1 of 3
What goes next to the 3?
Level 2 · Sums and pairs
How to solve XV sudoku
An X between two cells means they add to 10. A V means they add to 5.
XV sudoku puts small letters on the borders between cells. Two cells joined by an X add up to 10, and two cells joined by a V add up to 5. Normal sudoku rules still apply.
Each mark allows only a few digit pairs, so one placed digit fixes its partner. Many XV puzzles also say that all Xs and Vs are given, which turns every blank border into a clue as well.
How setters write it
The same rule, worded the ways you’ll see it on puzzles.
- Cells separated by an X sum to 10; cells separated by a V sum to 5.
- XV sudoku.
- All possible Xs and Vs are given.
Worth memorizing
- X pairs: 1-9, 2-8, 3-7, 4-6. 5 can never touch an X.
- V pairs: 1-4 and 2-3. Only 1 to 4 fit on a V.
- When a puzzle says all Xs and Vs are given, two neighbors with no mark can’t add to 5 or 10. That rules out a lot.
Solving tips
Know the X pairs
An X can only be 1 and 9, 2 and 8, 3 and 7, or 4 and 6. Place a 3 next to an X and the cell on the other side is 7.
Keep 5 away from X
The only way to make 10 with a 5 is 5 + 5, and neighbors in a row or column can't match. Any cell with an X on one of its edges is not 5.
V cells are 1 to 4
A V can only be 1 and 4 or 2 and 3. Both cells on a V are 4 or lower, and a V next to a placed 1 must hold 4.
Use the blank borders
When the rules say all Xs and Vs are given, two neighbors with no mark can't add to 5 or 10. A 3 with no mark beside it can't have a 7 or a 2 as that neighbor.
A worked example: an X and a V in one row
We name cells by row and column, so row 1, column 1 is r1c1. There is a V between r1c1 and r1c2 and an X between r1c2 and r1c3. The rules say all Xs and Vs are given, and no other marks sit between r1c3 and r1c6. r1c4 is 6 and r1c5 is 2.
- A V is 1 + 4 or 2 + 3. Row 1 already has a 2, so the V can't use 2 + 3. r1c1 and r1c2 are 1 and 4 in some order.
- r1c2 is 1 or 4. The X means r1c3 is 10 minus that, so 9 or 6. Row 1 already has a 6 at r1c4, so r1c3 is 9 and r1c2 is 1.
- That puts the 4 in r1c1.
- Now use a blank border. r1c5 is 2, and there's no mark between it and r1c6. So r1c6 can't be 3 (that would need a V) or 8 (that would need an X). The row already holds 1, 2, 4, 6 and 9, so r1c6 is 5 or 7.
- Check the other blank borders. 9 + 6 = 15 and 6 + 2 = 8. Neither makes 5 or 10, so the placed digits agree with the rule.
Practice
2 of 3
Tap every digit that could go on this V.
Select all that apply
3 of 3
Tap every digit that could go on this X.
Select all that apply
Reference table
| Digit | X partner (sum 10) | V partner (sum 5) |
|---|---|---|
| 1 | 9 | 4 |
| 2 | 8 | 3 |
| 3 | 7 | 2 |
| 4 | 6 | 1 |
| 5 | none | none |
| 6 | 4 | none |
| 7 | 3 | none |
| 8 | 2 | none |
| 9 | 1 | none |
Going deeper
Getting the most from blank borders
When all Xs and Vs are given, every blank border is a clue. A cell has up to four neighbors, and each unmarked neighbor removes up to two digits from it. A 1 with no marks around it rules out 4 and 9 from all four neighbors.
The 5 is special here. It has no X partner and no V partner, so a 5 can sit anywhere without breaking the negative rule. A cell with many blank borders and many digits ruled out can end up as 5.
Chains of marks
Marks link across cells. A cell between an X and a V must be 1 to 4 for the V, and its X partner must then be 6 to 9. So a V next to an X splits the three cells into low, low, high.
Two marks of the same letter on one cell force its two partners to match. If a cell has a V on two sides, both partners equal 5 minus that cell. In a straight line, those partners share a row or column, which is impossible. In an L shape, it works only when the two partners sit in different boxes, and then they hold the same digit.
The same is true for two Xs on one cell. Three cells in a row can never be joined by X, X or by V, V.
Which digits rule out the most
Each digit from 1 to 4 has both an X partner and a V partner. Each digit from 6 to 9 has only an X partner. So under the negative rule, a blank border next to a 1, 2, 3 or 4 removes two digits from the neighbor, and a blank border next to a 6, 7, 8 or 9 removes one.
That makes small digits worth placing early. A 2 surrounded by blank borders removes 3 and 8 from up to four neighbors, which is eight removals from a single digit. Scan around each new low digit before you move on.
Common mistakes
Ignoring the blank borders
In a puzzle where all Xs and Vs are given, most of the logic comes from the borders with no mark. Check every unmarked neighbor of a new digit and cross off its X and V partners.
Using the negative rule when it isn't stated
Some XV puzzles don't say all marks are given. Then a blank border tells you nothing. Read the rules before you use it.
Mixing up XV with X sudoku
X sudoku usually means both long diagonals hold 1 to 9. That rule has nothing to do with X marks on cell borders. Some puzzles use both, so read which one applies.
In mixed-rule puzzles
XV marks often appear with kropki dots, since both sit on cell borders. When a puzzle uses both and says all marks are given, one blank border can rule out a sum of 5, a sum of 10, a difference of 1 and a ratio of 2 all at once. With a nonconsecutive rule, a V has to be 1 and 4, because 2 and 3 are consecutive. Inside a killer cage, an X or V fixes part of the total. An X in a three-cell cage of 17 means the third cell is 7.
What the video shows
- Cells joined by an X sum to 10. Cells joined by a V sum to 5.
- 3 + 7 = 10 for the X.
- 2 + 5 = 7, so it can’t sit on a V.
- 1 + 4 = 5 for the V.
Questions solvers ask
What does X mean in XV sudoku?
The two cells on either side of the X add up to 10. The possible pairs are 1 and 9, 2 and 8, 3 and 7, or 4 and 6.
What does V mean in sudoku?
A V between two cells means they add up to 5. Only 1 and 4, or 2 and 3, can sit on a V.
Can 5 go next to an X in sudoku?
No. A 5 would need another 5 beside it to make 10, and digits can't repeat in a row or column.
What does "all Xs and Vs are given" mean?
Every pair of neighbors that adds to 10 has an X, and every pair that adds to 5 has a V. Neighbors with no mark can't add to 5 or 10.
What is the difference between X sudoku and XV sudoku?
X sudoku, also called diagonal sudoku, asks for 1 to 9 on both long diagonals. XV sudoku puts X and V marks on cell borders for sums of 10 and 5. The rules are separate, though a puzzle can use both.
Can two Vs touch the same cell?
Yes, when the two partner cells can hold the same digit. Both partners equal 5 minus the shared cell, so they can't share a row, column or box.
What if an XV sudoku doesn't say all marks are given?
Then a border with no mark tells you nothing. Only the Xs and Vs that are drawn count as clues.
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