Sudoku X: Diagonal Rules, XV Variant & Solving Tips

Sudoku X adds exactly two constraints to a normal sudoku. Both main diagonals must also contain the digits 1 to 9, once each. Everything else — the rows, the columns, the nine 3×3 boxes — is unchanged. That is the whole rule, and it is enough to make a puzzle that looks familiar behave very differently.

The name comes from the shape the two diagonals make across the grid. You will also see the variant called diagonal sudoku, X-sudoku, or Sudoku X. This page covers the rule change, the techniques the diagonals unlock, a worked example of a diagonal-only elimination, and the separate XV variant, which is about adjacent cells summing to five or ten.

The rule change, precisely

A classic sudoku has 27 units: nine rows, nine columns, nine boxes. Sudoku X has 29, because the two long diagonals are units in their own right. Each diagonal is nine cells and must hold each digit exactly once.

• The main diagonal runs from the top-left corner to the bottom-right corner: R1C1, R2C2, R3C3, R4C4, R5C5, R6C6, R7C7, R8C8, R9C9. A cell is on it when its row and column numbers are equal.
• The anti-diagonal runs from the top-right corner to the bottom-left: R1C9, R2C8, R3C7, R4C6, R5C5, R6C4, R7C3, R8C2, R9C1. A cell is on it when its row plus its column equals ten.
• The centre cell, R5C5, is the only cell on both diagonals, and therefore the only cell that belongs to five units at once: its row, its column, its box, and both diagonals.

That last bullet is the whole strategic heart of the variant, and it is worth stating as a count. An ordinary cell sits in three units. A diagonal cell that is not the centre sits in four. The centre sits in five. No other cell on the grid is constrained that many ways, which is why strong Sudoku X solvers look at the middle of the board first rather than last.

How the diagonals change solving

Most guides say "the diagonals give you more information" and stop there. The useful version is more specific: the diagonals add a fourth and fifth place to check for the same techniques you already use, and they add two kinds of elimination that a classic puzzle cannot express at all.

Diagonal-only eliminations

A diagonal-only elimination is a candidate removed because of a diagonal and for no other reason. The row, the column and the box all still permit the digit; the diagonal forbids it. These are the moves that separate a Sudoku X solve from a classic solve, and they are easy to spot once you know to look for them: take a cell on a diagonal, and check whether that digit already appears elsewhere on the same diagonal.

Note the asymmetry. A digit on the main diagonal eliminates itself from every other main-diagonal cell, but says nothing about the anti-diagonal. The two diagonals share exactly one cell, so they interact at the centre and nowhere else.

Hidden singles along a diagonal

Ask "where can 7 go on this diagonal?" and treat the answer exactly as you would a hidden single in a row. If only one diagonal cell can take the 7, it goes there — even when that cell has three other candidates from its row, column and box. This is the highest-yield diagonal technique and it is available early in most puzzles.

Diagonal pairs and triples

Two cells on the same diagonal that share the same two candidates lock those two digits to those two cells across the whole diagonal. In a classic puzzle you would apply the same logic to a row, a column or a box; here the diagonal is a unit of the same size, so the logic transfers without modification. The payoff is often larger, because a diagonal cuts across three boxes and can clear candidates in all of them.

Diagonal pointing

If a digit can only go in one box along a diagonal, then it cannot go anywhere else on that diagonal, and conversely a digit confined to a diagonal within one box can sometimes be eliminated from the rest of that box. This is the diagonal analogue of pointing pairs, and it appears most often in the middle band where the diagonals pass through the centre box at an angle.

The centre behaves like a hinge

Because R5C5 is the crossing point, the digit placed there is spent on both diagonals simultaneously. That makes it unusually informative and unusually fragile. If the centre is 5, then neither diagonal can contain another 5 anywhere, which is a very strong statement about sixteen cells at once. If the centre is still undecided, the two diagonals compete for it, and resolving the centre often resolves both.

A worked diagonal elimination

Here is a constructed teaching position rather than a real puzzle, so the arithmetic is easy to follow. Focus on the centre cell, R5C5.

Step 1 — what the row says. Row 5 already contains 1, 2, 3, 6, 7 and 8. So R5C5 cannot be any of those. The row leaves 4, 5 and 9.

Step 2 — what the column says. Column 5 contains a 4 at R2C5 and an 8 at R7C5. The 8 was already excluded by the row; the 4 is new. That leaves 5 and 9.

Step 3 — what the box says. The centre box, R4C4 to R6C6, contains neither 5 nor 9. So the box eliminates nothing here, and a classic solver stops with the pencil marks {5, 9}. This is the point where most players move on to another cell.

Step 4 — what the diagonal says. R5C5 sits on the anti-diagonal, because 5 + 5 = 10. Follow that anti-diagonal up and to the right: R4C6, R3C7, R2C8, R1C9. Suppose R3C7 is a 9. Then the anti-diagonal already contains a 9, so no other cell on it can be 9 — including R5C5.

Conclusion. The 9 is eliminated by the diagonal alone. R5C5 must be 5. A classic solver would have carried {5, 9} into the rest of the puzzle and might have needed a chain or a guess to resolve it. The diagonal resolves it immediately, in one step, with no search.

Here is the shape of the anti-diagonal on the grid. The marked cells are the nine cells that must each contain 1 to 9 exactly once, and the centre is the one cell that also belongs to the other diagonal.

The same move works in mirror image on the main diagonal: a digit sitting at R2C2 eliminates that digit from every other cell where the row and column numbers match, including the centre. In practice you scan both diagonals for any digit that appears once, then check which cells that digit now attacks.

One caution about the example. Diagonal eliminations are only sound if the puzzle actually has the diagonal constraint. Many apps publish a classic grid with an X drawn over it as decoration. If the rules do not say diagonals, the elimination above is invalid, and applying it will produce contradictions two moves later.

Standard techniques that still apply

Nothing is removed from the classic toolkit. Sudoku X is a sudoku with extra constraints, not a different logic system, so every technique you know continues to be valid — it simply has more places to fire.

• Naked singles. Still the workhorse. The diagonals create more of them, because more cells have their candidates trimmed.
• Hidden singles. Check rows, columns, boxes, and now both diagonals. Two extra units means two extra places per digit.
• Naked and hidden pairs, triples, quads. Identical logic, extra units. A pair on a diagonal is often more powerful than a pair in a row, because it removes candidates from three boxes at once.
• Pointing pairs and box-line reduction. Still valid inside the nine boxes. The diagonal adds a second form of pointing, described above.
• X-wing, swordfish and other fish. These work on rows and columns as before. They do not have a diagonal analogue, because there are only two diagonals and a fish needs several parallel units.
• XY-wing and chains. Unaffected. If a chain works in a classic grid, it works here, because the diagonal constraints only ever remove candidates, never add them.
• Uniqueness arguments. Still sound. The classic deadly rectangle swaps two pairs of digits between four cells, and that swap preserves the contents of every row, column, box and diagonal. The constraint set is unchanged by the swap, so the uniqueness argument survives.

That last point surprises people who assume extra constraints break uniqueness reasoning. They do not, and the reason is worth remembering: uniqueness arguments depend on a transformation that leaves every unit's contents intact. Diagonals are units, and the rectangle swap leaves them intact too.

Techniques the diagonals add

Beyond diagonal singles and pairs, three habits do most of the extra work.

Start at the centre. Not because the centre is always given, but because it is the cheapest cell to reason about. Five units constrain it, so its candidate list collapses faster than any other cell's. In many puzzles the centre falls out in the first two minutes, and the digit placed there then eliminates itself from sixteen other cells along the two diagonals.

Sweep each digit along both diagonals. Pick a digit, then trace the main diagonal and the anti-diagonal looking for where it is already placed. Each placement attacks eight cells on that diagonal. Doing this for all nine digits is a few minutes of work and usually produces two or three placements.

Use the diagonals as a cross-check. When a classic technique gives you a placement near a diagonal, verify it against the diagonal before you commit. Sudoku X puzzles are often generated so that the diagonal is the only thing disambiguating a cell, which means a classic-only solve can look complete while containing an error.

Difficulty: where Sudoku X actually sits

Diagonal puzzles feel harder than classic puzzles of the same given count, and the feeling is justified. Extra constraints reduce the search space, which sounds like it should make puzzles easier, but they also remove the slack that easy puzzles rely on: a grid with diagonals frequently has fewer alternative paths, so a wrong branch punishes you faster and a missed diagonal elimination stalls you for longer.

As a rough guide, treat a labelled easy Sudoku X as a classic medium, and a labelled hard one as a classic hard with a stricter solve path. The gap widens at the top end, where constructors use diagonals to force uniqueness in positions that would otherwise need guessing.

If you are new to the variant, start with puzzles that have plenty of givens on the diagonals themselves. A diagonal with five or six givens gives you immediate hidden singles; a diagonal with one given is a much longer project.

XV sudoku: the other variant in this family

XV sudoku is a separate idea that is often confused with Sudoku X because both use an X. In XV, the rules concern adjacent cells, not diagonals.

• A V between two orthogonally adjacent cells means those two digits sum to 5. The letter is the Roman numeral for five.
• An X between two adjacent cells means those two digits sum to 10. The letter is the Roman numeral for ten.
• No mark between two adjacent cells means their sum is neither 5 nor 10. This negative constraint is the part beginners forget, and it is where most of the puzzle's power lives.

Because every edge on the grid is in one of three states — marked V, marked X, or unmarked — an XV puzzle carries far more information per given than a classic grid. That is why XV puzzles often look sparse and still solve cleanly.

The rule that unlocks XV

A 5 cannot appear next to any mark at all. It has no partner that sums to 5, because 5 + 0 is not a digit, and it has no legal partner that sums to 10, because 5 + 5 would put two fives in the same row, column or box. So every edge touching a 5 must be unmarked.

Turn that around and you get a free elimination: every cell orthogonally adjacent to a V or an X cannot be 5. On a dense XV grid this single deduction clears the digit 5 from a large fraction of the board before you have placed anything. It is the equivalent of the centre-cell insight in Sudoku X — one observation that reorganises the whole solve.

Two smaller consequences follow. Any cell on a V edge is in the range 1 to 4, so a V edge near a cell whose row already contains 1, 2 and 3 forces the digit 4. And any cell on an X edge cannot be 5, which combines with a V edge on a neighbouring side to pin a digit quickly.

How to solve an XV puzzle

1. Mark the 5-free zones. Cross 5 out of every cell adjacent to a mark. Do this before anything else.
2. Collect the V edges. Those cells are 1 to 4. Where a V edge touches a cell with only two candidates, resolve it immediately.
3. Use the X edges for parity. Sum 10 pairs are one odd and one even digit in every case, so an X edge always links an odd to an even. That is a strong constraint on a row that is otherwise all odd or all even.
4. Apply the negative constraint. For an unmarked edge, cross out complement pairs: if a cell is 4, its unmarked neighbour cannot be 1 or 6. This is the step that separates a fast solve from a slow one.
5. Fall back to classic techniques. Naked singles, hidden singles, pairs and pointing all still apply, now with much thinner candidate lists.

A worked XV step

Take a cell R1C2 whose row and box have already reduced it to {5, 7}. Suppose the edge between R1C1 and R1C2 is marked X. An X edge cannot contain a 5, so R1C2 must be 7. The sum is 10, so R1C1 must be 3.

Now look at the edge between R2C1 and R1C1. If it is unmarked, then R2C1 cannot be 7, because 3 + 7 = 10 would require an X. That second elimination comes entirely from the negative constraint, and it is the kind of move players miss when they treat unmarked edges as carrying no information.

Sudoku X and XV are not the same puzzle

The overlap in naming causes real confusion, so keep the two ideas separate. Sudoku X constrains the two long diagonals. XV constrains pairs of adjacent cells. A grid can have either, both, or neither, and constructors do publish combined puzzles where both rule sets apply.

If you are handed a puzzle labelled "Sudoku X" with V and X marks in it, read the rules before you start: you may have a combined variant, in which case the diagonals and the adjacency sums both apply, or you may have a mislabelled XV puzzle with no diagonal constraint at all.

Common mistakes

1. Forgetting the anti-diagonal. Nearly everyone checks the top-left to bottom-right diagonal and forgets the other one. The anti-diagonal is the one that breaks most solves.
2. Applying diagonals to a classic grid. If the rules do not mention diagonals, a diagonal elimination is invalid and will corrupt your pencil marks.
3. Double-counting the centre. R5C5 is one cell, not two. It appears on both diagonals, which means the digit there is used twice, not that you can place two digits.
4. Treating a diagonal as a row. A diagonal is a unit of nine cells, but the cells are not contiguous in the grid. Counting a digit along a diagonal requires following the slope, and it is easy to skip a cell.
5. Ignoring unmarked edges in XV. The negative constraint is half the puzzle's information. Players who only read the V and X marks are solving with one hand tied.
6. Confusing the letters. V is five, X is ten. It is Roman numerals, not an X for "diagonal" — which is exactly why the two variants get mixed up.
7. Assuming XV means diagonal. Unless the rules say so, XV has no diagonal constraint.

Classic vs Sudoku X vs XV

If you want to test any of this on a live grid, our free sudoku game and the sudoku solver are the quickest way to check a position, and the core strategy guide covers the classic techniques this page assumes. Variants with their own rule twists are collected alongside them: jigsaw sudoku replaces the boxes with irregular regions, and killer sudoku adds cage sums.

Glossary

Unit

Any group of nine cells that must contain 1 to 9 once each: a row, a column, a box, or a diagonal.

Peer

Any cell sharing a unit with another cell. In Sudoku X a cell has more peers than in a classic grid.

Candidate

A digit that could legally go in a cell, given everything placed so far. Also called a pencil mark.

Naked single

A cell with exactly one candidate left.

Hidden single

A digit with exactly one possible cell left in a unit, even if that cell has other candidates.

Main diagonal

Top-left to bottom-right, where the row number equals the column number.

Anti-diagonal

Top-right to bottom-left, where the row number plus the column number equals ten.

Centre cell

R5C5, the only cell on both diagonals and therefore in five units.

Diagonal-only elimination

A candidate removed solely because of a diagonal, with the row, column and box all still allowing it.

Bivalue cell

A cell with exactly two candidates, the raw material for pairs and wings.

Pointing

A digit confined to one line within a box, which then eliminates it from the rest of that line, or the reverse.

Fish

Techniques such as X-wing and swordfish that exploit a digit confined to matching positions across parallel units. They apply to rows and columns, not to diagonals.

Band and stack

Three consecutive rows (a band) or three consecutive columns (a stack). Useful for scanning even in a 9×9 grid.

Given

A digit printed in the starting grid.

XV

A variant in which adjacent cells are marked V for a sum of 5 or X for a sum of 10, with unmarked pairs summing to neither.

Negative constraint

The rule that an unmarked edge carries information: its two digits cannot sum to 5 or 10.

Unique rectangle

A uniqueness technique that still works in Sudoku X, because the swap it relies on preserves every unit including the diagonals.

Frequently asked questions

What is Sudoku X?

Sudoku X is a sudoku variant with two extra constraints: both main diagonals must contain the digits 1 to 9 exactly once, in addition to every row, column and 3 by 3 box. That takes the grid from 27 units to 29 units and makes the centre cell the most constrained cell on the board.

Why is the centre cell so important in Sudoku X?

The centre cell at R5C5 is the only cell that lies on both diagonals, so it belongs to five units at once: its row, its column, its box and both diagonals. Its candidate list collapses faster than any other cell's, and the digit placed there is eliminated from sixteen other cells along the two diagonals.

What do the V and X marks mean in XV sudoku?

They are Roman numerals marking the sum of two orthogonally adjacent cells. A V means the pair sums to 5, so the digits are 1 and 4 or 2 and 3. An X means the pair sums to 10, so the digits are 1 and 9, 2 and 8, 3 and 7, or 4 and 6. An unmarked edge means the pair sums to neither 5 nor 10.

Is XV sudoku the same as Sudoku X?

No. Sudoku X constrains the two long diagonals. XV constrains pairs of adjacent cells through sum marks. They are separate variants that are often confused because both use an X, and a puzzle can combine the two rule sets if the rules say so.

Can a 5 sit next to a V or an X mark?

No. Five has no partner that sums to 5, and its only partner for a sum of 10 would be another 5, which cannot be adjacent in the same row, column or box. So every edge touching a 5 must be unmarked, which means every cell adjacent to a mark can have 5 eliminated immediately.

Do classic solving techniques still work in Sudoku X?

Yes, all of them. Naked and hidden singles, pairs and triples, pointing, fish, wings and chains remain valid because the diagonals only remove candidates. Uniqueness techniques also survive, since the digit swap they rely on leaves every unit, including both diagonals, with the same contents.