Basic Sudoku Solving
About
A block refers to a 3x3 block.
A cell or square refers to a 1x1 square.
A unit is a general term for either a row, column, or 3x3 block.
A subrow is the part of a row contained within a particular 3x3 block.
A subcolumn is the part of a column contained within a particular 3x3 block.
For each solving strategy there is a puzzle that utilizes that strategy. There might be a few other possible moves, but your goal is to find the play that uses the given strategy. The square will only be solvable with that strategy (amongst these basic strategies).
Hints are provided to narrow your search space. The spoilers give the solution and should provide enough information to help you know why that move was possible.
Note that not every puzzle is completely solvable so you should move on once you have determined the solution.
Finish Unit
The simplest strategy is to find a row, column, or 3x3 block that is missing exactly one cell. Find the row, column, and 3x3 block in the puzzle below that are each missing 1 number.
7
1
6
5
9
9
8
5
1
2
6
2
4
1
5
9
8
3
7
9
8
2
1
3
3
9
2
8
9
3
5
1
3
5
1
2
9
1
9
5
3
8
Spoiler
Row 3, column 3, and top-left block.One Spot in Unit
Another simple strategy is to find a number that can only fit in one location within a row, column, or 3x3 block. If a column (or row/block) is mostly full then check whether all but one of the remaining locations are impossible for a particular number. Some locations might not be possible because that number is already located in the same row or 3x3 block.
In the puzzle below, there is a column where a particular number can only fit into one location.
5
9
8
1
2
4
6
7
4
6
5
8
9
1
4
1
6
9
8
2
5
7
3
4
6
7
1
5
8
6
5
7
8
3
9
4
1
2
8
1
5
2
4
6
7
8
9
5
1
2
4
7
2
4
1
9
8
6
3
5
1
5
4
7
8
9
Hint
Look at column 4.Hint
The 2 needs to go somewhere in that column.Spoiler
The 2 can be placed in the last row of the 4th column. The number 2 is already in the first row and the 7th row. Every other cell is occupied.In the puzzle below, there is a row where a number can only fit into one location.
1
7
3
2
9
7
1
6
9
5
8
4
8
1
3
5
9
5
2
8
4
3
7
8
8
6
3
2
Hint
Look at row 7.Hint
The 2 needs to go somewhere in that row.Spoiler
The 2 can be placed in the 7th column of the 7th row. The number 2 is already in the leftmost block to prevent placing it in the first 3 columns. There is already a 2 in the 4th and 8th columns. Every other cell is occupied.In the puzzle below, there is a 3x3 block where a number can only fit into one location.
5
9
4
2
6
1
6
2
3
9
8
1
7
3
6
1
9
1
6
4
2
6
4
5
4
3
Hint
Look at the top-right block.Hint
The 4 needs to go somewhere in that block.Spoiler
The 4 can be placed in the 9th column of the 1st row. The number 4 is already in the 8th column and the 2nd row. Every other cell is occupied.One Digit in Spot
The final simple strategy is to find a cell where only one possible number can fit. If a column, row, or 3x3 block is mostly full then there are only a few possible values for each remaining cell. Check each cell to see if all but one of the possible values is actually impossible.
In the puzzle below, there is a cell somewhere that only has one possible value.
6
5
5
6
8
6
7
1
2
8
4
3
6
1
4
9
7
3
5
6
2
3
Hint
Look at column 7.Hint
Look at row 3 of column 7.Spoiler
The 9 can be placed in the 7th column of the 3rd row. No other number can be placed in that spot. 1, 2, 6, and 7 are in the row already. 3, 4, and 5 are in the column already. Finally, 8 is in the same block.Unique Subrow in Block
Once we have eliminated the most obvious solutions, we need to take things one step further. We will try to eliminate some possibilities even if we don't know exactly where a number is going to go.
Within each 3x3 block, a number must go in one of the 3 rows. If we can eliminate all but one possible row then we can use that information just like if we knew the exact location.
In the puzzle below we know the number 1 cannot appear in the top left cell because there is already a 1 in that row. We also know that the number 1 cannot appear in the bottom left cell because there has to be a 1 in that row in the bottom right block.
1
7
8
9
4
5
6
Look for a block where for one particular number only one subrow is possible. Then eliminate that number from the rest of the row outside of the block. This information might create a column/3x3 block where only one spot is possible for that number.
4
2
7
2
6
8
4
2
7
8
4
1
3
2
7
2
8
2
1
3
8
9
6
4
8
2
6
5
2
5
6
8
2
5
4
1
Hint
Look at column 6.Hint
The number 7 must go somewhere in that column.Spoiler
The 7 can be placed in the 6th column of the 6th row. The 1st, 3rd, and 5th rows of the column are impossible. We can eliminate the 8th row as a possibility because the 8th row of the bottom right block must have a 7. The 7th and 9th rows are impossible for that block.After identifying such a subrow, we might also create a location within that row where only one digit is possible.
9
6
1
4
4
1
6
3
8
7
9
6
4
1
8
3
1
7
4
5
6
5
4
1
6
3
7
2
6
5
4
3
1
6
8
5
3
4
5
4
1
6
8
8
4
6
Hint
Look at row 9.Hint
Look at column 4 of row 9.Spoiler
The 3 can be placed in the 9th row of the 4th column. That cell can only contain a 2 or a 3 because the column is filled with every other number. The bottom right block needs a 2 in the bottom row, though, so a 2 cannot be elsewhere in the row. A 3 is the only number left.Unique Subcolumn in Block
This technique is the same as the unique subrow but we are looking at columns instead.
In the puzzle below, the number 1 cannot appear in the top left cell because there is a 1 in that column. The 1 also cannot appear in the top right cell because there must be a 1 in the rightmost column of the bottom block.
7
4
8
5
1
9
6
Look for a block where for a particular number only one subcolumn is possible. Then eliminate that number from the rest of the column outside of the block. This information might create a row/3x3 block where only one spot is possible for that number.
7
8
4
9
6
9
8
4
3
3
4
8
2
9
1
3
9
6
4
7
3
8
6
9
5
7
8
3
2
6
1
5
4
6
3
9
Hint
Look at row 5.Hint
The 1 must go somewhere in that row.Spoiler
The 1 can be placed in the 9th column of the 5th row. A 1 can only be placed in the 1st or 9th columns because there is a 1 in the middle block. The top left block needs a 1 in the 1st column so a 1 cannot be in that column in the 5th row. Only the last column remains to place the 1 in the row.After identifying such a subcolumn, we might also create a location within that column where only one digit is possible.
9
4
3
5
8
3
7
2
9
1
6
5
3
6
3
1
5
7
8
7
2
9
8
3
6
8
4
5
6
3
3
6
8
9
8
6
5
2
3
3
6
Hint
Look at column 5.Hint
Look at row 2 of column 5.Spoiler
The 6 can be placed in the 5th column of the 2nd row. This cell can only contain a 4 or 6 because every other number is in the row already. It cannot be a 4, though, because there needs to be a 4 in the 5th column in the middle middle block.Unique Subrow in Row
This technique is similar to the unique subrow in a block. If we know that a number must be in a particular subrow then we can eliminate it from the rest of the block.
In the puzzle below the number 1 cannot be in the top left cell because there is already a 1 in that block. There also cannot be a 1 in the bottom left cell because there must be a 1 in the middle row of that block.
1
4
5
6
7
8
9
Look for a row where a number must be placed in 1 of the blocks. Then eliminate that possibility in the rest of block. Then look for a row or column where that number can only fit in one location.
1
6
7
8
4
6
1
9
1
7
6
6
2
8
3
5
1
5
6
4
3
8
3
2
7
8
5
1
6
9
4
8
2
6
2
8
Hint
Look at column 5.Hint
The 9 must go somewhere in that column.Spoiler
The 9 can be placed in the 5th column of the 8th row. A 9 must be placed in that column and can only be placed in the 5th or 8th rows as the rest of the column is occupied. A 9 needs to go somewhere in the 4th row, but the only options are in the middle block. Therefore the 9 in that block cannot be in the 5th row. The 9 must then go to the 8th row.Look for a row where a number must be placed in 1 of the blocks. Then eliminate that possibility in the rest of block. Then one of the cells within that block will only have one possibility left.
5
8
1
6
1
7
6
4
3
6
4
5
9
8
1
2
3
5
1
1
7
9
4
6
2
8
4
2
1
6
4
2
1
3
6
9
1
6
2
9
4
5
1
Hint
Look at row 3.Hint
Look at column 8 of row 3.Spoiler
The 7 can be placed in the 8th column of the 3rd row. That cell must be a 2, 3, or 7 because the rest of the digits are already in the row. There is a 3 in that block so the cell must be a 2 or 7. There must be a 2 in the 2nd row and it cannot be in the first two blocks. Since the 2 is in the 2nd row in the top right block it cannot be in the 3rd row so that cell must be a 7.Unique Subcolumn in Column
This technique is the same as the unique subrow in row but for columns.
In the puzzle below, the 1 cannot go in the top left cell because there is already a 1 in that block. There also cannot be a 1 in the top right cell because there must be a 1 somewhere in the middle column of that block.
1
4
7
5
8
6
9
Look for a column where a number must be placed in 1 of the blocks. Then eliminate that possibility in the rest of block. Then look for a row or column where that number can only fit in one location.
1
5
4
7
9
8
2
8
9
7
1
6
5
4
4
5
2
8
6
9
7
3
1
1
5
3
8
7
8
1
5
9
8
5
3
4
1
7
8
9
1
4
1
2
4
8
9
4
1
8
2
Hint
Look at row 6.Hint
The 2 must go somewhere in that row.Spoiler
The 2 can be placed in the 1st column of the 6th row. There are four locations for the 2 in that row. The 9th column is not possible because there is already a 2 in that column. Then look at the 4th column. There needs to be a 2, but it cannot go in the bottom block. Therefore the 2 in the middle middle block cannot be placed in the 5th or 6th columns. The only location left in the 6th row is the 1st column.Look for a column where a number must be placed in 1 of the blocks. Then eliminate that possibility in the rest of block. Then one of the cells within that block will only have one possibility left.
4
9
8
9
2
8
6
8
1
3
5
7
9
4
2
4
5
9
6
2
1
8
7
3
3
8
5
1
8
7
3
5
4
9
8
5
7
3
9
4
4
1
5
4
2
6
Hint
Look at row 7.Hint
Look at column 2 of row 7.Spoiler
The 1 can be placed in the 2nd column of the 7th row. This cell can only be a 1, 2, or 6 because the rest are already in the row. Examine the 3rd column. The 2nd row value cannot be a 2 or 6 because the 2 is in the row already and the 6 is in the same block. Therefore this column needs a 2 and a 6 in the bottom block. These numbers cannot then be in the 2nd column so this cell can only be a 1.Naked Pair
A naked pair is a pair of cells in the same unit that each contain one of the exact same pair of numbers. Each of the numbers must therefore go in one of these two cells so it cannot go anywhere else in that unit.
In the puzzle below, the 5th and 9th columns of the top row must contain the digits 5 and 9 in either order. Therefore the number 5 cannot appear anywhere else in the top row. So where must the 5 go in the top-left block?
6
7
8
4
6
7
8
9
4
4
3
3
2
2
1
1
In the puzzle below find a naked pair and then eliminate those possibilities from the rest of the unit. After doing this, there will be a unit with a number that only has one location.
8
7
1
3
3
1
8
7
4
1
5
3
7
8
6
1
8
9
6
3
1
8
6
8
5
2
1
7
8
5
3
1
2
6
9
2
9
8
5
3
7
1
1
5
8
4
Hint
Look at row 3.Hint
The 4 must go somewhere in that row.Hint
4,7 is the pair to look for.Spoiler
The 4 can be placed in the 2nd column of the 3rd row. There is a 4 in the 8th column so a 4 must go in either the 2nd or 5th column. In the fifth column, the 4th and 7th rows are both limited to 4 or 7. Since the 4 cannot be in the fifth column it must be in the 2nd.In the puzzle below find a naked pair and then eliminate those possibilities from the rest of the unit. After doing this, there will be a cell that only has one remaining possibility.
2
3
7
5
6
3
5
9
1
7
2
1
7
5
8
3
9
2
8
5
5
4
7
9
3
8
6
4
7
5
7
1
4
3
8
9
5
3
8
7
5
9
4
5
9
1
4
3
Hint
Look at row 4.Hint
Look at column 8 of row 4.Hint
1,6 is the pair to look for.Spoiler
A 7 can be placed in the 8th column of the 4th row. The cell can a 1 or 7. In that row the 3rd column and 5th column are both limited to 1 or 6 so a 1 is going to appear somewhere else in the row. The only number left is the 7.Hidden Pair
A hidden pair is a pair of digits can only appear in the same two places within a unit. If other numbers are possible for these cells then we can eliminate those options.
In the puzzle below, the 5 and 9 can only appear in the 5th and 9th columns of the top row. The top right cell therefore cannot be a 4. Once this location is eliminated then the 4 only has one place left in the top-right block.
6
7
8
4
5
9
1
2
9
5
In the puzzle below find a hidden pair and then eliminate the other possibilities for those cells. After doing this, there will be a unit with a number that only has one location.
4
2
5
7
3
3
8
7
5
2
4
6
3
2
8
5
6
3
8
5
2
7
2
3
6
4
9
4
2
6
3
2
9
3
8
2
1
6
4
2
Hint
Look at column 8.Hint
The 7 must go somewhere.Hint
4,5 is the pair to look for.Spoiler
The 7 can be placed in the 4th row of the 8th column. The numbers 4 and 5 can only go in rows 7 and 8 in the 8th column. Since the 7 cannot go in these locations the only possibility is row 4.In the puzzle below find two hidden pairs and then eliminate the other possibilities for those cells. After doing this, there will be a cell that only has one remaining possibility.
1
7
4
3
6
6
5
2
1
8
9
2
3
9
1
4
5
6
8
7
1
9
8
4
7
1
7
1
4
6
3
1
4
9
5
7
6
2
8
9
1
6
4
5
6
7
3
8
2
9
1
4
