The following presents a number of strategies which you will find useful when solving SUDOKU puzzles. Note that the
successful application of a strategy will open up new opportunities which permit the successful application of other
strategies within the list. Consequently, if the puzzle is not solved after the first pass through the list, you should
continue to repeat the process, until the puzzle is solved.
The graphic images included in the following use "pencil" marks to show what numbers could possibly be placed in each
cell of the puzzle. Many solvers prefer not to use these marks, but they are essential when explaining the solution
techniques.
When specific cells need to be referred to in this discussion, they will be designated as C4-R3 which means the 4th
cell in row 3.
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Single Candidate
This is the simplest strategy available for the solving of SUDOKU puzzles. So simple, it is really only included
here for the sake of completeness.
The cell highlighted in blue has the single number 7 "pencil marked" in. All of the other possible digits have already
been included in either the row, the column, or the 3x3 sub-square of which the highlighted cell is a member. It therefore
follows that 7 must be the solution value for this cell.
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Single Position
The single highlighted cell is a member of row 4 of the puzzle. An examination of this row reveals that the
highlighted cell is the only one which does contain a 4. Since every row must contain exactly one 4, it is clear that
it must be placed in this cell.
This is an example of a Single Position within a row. Single Position examples will also occur in
columns and 3x3 sub-squares.
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Naked Pair
Row 5 of this puzzle contains two highlighted cells, each of which contain "pencil marks" for 2 and 3 and
nothing else. This leads to the inescapable conclusion that one of the cells must contain a 2, and one must contain
a 3. We don't know at this stage which way it will be, but we do know that neither a 2 nor a 3 can appear elsewhere
in the 3x3 sub-square. Therefore, we can remove the "pencil mark" 3 from C7-R6.
This may not seem like a lot of progress, but it does uncover another valuable opportunity. C7-R6 and C7-R7 now
constitute another Naked Pair which allows the 4 and the 6 to be removed from C7-R4. This leads to a
Single Candidate situation for C7-R4 which can immediately be solved.
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Naked Triple
This is a simple extension of the Naked Pair idea. The three highlighted cells are all part of column 1,
and collectively, they contain only the numbers 1, 2 and 5. Consequently, these numbers cannot appear elsewhere in
column 1, so a 5 can be removed from C1-R7, and 2 and 5 can be removed from C1-R8. Fortunately, this uncovers another
Naked Pair which sets off a chain of simple solutions in the bottom left 3x3 sub-square.
Naked Triples can also appear in rows and sub-squares.
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Naked Quad
As a further extension of Naked Pairs and Naked Triples the highlighted cells in this case contain
only the numbers 1,5,6 and 7. These numbers can therefore be removed from elsewhere in the central 3x3 sub-square.
Fortunately, this leaves only a single 9 in C5-R6 which can immediately be solved.
Naked Quads can also appear in rows and columns, but are not encountered very frequently in published
puzzles.
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Candidate Line
If you refer to row 5, you will see that the only cells which contain an 8 are the two highlighted cells.
We don't know which one will contain the 8, but one of them certainly will. Therefore an 8 cannot appear elsewhere
in the center right 3x3 sub-square which means that the 8 can be removed from C8-R4. Not a great leap forward, but
every little bit helps.
Candidate Lines can be quite hard to spot, even when pointed out by a hint, but don't despair. With a bit
of practice you will become quite proficient at it.
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Hidden Pair
If you inspect row 7, you will find that the highlighted cells are the only ones which contain the numbers
5 and 6. It follows then that one of these cells must have 5 as a solution, and the other must have 6. Any other
numbers in these two cells are superfluous and may be removed.
As you might expect, Hidden Pairs can also be found in columns and sub-squares.
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Hidden Triple
This is a simple extension of the Hidden Pair idea. Referring to the bottom
right 3x3 sub-square, you will find that the three highlighted cells are the only ones which contain any of the
numbers 4, 5 and 9. Therefore, 4, 5 and 9 (in some order) must be the solutions for these three cells, and other
numbers in these cells are superfluous and may be removed.
This is a case of a Hidden Triple within a sub-square, but remember that they can also be found in rows
and columns.
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X-wing
In row 2, the only cells which contain a 5 are those which are highlighted. Therefore, one of them must
have 5 as the solution. Exactly the same observation may be made about row 6.
If you now rotate your thinking through 90 degrees, you will conclude that in both column 2 and column 9, the 5
must appear as the solution in one of the highlighted cells. Looking at column 2 in particular, the 5 in C2-R9
cannot be part of the solution and can be removed. Fortunately, this leaves a 4 as the Single Candidate
in C2-R9, and provides us with the solution for this cell.
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Swordfish
This is an extension of the X-wing idea. In row 1, 1 must be the solution for one of the highlighted
cells. Exactly the same can be said in connection with row 6 and row 9.
If you now rotate your thinking through 90 degrees, you will conclude that in all of columns 5, 6 7, the 1 must
appear as the solution in one of the highlighted cells. Any 1s in columns 5, 6 and 7, other than those in the
highlighted cells cannot be part of the solution and may be removed. There are a total of seven instances in which
a 1 can be removed, and we are lucky that in one of these cases (C7-R3) we are left with a Single Candidate
situation.
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Jellyfish
This is the final extension of the X-wing idea. In each of the rows 1, 2, 4 and 7, a 7 must appear as
the solution in one of the highlighted cells.
Rotating your thinking through 90 degrees will lead to the conclusion that in columns 1, 2, 5 and 6, the only
cells which can have 7 as a solution are those which are highlighted. Any 7s elsewhere in columns 1, 2, 5 and 6
cannot be part of the solution, and may be removed. There are four such examples, and one of them (C2-R9)
results in a Single Candidate situation.
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Astute solvers may find other, more obscure strategies which they can apply.
If you believe that you have found such a strategy,
please contact me with details so that I
can expand the capabilities of the program. In the mean time, the strategies listed here are guaranteed
to solve any SUDOKU puzzle created by Magnum Opus.
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