Rules
The grid is shaded blocks and white cells, like a crossword. You write a digit from 1 to 9 in every white cell. Nothing is filled in at the start.
- A crossword. Shaded blocks and white cells. You write a digit from 1 to 9 in every white cell. Nothing is filled in at the start.
- Runs. A run is an unbroken line of white cells across or down, at least two cells long.
- The clue. The digits in a run add up to the clue, and no digit repeats inside that run. The same digit may show up again in another run.
- The diagonal. The number in the upper right is the run to the right. The number in the lower left is the run underneath. A block can carry one of those numbers, or both.
A run is an unbroken line of white cells across or down, at least two cells long. The digits add up to the clue, and no digit repeats inside that run. The same digit may show up again in another run. In the window above, each run matches its clue, and the two 1s do not share a run: one sits in the top across run, the other in the right down run.
A clue block has a diagonal. The number in the upper right is the run to the right. The number in the lower left is the run underneath. This block carries 5 across and 11 down. Some blocks carry only one of those numbers. A block with neither is only a wall: no diagonal, no numbers. More than one filling fits these clues, so the digits stay off the picture. It is here to show the block, not to play.
Notes are small pencil marks you keep yourself. They are not digits, and Check ignores them.
One-way sums
Some sums can be made in only one way. A short run with a few sets is the next look.
One way only Easy
Some sums can be made from only one set of digits. Do these before anything else. The order inside the run is not fixed until you read the crossing run. Where two such runs cross, the shared cell is a digit that is in both sets.
In two cells, 3 is 1 and 2, 4 is 1 and 3, 16 is 7 and 9, and 17 is 8 and 9. In three cells, 6 is 1, 2 and 3, 7 is 1, 2 and 4, 23 is 6, 8 and 9, and 24 is 7, 8 and 9.
| Cells | Sums with one set |
|---|---|
| 2 | 3 is 1+2. 4 is 1+3. 16 is 7+9. 17 is 8+9. |
| 3 | 6 is 1+2+3. 7 is 1+2+4. 23 is 6+8+9. 24 is 7+8+9. |
| 4 | 10 is 1+2+3+4. 11 is 1+2+3+5. 29 is 5+7+8+9. 30 is 6+7+8+9. |
| 5 | 15 is 1+2+3+4+5. 16 is 1+2+3+4+6. 34 is 4+6+7+8+9. 35 is 5+6+7+8+9. |
A run of six is allowed on 8×8 medium and hard. There the one-way sums are 21, 22, 38 and 39. A 10×10 grid stays at five cells.
3 is only 1 and 2. 4 is only 1 and 3. The only shared digit is 1, so the corner is 1. The rest of this tiny window follows at once. The picture shows just that 1.
Once 1 is written, the other cell of the 3 must be 2. That is still the unique pair. A two-cell run with one cell filled is this step, or the short lists in the next section.
16 is only 7 and 9. 17 is only 8 and 9. The only shared digit is 9.
A few sets Easy
Most short clues have more than one set. Write the sets down, cross them with the other run, and keep only the digits that appear in both.
A two-cell 5 is 1+4 or 2+3, so it never holds a 5 or anything bigger. A two-cell 15 is 6+9 or 7+8. A digit you have already written leaves both of its runs. That is part of every step, not a separate one. A length-2 or length-3 run with one empty cell is this step, or the one-way sums, not the subtraction in section 3.
5 is 1+4 or 2+3. 13 is 4+9, 5+8 or 6+7. The only shared digit is 4. Neither run is a one-way sum, so you need the lists. The rest of the window follows in the same breath. The picture shows just the 4.
Place a digit
Write a digit when only one cell of a run can hold it, and subtract when one cell of a run is left.
The only cell for that digit Easy
Sometimes you know a one-way run must contain a digit, and only one of its cells can still hold that digit. Write it there, even if that cell could also be something else. This only starts from a one-way sum. It is an easy step. Not every easy grid needs it.
The across 6 is only 1, 2 and 3. The down 3 on the right is only 1 and 2, so that cell cannot be 3. The middle down is 6, which is 1+5 or 2+4, so the middle cell cannot be 3 either. The 3 has one home. The cell could also have been 1 or 2. You write 3 because no other cell in the run can take it.
One cell left Easy
When one cell in a run is still empty, subtract the digits you have already written from the clue. That difference is the missing digit, if the run has not repeated a digit and the difference is from 1 to 9.
On a run of two or three cells you will usually have written that digit already, from a one-way sum or a short list. This section is the long run: four or more cells, one empty, and the clue is not a one-way sum. The digits already in the run are ones you wrote. Nothing was printed in those cells at the start.
Three digits of a four-cell 12 are already written, and 12 − 6 = 6. The crossing column still allowed 6, 7, 8 or 9, so the subtraction is what picks 6. The other cell in that column is the same kind of subtraction. This window, before those digits were written, needs the longer list in the next section. Do not treat the picture as an easy puzzle, and do not treat the written digits as printed for you.
Easy grids on this site finish with this section and the three before it. You do not need the later sections to clear an easy 8×8 or an easy 10×10. Some hard grids finish from these steps plus the longer list, used together, and then they never need the two steps after that.
Longer runs Medium
A run of four or more has too many sets for the short list, unless the sum is one of the one-way sums above. Write the sets that still add up, cross out any digit already used, and keep a digit only when both runs still allow it.
On this site that step is what medium adds. Every medium grid uses it. Easy grids never need a list longer than three. A run of six appears on 8×8 medium and hard, not on 10×10 and not on 8×8 easy.
12 in four cells is 1+2+3+6 or 1+2+4+5. 15 in two cells is 6+9 or 7+8. The only shared digit is 6. The short lists never read the four-cell run, so the cell stays open until this list. The down 15 alone still allows 6, 7, 8 or 9. The rest of the window follows. The picture shows just the 6.
A hard grid may need nothing past this list and the one-way sums, used together. The one-cell step from a one-way run is not part of the medium list, so a hard grid can stall on each idea alone and still finish when you use them in turn.
In every set Hard
Look at the sets that still fit a run. If one digit is in every one of those sets, and only one cell of the run can still hold it, write that digit. The cell may have other digits open. This is the hard step that does not need a trial. Not every hard grid needs it.
Do not say that every way to make 9 uses a 2. Three sets make 9: 1, 2 and 6; 1, 3 and 5; and 2, 3 and 4. The crossings throw out 1, 3 and 5. Both sets that remain use a 2.
The middle down is 4, so that cell is 1 or 3. It cannot be 2. The bottom across is 10 in three cells, so it cannot hold an 8 or a 9, and the right down cannot be 2+8 or 1+9. That cell is 3, 4 or 6. It cannot be 2. The set 1, 3 and 5 needs a 5, and none of the three cells can be 5. The sets left are 2, 3 and 4, and 1, 2 and 6. Both contain 2, and the only cell that can be 2 is the left one. Its other open digits are 1, 3 and 4.
Some hard grids are finished by the earlier sections together, because a medium grid does not place the digit from section 3, and an easy grid does not read a run of four or more.
What if Hard
Use this only when the earlier sections have stopped. Pick one open cell. Try one digit. If the lists then leave a run with no set that fits, that digit is wrong. Erase it and write the digit that remains. Then go back to the short lists.
Try one cell, once. Do not try a second cell inside the first try. Do not leave the failed digit on the board. Easy and medium never need this. Some hard grids never need it either. On those, Hint stays with the earlier sentences and never mentions a crossing trial.
The death is often quiet. Check does not turn a cell red until a digit repeats, a run is already over its clue, or a full run has the wrong total. A trial can die while every cell still looks normal.
The 2 is the digit under test. The grid may not turn red. Each line is what the easy lists do with that 2 in place.
- The down clue is 8, so the cell below the 2 is 6.
- The across 7 is only 1, 2 and 4. The 2 is used, so the other two cells are 1 and 4 in some order.
- The bottom run is 19 and already contains a 6. The other two digits sum to 13, and they cannot include another 6. The pairs left for 13 are 4+9 and 5+8, so the middle of the bottom run is 4, 5, 8 or 9.
- The middle down is 11, and its top is only 1 or 4. A 1 would need a 10 below it. A 4 would need a 7, and 7 is not among 4, 5, 8 and 9. None of the pairs for 11 fit. The 2 dies.
1 is the digit the trial keeps. It is drawn like any digit you decided to write, not as the new answer. The next move is an ordinary short run: the cell beside it is 2. Then go back to the short lists.
Easy and medium never need this trial. Some hard grids never need it, or the step before it: the one-way sums and the longer list, used together, already finish them. On those, Hint stays with the earlier sentences.
Common mistakes
A repeat, a run that is already over its clue, and a full run with the wrong total are the breaks Check can show. Two other wrong digits stay quiet until you list the sets.
The repeat is the mistake. The total is wrong as well, because the run is full: 1+1 is 2, not 3. Two 1s are fine when they do not share a run, as in the finished window at the top.
The run does not have to be full. 9 is already more than 3, and more than 4. Both runs through that cell are over.
The across run shows 3 with a blank partner. That is not "over", and it is not a finished total. Check stays quiet. The sum 3 can only be 1 and 2, so erase the 3.
8+6 is 14. The clue is 15. The cells do not repeat, and neither digit alone exceeds 15. The run is full, and the total is wrong.
9 is in one of the pairs for 15. It is in neither set of the four-cell 12. The cell may not turn red. Listing the 12 is what removes the 9. Writing 7 fails the same quiet way.
Every clue is 3, so the pair is 1 and 2 in either order. A 1 in the corner is legal, and it is not forced. The other filling swaps the two diagonals. Do not treat the 1 as a broken cell.
What to play next
Easy adds nothing past the one-way sums, the short lists, the only cell for a digit, and one cell left. Medium adds the longer list. Hard adds a digit that sits in every remaining set, or one cell to test.
- EasyOne-way sumsStart with 3, 4, 16 and 17 in two cells, and 6, 7, 23 and 24 in three. Cross a short run that has only a few sets. When a one-way run needs a digit that only one cell can hold, write it. When one cell is left in a run, subtract.Play Easy →
- MediumLonger runsThe short lists have stopped. Write the sets for a run of four or more, and keep the digit that both runs still allow. This is the size where a run of six can appear.Play Medium →
- HardOne digit, then the listsLook for a digit that is in every set still alive, with only one cell that can hold it. If the grid still stalls, try one digit in one cell, erase it when a run has no set left, and go back to the lists.Play Hard →
8×8 and 10×10 each come in easy, medium and hard. A 10×10 grid never has a run longer than five. Try10×10 easyand10×10 hard. The daily puzzle is a 10×10 easy grid:today’s Kakuroopens the same size and setting as10×10 easy.
Print a Kakuro pack. An 8×8 sheet holds two puzzles. Each 10×10 has a page to itself. Large print is one grid on every size.
- GuideSudokuRows, columns and boxes, from singles to triples.
- GuideNonogramsOverlap the long clues, then read the line that crosses.
- GuideHashiCount bridges, never cross, connect every island.
- GuideBinairoSpot pairs, fill the gaps, balance every line.
- GuideLight UpLight every cell. Start beside the 0s and 4s.
- GuideSlitherlinkDraw one loop. Start from the 0s and the 3s.
- GuideStar BattleOne or two stars in every row, column and region.