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Can the highlighted cell be a 3?
Level 4 · Whole-grid rules
How to solve diagonal sudoku
Digits can’t repeat along a marked diagonal.
Diagonal sudoku marks one or both long diagonals of the grid. Digits can't repeat along a marked diagonal, so in a 9x9 grid each marked diagonal holds 1 to 9 exactly once.
A diagonal works like an extra row that runs through three boxes. It gives you another group to check for singles, and every digit on it rules out more cells.
How setters write it
The same rule, worded the ways you’ll see it on puzzles.
- Digits may not repeat along the marked diagonals.
- Sudoku X.
- Positive diagonal: no repeats bottom-left to top-right.
Worth memorizing
- A marked diagonal works like an extra row: in a 9×9, it holds 1 to 9 once.
- The center cell is on both diagonals when both are marked.
Solving tips
Treat it like a row
Scan a marked diagonal for missing digits just as you would a row. If it holds eight digits, the ninth cell gets the one that is missing.
The center cell is on both
When both diagonals are marked, the center cell lies on each of them. A digit placed there is ruled out of the other 16 diagonal cells.
Each diagonal crosses three boxes
One diagonal runs through the top-left, center and bottom-right boxes, and the other through the top-right, center and bottom-left. A digit on the diagonal in one of those boxes is ruled out of the diagonal cells in the other two.
Check which diagonal is marked
Some puzzles mark only one. The positive diagonal runs from bottom left to top right, and the negative diagonal runs from top left to bottom right.
A worked example: finishing a diagonal
Both diagonals are marked. We name cells by row and column, so row 1, column 1 is r1c1. The negative diagonal runs from r1c1 to r9c9. It holds 1 in r1c1, 2 in r2c2, 3 in r4c4, 4 in r5c5, 5 in r6c6, 6 in r8c8 and 7 in r9c9. r3c3 and r7c7 are empty. Column 3 has a 9 in r5c3.
- The negative diagonal must hold 1 to 9 once. It is missing 8 and 9, so r3c3 and r7c7 hold 8 and 9 in some order.
- r3c3 is in column 3, which already has a 9 in r5c3. So r3c3 can't be 9. It is 8, and r7c7 is 9.
- Now look at the positive diagonal, from r9c1 to r1c9. The center cell r5c5 is on both diagonals, and it holds 4. So no other cell on the positive diagonal can be 4.
- The positive diagonal passes through r7c3 and r3c7. r7c3 is in column 3, which has 8 in r3c3 and 9 in r5c3. r3c7 is in row 3 with the 8 and in column 7 with the 9 in r7c7. So neither cell can be 8 or 9, and neither can be 4.
- The positive diagonal still needs its own 8 and 9. They can't go in r5c5, r7c3 or r3c7, so they must go in two of its other six cells: r9c1, r8c2, r6c4, r4c6, r2c8 and r1c9. Mark those candidates now. The next 8 or 9 you place in those rows or columns may pin them down.
Practice
Reference table
| Box | Negative diagonal (top left to bottom right) | Positive diagonal (bottom left to top right) | Diagonal cells in the box |
|---|---|---|---|
| Top-left | r1c1, r2c2, r3c3 | none | 3 |
| Top-middle | none | none | 0 |
| Top-right | none | r1c9, r2c8, r3c7 | 3 |
| Middle-left | none | none | 0 |
| Center | r4c4, r5c5, r6c6 | r4c6, r5c5, r6c4 | 5 |
| Middle-right | none | none | 0 |
| Bottom-left | none | r7c3, r8c2, r9c1 | 3 |
| Bottom-middle | none | none | 0 |
| Bottom-right | r7c7, r8c8, r9c9 | none | 3 |
Going deeper
When a box pins a digit to the diagonal
Say the top-left box can only put its 5 in r1c1, r2c2 or r3c3. Those three cells are all on the negative diagonal. Whichever one gets the 5, the diagonal has its 5. So you can remove 5 from every other cell on that diagonal, in the center and bottom-right boxes.
This is the same idea as a digit pointing along a row. The diagonal is just another line of nine cells, and the box overlap works the same way. It also works with two cells. If the box's 5 can only go in r1c1 or r3c3, the rest of the diagonal still loses 5.
When the diagonal pins a digit to a box
It also works in reverse. Say the negative diagonal can only put its 2 in r4c4, r5c5 or r6c6, because every other diagonal cell is ruled out. All three cells are in the center box. So the center box's 2 is on the diagonal, and no other cell in the center box can be 2.
Use the table above to see which boxes this can happen in. The four middle-edge boxes have no diagonal cells, so the diagonals never affect them directly.
The center box holds five diagonal cells
With both diagonals marked, the center box has five diagonal cells: r4c4, r4c6, r5c5, r6c4 and r6c6. The four corner boxes each have three.
A digit in the center cell rules itself out of both diagonals at once. That makes r5c5 a good cell to solve early. Once it is known, 16 other diagonal cells lose that digit.
Every row and every column crosses each marked diagonal exactly once. So a digit placed anywhere in a row rules that digit out of the diagonal cell in that row. With both diagonals marked, each row crosses them at two cells, except row 5, where they meet at r5c5. Scan the diagonals this way each time you place a digit.
Common mistakes
Mixing up the two diagonals
The positive diagonal runs from bottom left to top right. The negative diagonal runs from top left to bottom right. If only one is marked, check which one before you use it.
Using diagonals in a normal puzzle
Classic sudoku has no diagonal rule. Digits can repeat on a diagonal unless the puzzle marks it and says so.
Forgetting the center cell is shared
When both diagonals are marked, r5c5 belongs to both. A digit there must be crossed off both lines. Beginners often cross it off only one.
In mixed-rule puzzles
Diagonals mix well with other global rules. With anti-king, the cells along a diagonal already can't repeat, so the extra limits come from cells beside the diagonal. With anti-knight, a digit on a diagonal also rules out cells a knight's move away, some of which sit on the other diagonal. With extra regions such as the four windows in hyper sudoku, each diagonal crosses those windows, which gives you more overlaps to scan. With killer cages, a cage that sits on a diagonal still can't repeat digits, and the diagonal can tell you which digits are left for it. Read the rules for which diagonals count, since some puzzles mark only one. With odd-even marks, a marked diagonal needs five odd digits and four even ones, just like a row.
What the video shows
- Digits can’t repeat along a marked diagonal.
- Two 3s on the diagonal.
- Different digits are fine.
Questions solvers ask
What is Sudoku X?
Sudoku X is another name for diagonal sudoku. Both long diagonals must hold 1 to 9 once, along with the normal rules.
Can numbers repeat on the diagonal in sudoku?
Not on a marked diagonal. In classic sudoku the diagonals have no rule, so digits can repeat on them.
What is a positive diagonal in sudoku?
It is the diagonal from the bottom-left corner to the top-right corner. The negative diagonal runs from top left to bottom right.
How many cells are on a sudoku diagonal?
Nine. The two long diagonals share the center cell, so together they cover 17 cells.
Do short diagonals count in diagonal sudoku?
Usually only the two long diagonals, from corner to corner, carry the rule. Other diagonal lines belong to different rules, so check the puzzle's rules.
What is the difference between Sudoku X and hyper sudoku?
Sudoku X adds the two long diagonals as places where digits can't repeat. Hyper sudoku adds four shaded 3x3 windows. Some puzzles use both.
How does diagonal sudoku work on a 6x6 grid?
The same way. Each marked diagonal has six cells, and it must hold 1 to 6 once.
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