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SUDOKU


Background
Sudoku (Japanese: æ•°ç‹¬,, sÅ«doku),
sometimes spelled Su Doku, is a placement puzzle,
also known as Number Place in the United States. The aim of
the puzzle is to enter a number from 1 through 9 in each
cell of a grid, most frequently a 9×9 grid made up of 3×3
subgrids (called "regions"), starting with various numbers
given in some cells (the "givens"). Each row, column and
region must contain only one instance of each number.
Completing the puzzle requires patience and modest logical
ability (although some puzzles can be very difficult). Its
classic grid layout is reminiscent of other newspaper
puzzles like crosswords and chess problems. First published
in the United States, Sudoku initially became popular in
Japan in 1986 and had attained widespread international popularity
by 2005.
The word Sudoku means "single number" in
Japanese. The numerals in Sudoku puzzles are used for
convenience; arithmetic relationships between numerals are
not important. Any set of distinct symbols will do; letters,
shapes, or colors may be used without altering the rules.
Indeed, Penny Press uses letters in their version called
Scramblets. Dell Magazines, the puzzle's originator, has
been using numerals for Number Place in its magazines since
they first published it over 25 years ago. Numerals are used
throughout this article.
The attraction of the puzzle is that the
completion rules are simple, yet the line of reasoning
required to reach the completion may be difficult. Published
puzzles often are ranked in terms of difficulty. This also
may be expressed by giving an estimated solution time.
While, generally speaking, the greater the number of givens,
the easier the solution, the opposite is not necessarily
true. The true difficulty of the puzzle depends upon how
easy it is to logically determine subsequent numbers.
Sudoku is recommended by some teachers as
an exercise in logical reasoning. The level of difficulty of
the puzzles can be selected to suit the audience. The
puzzles are often available free from published sources and
also may be customgenerated using
software.
The strategy for solving a puzzle may be regarded as
comprising a combination of three processes: scanning,
marking up, and analyzing.



The 3x3 region in the
topright corner must contain a 5. By hatching across and up from 5s
located elsewhere in the grid, the solver can eliminate all of the
empty cells in the topleft corner which cannot contain a 5. This
leaves only one possible cell (highlighted in green). 

Scanning

Scanning is performed at the outset and periodically
throughout the solution. Scans may have to be performed several times in
between analysis periods. Scanning comprises two basic techniques,
crosshatching and counting, which may be used alternately:
Crosshatching: 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 digits 19.

Counting 19 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 crosshatching and counting
(Contingency example at Puzzle Japan). Particularly
challenging puzzles may require multiple contingencies to be
recognized, perhaps in multiple directions or even
intersecting  relegating most solvers to marking up (as
described below). 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 basic scanning alone.
Marking up
Scanning comes to a halt when no further
numbers can be discovered. From this point, it
is necessary to engage in some logical analysis.
Many 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.
Analyzing
There are two main analysis approaches 
elimination and whatif.

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. One of the most common 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.
For example, cells are said to be matched
within a particular row, column, or region if
two cells contain the same pair of candidate
numbers (p,q) and no others, or if three cells
contain the same triple of candidate numbers (p,q,r)
and no others. These are essentially
coincident contingencies. These numbers (p,q,r)
appearing as candidates elsewhere in the same
row, column, or region in unmatched cells can
be deleted.

In the whatif 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, 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 if
yes, then that candidate can be eliminated.
The whatif approach requires a pencil and
eraser. This approach may be frowned on by
logical purists as too much trial and error
but it can arrive at solutions fairly rapidly.
Ideally one needs to find a combination of
techniques which avoids some of the drawbacks of
the above elements. The counting of regions,
rows, and columns can feel boring. Writing
candidate numbers into empty cells can be
timeconsuming. The whatif approach can be
confusing unless you are well organized. The
Holy Grail is to find a technique which
reduces counting, marking up, and rubbing out.



