Stuck on a Sudoku? How to Find the Next Move
Hiroki Obara · Developer
A sudoku puzzle is normally built to have exactly one solution. If there is only one solution, then at every point in the puzzle there should be some cell you can fill for a clear reason. When you feel stuck, it is usually not that there is no next move. It is that you have not looked in the right place yet.
This post is about what to check, and in what order, when you stop making progress. I covered the techniques themselves in How to Solve Sudoku: The First Techniques Beginners Should Learn, so here I focus on the order of checking. The words are the same as in that post: a row runs across, a column runs down, a box is one of the 3 × 3 squares with a thick border, and the digits that could still go in an empty cell are its candidates.
The order to check
This is the order I recommend, from light checks to heavier ones.
1. Go digit by digit through the boxes. For each digit from 1 to 9, look at every box that does not have it yet and see whether only one cell is left for it. As you get used to sudoku, it is easy to look only at the common digits and skip the rare ones. When you are stuck, go through all nine, in order.
2. Go digit by digit through the rows and columns. Do the same thing for rows and columns. This is where things are missed most often, because once your eyes are trained on boxes, you stop looking at whole rows and columns. Start with the rows and columns that have the fewest empty cells.
3. In crowded areas, find the digit for a cell. Pick a box or row with only two or three empty cells, choose one cell, and collect every digit in its row, column and box. If eight different digits are there, the ninth goes in the cell.
4. Write pencil marks, or rewrite them. If none of that works, it is time for pencil marks. If you have not written any, start with the boxes that have the fewest empty cells. If you already have them, check that they are right (see the next section).
5. Look for techniques that use pencil marks. Are a digit’s candidates in a box lined up in one row? Do two cells share the same pair of candidates? These are techniques 4 and 5 from the earlier post.
An example: only one place in the row
Here is a position partway through a puzzle I made myself. I checked with a solver that the puzzle has exactly one solution. A dot is an empty cell.
9 ・ 2 | 5 8 ・ | ・ 7 ・
・ ・ ・ | 1 ・ 2 | 5 ・ 9
・ 6 5 | 9 ・ 7 | ・ ・ 2
5 ・ 6 | 7 9 1 | ・ 2 4
2 ・ 9 | 8 5 ・ | 7 ・ ・
・ ・ ・ | 2 ・ 3 | ・ 9 5
・ 5 ・ | 3 2 8 | 9 6 7
6 2 3 | 4 7 9 | 1 5 8
・ 9 ・ | 6 1 5 | 2 4 3In this position, no cell can be settled by looking for a digit’s place in a box, or by looking for the digit that fits a cell. The only move is one you find by looking along a row.
Take the 6 in row 2. Row 2 has no 6 yet. The left box already has a 6 in row 3, so the three left cells of row 2 are out. In the right box, the only empty cell in row 2 is in column 8, but column 8 has a 6 in row 7. That leaves row 2, column 5.
If you look for the 6 in the top middle box instead, two cells remain, row 1 column 6 and row 2 column 5, and you cannot decide. The same 6 can be found or not found depending on whether you look at the box or the row. That is why step 2 should not be skipped.
Finding mistakes in pencil marks
If you are using pencil marks and get stuck, suspect the marks themselves. Mistakes go in two directions.
Erasing too much. You removed a digit that actually belongs in a cell. This is the worst kind, because the right answer has disappeared from your marks, and no amount of searching will find it. To check, look at each row, column and box, and make sure every digit from 1 to 9 is either placed or appears in at least one mark. If a row has no 7 and none of its marks include 7, you erased a 7 somewhere you should not have.
Forgetting to erase. You placed a digit but did not remove it from the marks in the same row, column and box. This only makes the candidates look more numerous, but that is enough to stop you. Check the marks around the digits you placed most recently.
If you still cannot find the mistake, erasing all the marks and writing them again is an option. It takes time, but it is safer than reasoning from marks you cannot trust.
Stepping away
If you have gone through the whole order and still see nothing, I think it is fine to step away for a while. When you are stuck, you tend to look at the same places in the same order. Coming back later, you start from a different row or column, and sometimes the thing you missed is right there.
One more thing: I do not recommend guessing a digit. As I wrote in the earlier post, a lucky guess is not solving, and a wrong one breaks everything that follows.
When you are stuck in Sudoku365
Sudoku365 is a sudoku app I make for iPhone, iPad and Android. Here is exactly how its tools behave when you are stuck.
Highlighting. When you select a cell, its row, column and box are highlighted, along with every cell holding the same digit. This helps with steps 1 and 2, when you pick one digit and scan the grid. With Hide placed number buttons on in settings (the default), the button for a digit disappears once all nine are placed, so you can see which digits are still worth looking for.
UNDO. Each press goes back one move. Writing a pencil mark counts as a move. Marks that were cleared automatically when you placed a digit come back too. Your mistake count, however, does not go back.
Mistakes. With Show incorrect numbers on in settings (the default), a wrong digit shows up in red right away and counts as a mistake. Three mistakes end the game. Even with Remove numbers from notes switched on, placing a wrong digit does not clear marks in related cells, so a mistake cannot wipe out correct pencil marks.
HINTS. Pressing the hint button fills the selected cell with the correct digit. A digit placed by a hint cannot be changed and cannot be undone. You start with 10 hints and get one more for each puzzle you clear, or for watching an ad. I think hints are best used after you have gone through the whole checking order and still found nothing. When a hint fills a cell, ask yourself why that digit goes there. It will help the next time you get stuck in the same kind of spot.
If you stop partway through, CONTINUE on the home screen takes you back to the same grid, so stepping away costs nothing in the app.
The app is free with ads, and a one-time Remove Ads purchase turns them off. For pencil marks, see How to Write Sudoku Pencil Marks. For the rules, see Sudoku Rules: What First-Timers Need to Know.
The places to check when you are stuck are mostly the same every time: digits 1 to 9, then boxes, rows and columns. Go through them in order from where you stopped, and the next move is usually waiting somewhere.
