How to solve Sudoku X

Sudoku X, also called Diagonal Sudoku, is ordinary Sudoku with one extra rule. The two long diagonals count as units too, which gives every puzzle two more places to look for a move.

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The rules

Fill the grid so that every row, every column and every 3×3 box holds the digits 1 to 9 once each.

In Sudoku X, each of the two long diagonals also holds 1 to 9 once. The diagonals run corner to corner and cross in the centre cell.

Sudoku X strategies

These eight techniques put the diagonals to work. Everything in How to solve Sudoku still applies, with each diagonal counted as one more unit, except unique rectangles: a rectangle with a corner on a diagonal isn't a deadly pattern, so only use rectangles whose four corners are all off the diagonals.

A few terms first. The down diagonal runs from the top left corner to the bottom right. The up diagonal runs from the bottom left corner to the top right. They cross in the centre cell, row 5, column 5, which is on both. In the diagrams both diagonals are drawn as thin grey lines.

The diagrams use the same colours as our other guides. Green marks the cell the step is about and the answer it gives you. Blue marks the cells it depends on. Purple marks the cells you're scanning. Red with a cross is a candidate you can remove.

Whenever you look for a move, check both diagonals just as you would a row or column.

Singles on the diagonals

Hidden singles on a diagonal

Seeing it. Pick a digit the diagonal doesn't have yet and ask where it can go. In the diagram, the down diagonal has no 1. The 1s in the grid, marked blue, rule out every purple cell. The 1 in row 1, column 9 covers its row and column and also the up diagonal, which crosses the centre. The others cover their rows, columns and boxes.

Using it. Only row 8, column 8 is left, so it's the diagonal's 1. Sweep each digit along both diagonals the same way you sweep it across the boxes. A diagonal that is still empty is worth checking, because nine cells spread over nine rows, nine columns and three boxes are easy for the rest of the grid to rule out.

On the down diagonal, the 1s in the grid rule out every cell except row 8, column 8

Singles on the diagonals

Count the diagonal too

Seeing it. A cell on a diagonal sees six more cells than an ordinary cell, so it's often down to one candidate sooner than you'd expect. In the diagram, the green cell is row 1, column 9. Its row, column and box hold every digit except 5 and 7.

Using it. Now look along its diagonal. The up diagonal already has a 5, in row 9, column 1, so the green cell can't be 5 and must be 7. When you list the candidates for a cell on a diagonal, always include the digits on that diagonal.

Row, column and box leave 5 or 7 in row 1, column 9, and the 5 on the up diagonal leaves 7

Singles on the diagonals

The centre cell

Seeing it. The centre cell is on both diagonals as well as its row, column and box, so it sees 32 other cells instead of 20. That makes it one of the first cells to solve. In the diagram, row 5, column 5 and the centre box leave 1, 4, 6 or 8.

Using it. The diagonals finish the job. The down diagonal has 1, 8 and 4, and the up diagonal has another 1, all marked blue. Only 6 is left. Check the centre again whenever a digit lands on either diagonal.

The diagonals rule out 1, 4 and 8 in the centre cell, which leaves 6

Boxes and diagonals

A box that points along a diagonal

Seeing it. This is the pointing pair from ordinary Sudoku, with a diagonal in place of a row. Look for a box whose only places for a digit are all on one diagonal. In the diagram, box 7 can only have its 8 in row 7, column 3 or row 9, column 1. Both are on the up diagonal.

Using it. Whichever of the two takes the 8, the up diagonal has its 8 inside box 7. So no other cell on that diagonal can be 8. The centre cell was 2 or 8, so it loses the 8 and becomes 2. Only the four corner boxes and the centre box contain diagonal cells, so those are the boxes to check.

Box 7 can only have its 8 in two cells on the up diagonal, so the centre cell can't be 8

Boxes and diagonals

A diagonal inside one box

Seeing it. Now turn it round. Look for a digit whose only places on a diagonal are inside one box. In the diagram, the down diagonal can only take 7 in row 1, column 1 or row 2, column 2, both in box 1.

Using it. The diagonal's 7 is one of those two cells, so box 1's 7 is too. Remove 7 from the rest of box 1. Here that's the green cell, row 2, column 1. A diagonal crosses only three boxes, so its candidates often bunch up like this once a few digits are in.

The down diagonal's 7 can only be in box 1, so row 2, column 1 can't be 7

Pairs and links

Pairs on a diagonal

Seeing it. Naked pairs and triples work on a diagonal exactly as they do in a row. In the diagram, the blue cells on the up diagonal, row 2, column 8 and row 4, column 6, can only be 5 or 8.

Using it. Those two cells will take the diagonal's 5 and 8 between them, so no other cell on the diagonal can be 5 or 8. The green cell, row 1, column 9, loses its 8 and is left with 1 or 7. Three cells that share three digits work the same way.

Two cells on the up diagonal can only be 5 or 8, so 8 comes off row 1, column 9

Pairs and links

Hidden pairs on a diagonal

Seeing it. Look for two digits that fit in only the same two cells of a diagonal. In the diagram, the up diagonal still needs 2 and 4, and only the green cells, row 2, column 8 and row 7, column 3, can take either of them.

Using it. Those two cells must be 2 and 4, in some order, so every other candidate in them can go. Row 2, column 8 loses 1, 3 and 9, and row 7, column 3 loses 3, 7 and 9. Hidden pairs are hard to spot by eye, so look for them on a diagonal that has only a few empty cells left.

Only two cells on the up diagonal can take 2 and 4, so their other candidates come off

Pairs and links

Two places on a diagonal

Seeing it. Two cells on the same diagonal never share a row or a column. So when a digit has only two places on a diagonal, they sit at opposite corners of a rectangle. In the diagram, the up diagonal can only take 8 in row 6, column 4 or row 9, column 1.

Using it. One of the two blue cells must be 8. The green cell, row 9, column 4, shares a row with one and a column with the other, so it can't be 8 whichever it is. The same goes for the fourth corner of the rectangle, row 6, column 1, though here it's already filled. If the two places share a box, the other cells of that box lose the digit too.

The up diagonal's 8 must be in one of the two blue cells, so row 9, column 4, which sees both, can't be 8