Diagonal Sudoku (Sudoku X): Rules and Solving Tips

Diagonal Sudoku, also called Sudoku X, is the gentlest way to try a Sudoku variant. It keeps every rule you already know and adds just one more: the two long diagonals must also hold the digits 1 to 9. That small change gives you two extra lines to scan, a very special centre cell and a few new patterns to spot. Here are the rules, how the diagonals change your solving, and a calm strategy for finishing one.

What is diagonal Sudoku?

The rules of Sudoku X

The rules of diagonal Sudoku are the classic ones plus a single addition:

1. Fill every empty cell with a digit from 1 to 9.
2. Each row contains 1 to 9 exactly once.
3. Each column contains 1 to 9 exactly once.
4. Each 3×3 box contains 1 to 9 exactly once.
5. Each of the two main diagonals (top-left to bottom-right, and top-right to bottom-left) contains 1 to 9 exactly once.

That’s it. In printed puzzles and apps the diagonal cells are usually shaded or crossed by a faint line, so you can see at a glance which cells carry the extra rule. A well-made Sudoku X has exactly one solution, just like a good classic puzzle.

One subtle point is worth knowing. The diagonals are not a decoration; they are part of what makes the answer unique. The extra rule lets setters use fewer clues, because the diagonals do some of the work the missing clues would have done. So if you ignored them and solved the grid as a classic Sudoku, the givens might well allow more than one answer. You have to use them.

Which cells lie on the diagonals

It helps to name cells by row and column: r1c1 is row 1, column 1 (the top-left corner) and r9c9 is row 9, column 9 (the bottom-right corner). With that notation the two diagonals are easy to describe:

A few facts fall straight out of that table, and every one of them is useful while solving:

• There are 17 diagonal cells, not 18. Each diagonal has nine cells, but they share the centre cell r5c5.
• Each diagonal visits three boxes, three cells in each. The main diagonal runs through the top-left, centre and bottom-right boxes; the anti-diagonal through the top-right, centre and bottom-left boxes.
• The centre box holds five diagonal cells: r4c4, r4c6, r5c5, r6c4 and r6c6, a small X of its own.
• The four edge boxes (top-middle, middle-left, middle-right and bottom-middle) contain no diagonal cells at all. They behave exactly as in a classic puzzle.
• Every diagonal cell is in a different row and a different column from the other cells on its diagonal. A diagonal and a row only ever meet in one cell.

How the diagonals change solving

In classic Sudoku every cell “sees” 20 other cells: the 8 others in its row, the 8 others in its column and the 4 cells of its box that aren’t already counted. A digit placed in a cell rules that digit out of all 20. In Sudoku X, cells on a diagonal see more.

The centre cell is the most powerful square on the board

The centre cell r5c5 sits in row 5, column 5, the centre box and both diagonals. It sees 32 of the other 80 cells, so it is squeezed from more directions than any other cell. Two things follow:

• It is often one of the first cells you can fill. With so many neighbours, the candidates in r5c5 tend to run out quickly. Check it early and check it again after every few placements.
• Once it is filled, it reaches everywhere. A digit in r5c5 is removed from row 5, column 5, the centre box and all 16 other diagonal cells at once. That single placement can open singles in all four corner boxes.

The corners matter too

The four corner cells, r1c1, r1c9, r9c1 and r9c9, each sit at the end of a diagonal. In classic Sudoku r1c1 and r9c9 have nothing to do with each other. In Sudoku X they are on the same line, so they can never hold the same digit. Corner boxes, in general, feel more connected than in a classic puzzle, because each one is tied by a diagonal to the opposite corner and to the centre.

Diagonal singles

The most common new move in Sudoku X is a single found on a diagonal. It works exactly like a single in a row or column; the only change is the shape of the line you scan.

Hidden single on a diagonal

Pick a digit, say 4, and look along one diagonal. For each of its empty cells, ask: “Is there already a 4 in this cell’s row, column or box, or elsewhere on this diagonal?” If only one cell on the diagonal can still take a 4, then the 4 must go there. That is a hidden single on the diagonal.

Because the nine diagonal cells sit in nine different rows and nine different columns, every 4 in the grid rules out at least one diagonal cell (the one in its row) and usually two (the one in its column too). Diagonals are therefore surprisingly easy to narrow down: a handful of givens of the same digit can eliminate most of a diagonal on its own.

Naked single helped by a diagonal

The other kind of single is a cell with only one possible digit left. In Sudoku X, a diagonal cell has more neighbours, so it reaches that point sooner. When you check a diagonal cell, remember to look along its diagonal as well as its row, column and box. A cell that seems to have three options in classic terms may have only one once you count the digits on its diagonal.

If hidden and naked singles are still a little hazy, our Sudoku strategies guide explains both with real diagrams. Everything there works on a diagonal too.

Where diagonals meet boxes

Each diagonal passes through three boxes, three cells at a time. Those overlaps create a diagonal version of a classic technique called locked candidates, and it works in both directions.

From box to diagonal (pointing)

Suppose that in the top-left box, the only cells that can hold a 7 are r1c1, r2c2 and r3c3, which are all on the main diagonal. The box must have a 7 somewhere, so the main diagonal’s 7 is inside the top-left box. That means no other cell on the main diagonal can be a 7: you can remove 7 from r4c4 to r9c9. The same works if only two of those cells can hold a 7, as long as both are on the diagonal.

From diagonal to box (claiming)

Now turn it around. Suppose that along the anti-diagonal, the only cells that can hold a 2 are r7c3 and r8c2, both in the bottom-left box. The diagonal must have a 2, so it will be in one of those two cells, which means the bottom-left box’s 2 is on the diagonal. You can remove 2 from every other cell of the bottom-left box.

The busy centre box

The centre box is crossed by both diagonals, so it has two of these overlaps at once: r4c4, r5c5 and r6c6 on the main diagonal, and r4c6, r5c5 and r6c4 on the anti-diagonal. Only the four middle-edge cells of the box, r4c5, r5c4, r5c6 and r6c5, are off the diagonals. When a digit in the centre box is limited to one diagonal, it often clears that digit from a corner box far away. It is a good place to look when the corners have gone quiet.

A step-by-step solving strategy

There is no single Sudoku X method, but this order works well for most puzzles, from gentle to tough:

1. Mark the diagonals clearly

If you are solving on paper and the diagonals aren’t shaded, draw two light lines from corner to corner. Mistakes in Sudoku X almost always come from forgetting that a cell is on a diagonal.

2. Start with the centre cell

Write out every digit that r5c5 could still be, checking row 5, column 5, the centre box and both diagonals. With 32 neighbours it is often solved early, and it feeds every corner box.

3. Scan both diagonals for hidden singles

Go through the digits 1 to 9 and look for any digit that has only one home on a diagonal. Then do your usual scan of boxes, rows and columns. Diagonal singles and classic singles feed each other, so switch between them freely.

4. Work the corner boxes

The four corner boxes each hold three diagonal cells. Once a diagonal has several digits placed, those three cells become tightly limited, and the rest of the corner box often follows.

5. Use pencil marks on the diagonals

When the singles run out, write candidates in the 17 diagonal cells first. Diagonals are where Sudoku X differs from classic Sudoku, so that is where hidden progress usually sits. Look for the box-and-diagonal overlaps described above, and for pairs: if two cells on a diagonal can only be 3 or 8, no other cell on that diagonal can be 3 or 8. Our guide to Sudoku pencil marks shows a tidy way to keep notes.

6. Check each new digit against the diagonal

Every time you place a digit in a diagonal cell, clear that digit from your notes along the whole diagonal, not just its row, column and box. Then glance at the centre cell again. Progress in Sudoku X tends to arrive in bursts, and the centre is often where the next burst starts.

Common mistakes to avoid

• Forgetting the diagonals. The most common error is placing a digit that is fine for the row, column and box but repeats on a diagonal. Before writing in any diagonal cell, run your eye along its diagonal.
• Treating the diagonals as optional. They are part of the puzzle’s logic. If you ignore them, you may reach a point where the classic rules allow two answers and feel forced to guess.
• Confusing the two diagonals. r1c1 and r9c9 are on the main diagonal; r1c9 and r9c1 are on the other. Only r5c5 is on both. Cells such as r1c1 and r9c1 share a column but not a diagonal.
• Inventing extra diagonals. Only the two long corner-to-corner lines count. Shorter diagonals inside a box, or diagonals that “wrap around” the edge, are not part of the rule in a standard Sudoku X.
• Not updating notes along the diagonal. If you keep pencil marks, a digit placed on a diagonal must be cleared from all the other diagonal cells too.
• Using uniqueness shortcuts on diagonal cells. As noted above, unique-rectangle reasoning isn’t safe when the rectangle touches a diagonal.

A short note on history

Diagonal rules are older than modern Sudoku itself. In 1895 the French newspaper La France printed a number puzzle, a kind of magic square, in which rows, columns and broken diagonals had to contain only the numbers 1 to 9. It was a relative of Sudoku rather than Sudoku as we know it, but it shows how natural the idea of a diagonal rule is. You can read more about those early puzzles in our history of Sudoku.

For modern puzzles, Wikipedia describes the diagonal rule as the most common of the extra constraints added to classic Sudoku, and notes that Dell Magazines’ large 16×16 “Number Place Challenger” puzzles use it. Who first published a 9×9 Sudoku X, and exactly when, is not clearly recorded, so we won’t guess. Today it appears in newspapers, puzzle books and apps under both names, Sudoku X and diagonal Sudoku.

If you enjoy one extra rule, you might like the other variants that add regions, such as Hyper and Jigsaw, in our tour of Sudoku variants. And if you’d rather have more grid than more rules, try the five overlapping grids of Samurai Sudoku.

How to practise for Sudoku X

A quick, honest note: Sudoku.my currently offers classic 9×9 Sudoku, not diagonal Sudoku. The good news is that classic puzzles are excellent training for Sudoku X, because every diagonal technique is a classic technique applied to a new line. Here is how to use the app to get ready:

• Make singles automatic. Easy puzzles can be solved with hidden and naked singles alone, the same moves you’ll use most on the diagonals.
• Practise locked candidates. The box-and-diagonal overlaps are the same idea as box-and-row overlaps. Medium puzzles bring in locked candidates, and hard puzzles add pairs and triples.
• Learn from the hints. Each hint explains the logical step it suggests, so when you are stuck you learn the pattern rather than just getting a digit.
• Keep a daily rhythm. The daily challenge gives you one fresh puzzle every day. Like a good Sudoku X, every puzzle on Sudoku.my has exactly one solution and can be solved by logic, with no guessing.

Miko the cat suggests a small exercise for classic puzzles: after you finish one, look along its two diagonals. Usually a few digits repeat, which is a nice reminder of how much extra information the X rule adds when a setter builds a puzzle around it.

#Diagonal Sudoku#Sudoku X#Variants