2. Write in the numbers (clues) that are given to you on the lines at the bottom of each cell.
3. Visually scan the puzzle. See if you can solve any cells using the strategy described in the scanning example. Click scanning example.
4. At the top of each empty cell, write all of the possible numbers (pencil marks) that could be in the cell. Try to use a different color ink. Red is a good choice.
5. Using the process of elimination:
Do not write any numbers that are already in the same column as the empty cell.
Do not write any numbers that are already in the same row as the empty cell. Box: a box is a group of 3x3 cells.
Do not write any numbers that are already in the same box as the empty cell. 6. Repeat step 5, in each empty cell of the box. 7. Go to the next box, and repeat step 5. 8. As you go from empty cell to empty cell, you will eventually see that only one number can fill the empty cell.
When this happens, write that number at the bottom of the cell and cross out the possible numbers that you have written at the top of that cell. When you write a number at the bottom of a cell, that cell is no longer an empty cell. Then look to see if you can eliminate
any of the possible numbers that you have already written at the top of each empty cell. Cross out those numbers. 9. When you have eliminated all but one of the possible numbers in a cell write it at the bottom of the cell. Continue in this manner until the puzzle is solved.
Strategy
Example # 1
In the example below, "2" has been placed in the formerly empty cell (row 1 column 2). The possible number "2" in the empty cell (row 1 column 1) has been eliminated and crossed out (2).
1 2 3
_
2
_
_
_
_
_
_
_
Example # 2
In the example below, the pair "5" and "7" are the possible numbers for cells (row 3 column 1) and (row 3 column 2).
Since only "5" or "7" can be used to fill these two cell, you can eliminate "7" in cell (row 1 column 3).
"9" can now fill that empty cell (row 1 column 3)
Step 1
_
_
7 9
_
_
_
_
5 7
_
5 7
_
_
Step 2
_
_
7 9
9
_
_
_
5 7
_
5 7
_
_
Example # 3
In the example below, "1", "2", "3" and "4" are the possible numbers for cell (row 2 column 1).
Since "3" is not a possible number in any of the other cells in the box, only "3" can be used to fill cell (row 2 column 1).
Step 1
1 2
_
1 2 5 7
_
1 2 5 7
_
1 2 3 4
_
6
1 2 4 5 7
_
8
2 4 7
_
9
Step 2
1 2
_
1 2 5 7
_
1 2 5 7
_
12 3 4
3
6
1 2 4 5 7
_
8
2 4 7
_
9
Example # 4
In the example below, there is only one cell that has "8" as a possible number (row 1 column 2).
Step 1.
6
1 2 4 7 8 9
_
1 2 4 7
_
1 3 5
_
1 2 9
_
1 2 5
_
3 4 5
_
2 4 7
_
2 4 5 7
_
5
2 3 4 7 9
_
4 7 9
_
8
2 3 6 9
_
6 9
_
1
2 3 4 6 7
_
4 6 7
_
1 2 3
_
1 2 3
_
1 2 3
_
1 2 3 5
_
4
7
9
2 3 5 6
_
8
Step 2.
In Box 1, there is only one cell that has "9" as a possible number (row 2 column 2).
6
1247 8 9
8
1 2 4 7
_
1 3 5
_
1 2 9
_
1 2 5
_
3 4 5
_
2 4 7
_
2 4 5 7
_
5
2 3 4 7 9
_
4 7 9
_
8
2 3 6 9
_
6 9
_
1
2 3 4 6 7
_
4 6 7
_
1 2 3
_
1 2 3
_
1 2 3
_
1 2 3 5
_
4
7
9
2 3 5 6
_
8
Step 3.
In Box 3, "1", "2" and "3" are the possible numbers in (row 1 column 7), (row 1 column 8) and (row 1 column 9).
Since "1", "2" and "3" must be in row one, you can eliminate "1", "2" and "3" in (row 2 column 7).
6
1247 8 9
8
1 2 4 7
_
1 3 5
_
12 9
9
1 2 5
_
3 4 5
_
2 4 7
_
2 4 5 7
_
5
2 3 4 7 9
_
4 7 9
_
8
2 3 6 9
_
6 9
_
1
2 3 4 6 7
_
4 6 7
_
1 2 3
_
1 2 3
_
1 2 3
_
123 5
5
4
7
9
2 3 5 6
_
8
Step 4.
You can eliminate "1", "2" and/or "3" in (row 1 column 3), (row 1 column 5) and (row 3 column 8)
6
1247 8 9
8
12 4 7
_
1 3 5
_
12 9
9
1 2 5
_
3 4 5
_
2 4 7
_
2 4 5 7
_
5
23 4 7 9
_
4 7 9
_
8
2 3 6 9
_
6 9
_
1
2 3 4 6 7
_
4 6 7
_
1 2 3
_
1 2 3
_
1 2 3
_
123 5
5
4
7
9
23 5 6
_
8
Example # 5
In the example below, the only place for a "9" in Box 3 is row 2. A "9" must be in one of the cells on row 2 of Box 3 (row 2 column 7), (row 2 column 8) or (row 2 column 9).
You can eliminate "9" in (row 2 column 1), (row 2 column 3) and (row 2 column 6)
Step 1.
2 4 6 9
_
2 3 6
_
2 3 4 6 9
_
2 4 5 6 9
_
1
2 3 46 9
_
7
2 5 6
5
8
2 4
_
2 4 6
_
1
4 7
7
34 5 69
5
6 9
_
2 9
_
2 45 6
_
3
246 8 9
8
7
5
2 4 6 9
_
3 4 9
_
2 3 4 9
_
2 4 6
_
4
4
1
Step 2.
The only place for a "3" in Box 3 is row 2.
A "3" must be in one of the cells on row 2 of Box 3 (row 2 column 8) or (row 2 column 9).
You can eliminate "3" in (row 2 column 3)
2 4 6 9
_
2 3 6
_
2 3 4 6 9
_
2 4 5 6 9
_
1
2 3 46 9
_
7
2 5 6
5
8
2 4
_
2 4 6
_
1
4 7
7
34 5 69
5
6 9
_
2 9
_
2 45 6
_
3
246 8 9
8
7
5
2 4 6 9
_
3 4 9
_
2 3 4 9
_
2 4 6
_
4
4
1
Step 3.
"2" can now fill the cell (row 2 column 3). You can eliminate the "2" in the other cells of Box 1 and on row 2 of Box 3.
"6" can now fill the cell (row 2 column 6).
2 4 6 9
_
2 3 6
_
2 3 4 6 9
_
2 4 5 6 9
_
1
2 346 9
2
7
2 5 6
5
8
2 4
_
2 4 6
_
1
4 7
7
34 5 69
5
6 9
6
2 9
_
2 45 6
_
3
246 8 9
8
7
5
24 6 9
_
3 4 9
_
2 3 4 9
_
2 4 6
_
4
4
1
Example # 6
In order to view this example on the screen, this vertical sudoku has been converted into a horizontal
example.
8
1 5 6 9
_
2 345 9
2
3 5 6 7 9
_
1 567 9
1
3 5 7 9
_
3 4 5 6 9
_
5 6 9
_
3 4 5 9
_
3 4 5 7
_
57 8
8
6
1
78 9
9
3 78 9
3
4 5 7
_
2
4 5 7 8
_
5 6 7 9
_
3
1
2 5 6 7 9
_
4
257 8 9
8
2 56 79
2
5 6 7 9
_
2 5 7 9
_
Step 1:
The pair "5" and "6" must be on row 2 of Box 1 cells (row 2 column 1) and/or (row 2 column 3).
You can eliminate the "5" and "6" in Box 1 (rows 1 and 3) and in cell (row 2 column 9).
2 345 9
2
3 5 7 9
_
3 4 5 9
_
1 5 6 9
_
1 567 9
1
5 6 9
_
8
3 5 6 7 9
_
3 4 5 6 9
_
6
3 78 9
3
4 5 7 8
_
57 8
8
78 9
9
2
3 4 5 7
_
1
4 5 7
_
1
257 8 9
8
2 5 7 9
_
3
4
5 6 7 9
_
5 6 7 9
_
2 5 6 7 9
_
2 56 79
2
Step 2:
Eliminating the "5" and "6" in cell (row 2 column 9), allows you to fill that cell with "7".
You can now eliminate "7" in cells (row 1 column 9), (row 3 column 7) and (row 3 column 8).
2 345 9
2
35 7 9
_
3 4 5 9
_
1 5 6 9
_
1 567 9
1
5 6 9
_
8
3 56 7 9
_
3 4 5 6 9
_
6
3 78 9
3
4 5 7 8
_
57 8
8
78 9
9
2
3 4 5 7
_
1
4 5 7
_
1
257 8 9
8
2 5 7 9
_
3
4
56 7 9
7
5 6 7 9
_
2 5 6 7 9
_
2 56 79
2
Example # 7
Step 1:
The pair "2" and "6" on row 2 (row 2 column 3) and (row 2 column 9) can be used to eliminate the "2" and "6" in (row 2 column 5), (row 2 column 6) and (row 2 column 8).
3 6 8
_
2 34 8
_
5
9
1
2 6 7
_
3 6 8
_
234 7 8
7
2 4 7
4
1 4 8 9
_
1 2 8 9
_
1 2 4 6 8 9
_
3
2 5 7 8
_
2 5 7 8
_
147 8 9
_
6
1 2 4 7 8 9
_
7
2 3 6 8 9
_
2 3 6 9
_
4
2 6 8
_
2 6
_
1 28
1
2 3 8 9
_
5
Step 2:
A "2" must be in column 9, cell (row 1 column 9) or cell (row 2 column 9).
(not shown is cell (row 7 column 9) which has "3" and "6" as possible numbers the other cells in column 9 rows 4-6 and rows 8-9 are filled)
The "2" in the other cells of Box 3 can be eliminated.
3 6 8
_
2 34 8
_
5
9
1
2 6 7
_
3 6 8
_
234 7 8
7
2 4 7
4
1 4 8 9
_
1 2 8 9
_
1 2 4 6 8 9
_
3
2 5 7 8
_
2 5 7 8
_
147 8 9
_
6
1 2 4 7 8 9
_
7
2 3 6 8 9
_
2 3 6 9
_
4
26 8
8
2 6
_
1 28
1
2 3 8 9
_
5
Step 3:
There is only one cell in row 3 that can have a "2" cell (row 3 column 6).
3 6 8
_
2 34 8
_
5
9
1
2 6 7
_
3 6 8
_
234 7 8
7
2 4 7
4
1 4 8 9
_
1 2 8 9
_
1 2 4 6 8 9
_
3
2 5 7 8
_
2 5 7 8
_
147 8 9
_
6
1 2 4 7 8 9
_
7
2 3 6 8 9
_
2 3 6 9
_
4
26 8
8
2 6
_
1 28
1
2 3 8 9
_
5
Step 4:
The "2" in cell (row 1 column 5) and cell (row 1 column 6) can be elimianted .
3 6 8
_
2 34 8
_
5
9
1
2 6 7
_
3 6 8
_
234 7 8
7
2 4 7
4
1 4 8 9
_
1 2 8 9
_
1 2 4 6 8 9
_
3
2 5 7 8
_
2 5 7 8
_
147 8 9
_
6
1 2 4 789
2
7
2 3 6 8 9
_
2 3 6 9
_
4
26 8
8
2 6
_
1 28
1
2 3 8 9
_
5
Example # 8
Step 1:
On Row 1, the triple "2", "3" and "5" are the possible numbers in (row 1 column 3), (row 1 column 7) and (row 1 column 8). Since "2", "3" and "5" must be in row one (columns 3, 7 and 8) you can eliminate "2", "3" and "5" in (row 1 column 2).
6
2 3 4 5 9
_
2 3 5 9
_
3 4 9
_
1 3 5 9
_
1 3 5 9
_
8
1 2
_
7
8 9
8
7 9
7
7 9
9
2 4
_
1 6
_
2 4
_
5
1 6
_
3
2 3 5
_
2 3 5
_
1
3 5 6 8
_
356 7 8
7
6 8
_
9
2 6
_
4
Step 2:
The number "4" can now fill the cell (row 1 column 2).
6
23 4 59
4
2 3 5 9
_
3 4 9
_
1 3 5 9
_
1 3 5 9
_
8
1 2
_
7
8 9
8
7 9
7
7 9
9
2 4
_
1 6
_
2 4
_
5
1 6
_
3
2 3 5
_
2 3 5
_
1
3 5 6 8
_
356 7 8
7
6 8
_
9
2 6
_
4
Example # 9
In the example below,
the pair "5" and "7" in cell (row 1 column 3)
the pair "4" and "7" in cell (row 2 column 3)
the triple "4", "5" and "7"in cell (row 3 column 3)
result in "4", "5" and "7" as the only possible numbers for column 3.
"4", "5" and "7" can therefore be eliminated from column 1 and column 2.
Step 1
1 2 5 7 9
_
8
1 5 7 9
_
1 2 6 7 9
_
1 2 6 7
_
1 4 7 9
_
1 4 7 9
_
3
1 4 5 7 9
_
Step 2
1 2 57 9
_
8
1 5 7 9
_
1 2 6 7 9
_
1 2 6 7
_
1 4 7 9
_
1 47 9
_
3
1 4 5 7 9
_
Example # 10
In the example below,
There are only two cells that can have the possible numbers "2" and "4".
The pair "2" and "4" must be in cells (row 1 column 2) and (row 3 column 1)
You can eliminate the other possible numbers in those two cells.
Step 1
7
1 2 3 4 6
_
9
1 6 8
_
1 3 6 8
_
1 3 8
_
2 4 6 8
_
5
3 8
_
Step 2
7
1 2 3 4 6
_
9
1 6 8
_
1 3 6 8
_
1 3 8
_
2 4 68
_
5
3 8
_
Example # 11
Step 1:
Example #11 is a variation on Example #10, using a row instead of a box.
On Row 2, there are only two cells that can have the possible numbers "6" and "7".
The pair "6" and "7" must be in cells (row 2 column 3) and (row 2 column 8).
You can eliminate the other possible numbers in those two cells.
3 4 5 7 9
_
3 5 8 9
_
3 5 6 7 8
_
2
1 3 5 8 9
_
3 5 6 7 8
_
3 7 9
_
1 38 9
1
4
1
3 4 5 689
4
2
3 67 9
_
3 56 8 9
_
5
5
3 6 7 9
_
3 6 8 9
_
7
7
3 8
_
6 7 9
_
6 7 8 9
_
1 3 8
_
1 6 7 9
_
4
5
1 2 6 9
2
26 8 9
_
Step 2:
3 4 5 7 9
_
3 5 8 9
_
3 5 6 7 8
_
2
1 3 5 8 9
_
35 6 7 8
_
3 7 9
_
1 38 9
1
4
1
3 4 5 689
4
2
3 67 9
_
3 56 8 9
_
5
5
3 6 7 9
_
3 6 8 9
_
7
7
3 8
_
6 7 9
_
6 7 8 9
_
1 3 8
_
1 6 7 9
_
4
5
1 2 6 9
2
26 8 9
_
Example # 12
Step 1:
Example #12 is a variation on Example #8.
On Row 3, there are only three cells that can have the possible numbers "3", "7" and "9".
The triple "3", "7" or "9" must be in cells (row 3 column 2) and (row 3 column 3) and the pair "3" or "9" must be in cell (row 3 column 5).
You can eliminate "7" in cell (row 3 column 4), and "7" and "9" in cells (row 3 column 6) and (row 3 column 8).
4
1
8
6
2 37 9
2
3 7 9
_
5
3 7 9
_
3 7 9
_
6 7
_
2
5 7 9
_
4 7 8
_
3 5 7 8 9
_
1
4 6 7 8
_
3 78 9
_
7 8 9
_
3 5 6
_
3 6 7 9
_
3 6 7 9
_
3 4 5 8
_
3 4 7 8 9
_
3 7 8 9
_
1 34 68
1
3 4 6 7 8 9
_
2
Step 2:
4
1
8
6
2 37 9
2
3 7 9
_
5
3 7 9
_
3 7 9
_
6 7
_
2
5 7 9
_
4 7 8
_
3 5 7 8 9
_
1
4 6 7 8
_
3 78 9
_
7 8 9
8
3 5 6
_
3 6 7 9
_
3 6 7 9
_
3 4 5 8
_
3 4 7 8 9
_
3 7 8 9
_
1 34 68
1
3 4 6 7 8 9
_
2
Example # 13
Step 1:
Cells (row 1 column 2) and (row 3 column 8) contain a "5" and "6" pair.
You can eliminate "5" and "6" in cell (row 3 column 1), cell (row 3 column 3), cell (row 1 column 8) and cell (row 1 column 9).
1 3 5 6
_
5 6
_
4 5 6
_
3 4 5 8
_
5 9
_
7
3 5 6 8
_
2
1 5 6 9
_
7
1 4
_
2 3 5 9
_
2 3 4 5 9
_
6
2 3 5 8 9
_
3 5 9
_
1 8
_
3 5 8 9
_
8
2 5 6
_
2 3 5 6 9
_
1 2 5
_
2 3 5
_
1 2 5 9
_
7
56
_
4
Step 2:
1 3 5 6
_
5 6
_
4 5 6
_
3 4 5 8
_
5 9
_
7
3 56 8
_
2
1 56 9
_
7
1 4
_
2 3 5 9
_
2 3 4 5 9
_
6
2 3 5 8 9
_
3 5 9
_
1 8
_
3 5 8 9
_
8
2 56
_
2 3 5 6 9
_
1 2 5
_
2 3 5
_
1 2 5 9
_
7
56
_
4
The search for a pair of candidates in two cells in a unit
Unit: a unit is a box, row or column. Candidate: a candidate is a possible number or pencil mark.
Sometimes the search for a pair of candidates in two cells is obvious (see Examples #2, 6 and 7).
The pair will be in two cells, but a one or both may be in other cells in the unit in addition to other candidates.
You can then eliminate members of the pair from those cells.
Sometimes the search for a pair of candidates in two cells is not obvious (hidden pair) (see Examples #10 and 11).
In this case, the pair will be in only two cells of the unit. One or both of the pair will not be in any other
cell in the unit. There will be other candidates in the cells with the pair. Once you find the pair, you can then
eliminate the other candidates in those two cells.
To print puzzles that require the use of this strategy, click the link below.
