How To Play SuDoku Puzzles
The puzzle is most often a 9×9 grid, made up of 3 × 3 subgrids called
"regions" (other terms include "boxes", "blocks", and the like when referring to
the standard variation). Some cells already contain numbers, known as "givens"
(or "clues"). The goal is to fill in the empty cells, one number in each, so
that each column, row, and region contains the numbers 1 – 9 exactly once. Each
number in the solution therefore occurs only once in each of three "directions".
Solving the puzzle
Scanning is performed at the outset and periodically throughout the solution.
Scans will have to be performed several times in between analysis periods. There
are two basic techniques for scanning:
- Cross-hatching: the scanning of rows (or columns) to identify
which line in a particular region may contain a certain number by a process
of elimination. This process is then repeated with the columns (or rows).
For fastest results, the numbers are scanned in order of their frequency. It
is important to perform this process systematically, checking all of the
- Counting 1–9 in regions, rows, and columns to identify missing numbers.
Counting based upon the last number discovered may speed up the search. It
also can be the case (typically in tougher puzzles) that the value of an
individual cell can be determined by counting in reverse—that is, scanning
its region, row, and column for values it cannot be to see which is left.
Advanced solvers look for "contingencies" while scanning — that is, narrowing
a number's location within a row, column, or region to two or three cells. When
those cells all lie within the same row (or column) and region, they can be used
for elimination purposes during cross-hatching and counting. Particularly
challenging puzzles may require multiple contingencies to be recognized, perhaps
in multiple directions or even intersecting — relegating most solvers to marking
up (please see below). Sudoku Puzzles which can be solved by scanning alone
without requiring the detection of contingencies are classified as "easy"
puzzles; more difficult puzzles, by definition, cannot be solved by scanning
Scanning comes to a halt when no further numbers can be discovered. From this
point, it is necessary to engage in some logical analysis. You may find it
useful to guide this analysis by marking candidate numbers in the blank cells.
There are two popular notations: subscripts and dots.
- In the subscript notation the candidate numbers are written in
subscript in the cells. The drawback to this is that original puzzles
printed in a newspaper usually are too small to accommodate more than a few
digits of normal handwriting. If using the subscript notation, solvers often
create a larger copy of the puzzle or employ a sharp or mechanical pencil.
- The second notation is a pattern of dots with a dot in the top
left hand corner representing a 1 and a dot in the bottom right hand corner
representing a 9. The dot notation has the advantage that it can be used on
the original puzzle. Dexterity is required in placing the dots, since
misplaced dots or inadvertent marks inevitably lead to confusion and may not
be easy to erase without adding to the confusion. Using a pencil would then
Another technique that you may find easier is to mark up those numbers that a cell cannot be. Thus a cell will start empty and as more constraints become known it will slowly fill. When only one marking is missing, that has to be the value of the cell.
There are two approaches to analysis: "candidate elimination" and "what-if".
- In elimination, progress is made by successively eliminating candidate
numbers from one or more cells to leave just one choice. After each answer
has been achieved, another scan may be performed—usually checking to see the
effect of the latest number. There are a number of elimination tactics, all
of which are based on the simple rules given above, which have important and
useful corollaries, including:
- A given set of n cells in any particular block, row, or column can
only accommodate n different numbers. This is the basis for the
"unmatched candidate deletion" technique, discussed below.
- Each set of candidate numbers, 1–9, must ultimately be in an
independently self-consistent pattern. This is the basis for advanced
analysis techniques that require inspection of the entire set of
possibilities for a given candidate number. Only certain "closed circuit" or
"n×n grid" possibilities exist (which have acquired peculiar names such as
"X-wing" and "Swordfish", among others). If these patterns can be
identified, elimination of candidate possibilities external to the grid
framework can sometimes be achieved.
- One of the most common elimination tactics is "unmatched candidate
deletion". Cells with identical sets of candidate numbers are said to be
matched if the quantity of candidate numbers in each is equal to the number
of cells containing them; essentially, these are perfectly coincident
contingencies. For example, cells are said to be matched within a particular
row, column, or region (scope) if two cells contain the same pair of
candidate numbers (p,q) and no others, or if three cells contain the same
triplet of candidate numbers (p,q,r) and no others. The placement of these
numbers anywhere else in the matching scope would make a solution for the
matched cells impossible; thus, the candidate numbers (p,q,r) appearing in
unmatched cells in the row, column or region scope can be deleted. This
principle also works with candidate number subsets—if three cells have
candidates (p,q,r), (p,q), and (q,r) or even just (p,r), (q,r), and (p,q),
all of the set (p,q,r) elsewhere in the scope can be deleted. The principle
is true for all quantities of candidate numbers.
- A second related principle is also true — if the number of cells - in a
row, column or region scope - where a set of candidate numbers only appear
is equal to the quantity of candidate numbers, the cells and numbers are
matched and only those numbers can appear in matched cells. Other candidates
in the matched cells can be eliminated. For example, if (p,q) can only
appear in 2 cells (within a specific row, column, region scope), other
candidates in the 2 cells can be eliminated.
- The first principle is based on cells where only matched numbers appear.
The second is based on numbers that appear only in matched cells. The
validity of either principle is demonstrated by posing the question 'Would
entering the eliminated number prevent completion of the other necessary
- In the what-if approach, a cell with only two candidate numbers
is selected, and a guess is made. The steps above are repeated unless a
duplication is found or a cell is left with no possible candidate, in which
case the alternative candidate is the solution. In logical terms, this is
known as reductio ad absurdum. Nishio is a limited form of this approach:
for each candidate for a cell, the question is posed: will entering a
particular number prevent completion of the other placements of that number?
If the answer is yes, then that candidate can be eliminated. The what-if
approach requires a pencil and eraser. This approach may be frowned on by
logical purists as trial and error - and most published puzzles are built to
ensure that it will never be necessary to resort to this tactic - but it can
arrive at solutions fairly rapidly.
Ideally you need to find a combination of techniques which will avoid some of
the drawbacks of the above elements. The counting of regions, rows, and columns
can be very boring. Writing candidate numbers into empty cells can be very
time-consuming. The what-if approach can be confusing. We suggest you find a technique which
will minimize counting, marking up, and rubbing out.