Instructions
for Sudoku
Sudoku, also known as Number
Place, is a logicbased placement puzzle. The aim of the puzzle is to
enter a numerical digit from 1 through 9 in each cell of a
9×9 grid made up of 3×3 subgrids (called
"regions"), starting with various digits given in some cells (the
"givens").
Each row, column, and region must contain only one instance
of each numeral. Completing the puzzle requires patience and logical
ability. Although first published in a U.S. puzzle magazine in 1979,
Sudoku initially caught on in Japan in 1986 and attained international
popularity in 2005.
Introduction
The numerals in Sudoku puzzles are used for convenience; arithmetic
relationships between numerals are absolutely irrelevant. Any set of
distinct symbols will do; letters, shapes, or colours may be used
without altering the rules (Penny Press' Scramblets and Knight Features
Syndicate's Sudoku Word both use letters). Dell Magazines, the puzzle's
originator, has been using numerals for Number Place in its magazines
since they first published it in 1979. Numerals are used throughout
this article.
The attraction of the puzzle is that the rules are simple, yet the line
of reasoning required to reach the solution may be complex. 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 may also be customgenerated using software.
Gameplay
The puzzle is most frequently 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; even
"quadrants" is sometimes used, despite this being an inaccurate term
for a 9×9 grid). Some cells already contain numerals, known
as "givens" (or sometimes as "clues"). The goal is to fill in the empty
cells, one numeral in each, so that each column, row, and region
contains the numerals 1–9 exactly once. Each numeral in the
solution therefore occurs only once in each of three "directions" or
"scopes", hence the "single numbers" implied by the puzzle's name.
Solution methods
The strategy for solving a puzzle may be regarded as comprising a
combination of three processes: scanning, marking up, and analysing.
Scanning
Scanning is performed at the outset and throughout the solution. Scans
only have to be performed one time in between analysis periods.
Scanning consists of two basic techniques: Crosshatching: the scanning
of rows (or columns) to identify which line in a particular region may
contain a certain numeral by a process of elimination. This process is
then repeated with the columns (or rows). For fastest results, the
numerals are scanned in order of their frequency. It is important to
perform this process systematically, checking all of the digits
1–9. Counting 1–9 in regions, rows, and columns to
identify missing numerals. Counting based upon the last numeral
discovered may speed up the search. It also can be the case (typically
in tougher puzzles) that the easiest way to ascertain the value of an
individual cell is by counting in reverse—that is, by
scanning the cell's region, row, and column for values it cannot be, in
order to see which is left. Advanced solvers look for "contingencies"
while scanning—that is, narrowing a numeral'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 stops when no further numerals 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 numerals in the
blank cells. There are two popular notations: subscripts and dots.
 In the subscript notation the candidate numerals 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 uses a pattern of dots within each square, where
the position of the dot represents a number from 1 to 9. Dot schemes
differ and one method is illustrated here. 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 sharp pencil with an eraser end is
recommended.
An alternative technique, that some find easier, is to "mark up" those
numerals that a cell cannot be. Thus a cell will start empty and as
more constraints become known it will slowly fill. When only one mark
is missing, that has to be the value of the cell. One advantage to this
method of marking is that, assuming no mistakes are made and the marks
can be overwritten with the value of a cell, there is no longer a need
for any erasures. When using marking, additional analysis can be
performed. For example, if a digit appears only one time in the
markups written inside one region, then it is clear that the digit
should be there, even if the cell has other digits marked as well. When
using marking, a couple of similar rules applied in a specified order
can solve any Sudoku puzzle, without performing any kind of
backtracking.
Analysis
The two main approaches to analysis are "candidate elimination" and
"whatif".
 In "candidate elimination", progress is made by successively
eliminating candidate numerals 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
contingencies. One method of candidate elimination works by identifying
"matched cells". Cells are said to be matched within a particular row,
column, or region (scope) if two cells contain the same pair of
candidate numerals (p,q) and no others, or if three cells contain the
same triplet of candidate numerals (p,q,r) and no others. The placement
of these numerals anywhere else within that same scope would make a
solution for the matched cells impossible; thus, the candidate numerals
(p,q,r) appearing in unmatched cells in that same row, column or region
(scope) can be deleted. This principle also works with candidate
numeral subsets, that is, 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 within that same scope can be deleted. The principle
is true for all quantities of candidate numerals. A second related
principle is also true. If, within any set of cells (row, column or
region), a set of candidate numerals can only appear within a number of
cells equal to the quantity of candidate numerals, the cells and
numerals are matched and only those numerals can appear in the matched
cells. Other candidates in the matched cells can be eliminated. For
example, if the 2 numerals (p,q) can only appear in 2 cells within a
specific set of cells (row, column or region), all other candidates in
those 2 cells can be eliminated. The first principle is based on cells
where only matched numerals appear. The second is based on numerals
that appear only in matched cells. The validity of either principle is
demonstrated by posing the question, 'Would entering the eliminated
numeral prevent completion of the other necessary placements?' If the
answer to the question is 'Yes,' then the candidate numeral in question
can be eliminated. Advanced techniques carry these concepts further to
include multiple rows, columns, and regions.
 In the "whatif" approach, a cell with only two candidate numerals 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 numeral prevent completion of the other
placements of that numeral? If the answer is 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 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 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 numerals into empty
cells can be timeconsuming. The whatif approach can be confusing
unless you are well organised. The proverbial Holy Grail is to find a
technique which minimizes counting, marking up, and rubbing out.
Difficulty ratings
Published puzzles often are ranked in terms of difficulty.
Surprisingly, the number of givens has little or no bearing on a
puzzle's difficulty. A puzzle with a minimum number of givens may be
very easy to solve, and a puzzle with more than the average number of
givens can still be extremely difficult to solve. The difficulty of a
puzzle is based on the relevance and the positioning of the given
numbers rather than the quantity of the numbers.
Computer solvers can estimate the difficulty for a human to find the
solution, based on the complexity of the solving techniques required.
This estimation allows publishers to tailor their Sudoku puzzles to
audiences of varied solving experience. Some online versions offer
several difficulty levels.
Most publications sort their Sudoku puzzles into four rating levels,
although the actual cutoff points of the levels and indeed the names
of the levels themselves can vary widely. Typically, however, the
titles are some set of synonyms of "easy", "intermediate", "hard", and
"challenging".
Source Wikipedia :
http://en.wikipedia.org/wiki/Sudoku

