top-right to bottom-left) contains 1 to 9 once.

That's it. There are 29 units instead of 27, and a valid diagonal sudoku still has exactly one solution.

Why the diagonals change the game

In a classic sudoku every cell has 20 peers: the other cells in its row, column and box. In diagonal sudoku, the 17 cells on a diagonal get more:

• Off-diagonal cells keep their 20 peers. Nothing changes for them.
• Cells on one diagonal have 26 peers. The diagonal adds eight cells, but two of those are already in the same box.
• The center cell (row 5, column 5) sits on both diagonals and has 32 peers. It is the most constrained cell on the board.

More peers means fewer candidates. Diagonal cells tend to run out of candidates sooner than the cells around them, so they are a good place to start looking.

Strategy 1: start with the center

Because the center cell sees its row, its column, its box and both diagonals, it often has only one or two candidates early in the puzzle. Check it first.

Here is how that can play out. Say that around the center cell:

• Row 5 already has 1 and 3.
• Column 5 already has 2 and 8.
• The main diagonal already has 4 and 6.
• The anti-diagonal already has 7 and 9.

That rules out eight digits. Only 5 is left, so the center is a 5, a naked single you would never find if you forgot to look along the diagonals.

Strategy 2: scan the diagonals for hidden singles

A diagonal is a unit, so a hidden single works on it the same way it works on a row. Pick a digit that is missing from a diagonal and ask where on that diagonal it can still go.

For example, suppose the main diagonal already holds 1, 2, 3, 5, 6, 8 and 9. The only empty diagonal cells are row 4, column 4 and row 7, column 7, and the missing digits are 4 and 7. If row 7 already has a 4 somewhere, then row 7, column 7 can't be 4. It must be 7, and row 4, column 4 must be 4. Two digits from one glance.

Do this for both diagonals every time you finish a round of row, column and box scanning. It's easy to skip, and it's where many diagonal puzzles hide their next move.

Strategy 3: use box and diagonal intersections

Each diagonal passes through three boxes. The main diagonal crosses the top-left, center and bottom-right boxes; the anti-diagonal crosses the top-right, center and bottom-left boxes. Where a diagonal and a box overlap they share three cells, and that overlap works just like the overlap between a box and a row in pointing pairs.

• Box to diagonal. If, inside the top-left box, the digit 5 can only go in row 1, column 1 or row 3, column 3 (both on the main diagonal), then the main diagonal gets its 5 from that box. Remove 5 from every other cell on the main diagonal.
• Diagonal to box. If, along the main diagonal, 5 can only go in cells that are inside the top-left box, then the box gets its 5 from those diagonal cells. Remove 5 from the other six cells of that box.

A diagonal and a row or column share only one cell, so there is no similar trick between them. The box overlaps are the ones worth watching.

Strategy 4: write diagonal-aware notes

If you take notes, the one habit that matters most is remembering the diagonals when you fill them in. For any cell on a diagonal, rule out every digit already placed on that diagonal, not just in its row, column and box. Forget this and your notes will carry candidates that are already impossible, which hides singles and makes pairs look wrong.

The same goes for cleanup. When you place a digit on a diagonal, erase it from the notes of every other cell on that diagonal too.

Common mistakes

• Treating the diagonals as a clue rather than a rule. They are a hard constraint. A digit that repeats on a diagonal is an error even if its row, column and box are fine.
• Mixing up which cells are on a diagonal. Only cells where the row number equals the column number (main) or where they add up to 10 (anti) count. Row 2, column 3 is not on a diagonal.
• Forgetting the center counts twice. The center cell belongs to both diagonals, so a digit there is ruled out along both.

Playing diagonal sudoku on Kodiak

Kodiak Sudoku has a diagonal mode with Easy, Medium and Hard puzzles. The two diagonals are drawn as faint lines across the grid so you always know which cells are involved. When you select a cell on a diagonal, the highlighted peers include the rest of that diagonal, and a digit that repeats on a diagonal is flagged as a conflict, the same way a repeat in a row is.

If you'd rather warm up on the classic rules first, try today's daily puzzle or pick a difficulty on the play page.