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If you are looking for Sudoku Tips, Information on How to Solve Sudoku Puzzles, or a Sudoku Strategy that you can benefit from, below are some great Sudoku Intructions we've extracted from wikipedia.
Sudoku
Solution methods:
The strategy for solving a puzzle may be regarded as comprising a combination
of three processes: scanning, marking up, and analysing.
The 3×3 region in the top-right 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 top-right corner which cannot
contain a 5. This leaves only one possible cell (highlighted in green).
Sudoku 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:
Sudoku Cross-hatching:
Is 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.
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Here's the BIG SECRET TO WINNING at SUDOKU: 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 cross-hatching and counting (Contingency example at Puzzle Japan). |
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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.
Candidates for each empty cell have been entered. Some cells have only one candidate once obvious invalids have been excluded. Also, some mark with dots instead of numbers, simply using the position of the dot within the cell to distinguish them. (Click the image to see a larger one)
Sudoku 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.
A method for marking likely numerals in a single cell by the placing
of pencil dots. To reduce the number of dots used in each cell, the
marking would only be done after as many numbers as possible have been
added to the puzzle by scanning. Dots are erased as their corresponding
numerals are eliminated as candidates.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 mark-ups written inside one region, then it is clear that the digit should be there, even if the cell has other digits marked as well.
Sudoku Analysis:
The two main approaches to analysis are "candidate elimination"
and "what-if".
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 "what-if" 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 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 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 time-consuming. The what-if 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.
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Sudoku Computer
Solutions:
Although a simple Ariadne's thread (depth-first search) algorithm can
solve (or prove invalid) any Sudoku puzzle, it is computationally inefficient
and as such rarely employed on its own. There are two general approaches
taken in the creation of serious Sudoku-solving programs:
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emulate the human solving method as closely as possible, which allows for determining the difficulty level of an inputted puzzle;
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solve puzzles as efficiently as possible, which allows for faster computation.
Either way, a computer program is capable of exhaustively searching a Sudoku puzzle for solutions, thereby determining whether it is valid (has exactly one solution) or not, with great ease relative to a human attempting the same.
Human-style solvers will typically operate by maintaining a mark-up matrix identical to that a human solver may use (see "Marking up" under "Solution methods" above), and search for contingencies, matched cells, and other elements a human solver can utilize in order to determine and exclude cell values, resorting to Ariadne's thread only as a last resort. Each type of operation performed can be assigned a difficulty value; the sum of these values can be construed as a difficulty level of the puzzle.
Many rapid-style solvers still employ backtracking searches, but with various shortcuts and optimizations to reduce the depth of the search tree; which techniques are superior is under frequent debate. Another alternative uses finite domain constraint programming. A constraint program specifies the constraints of the puzzle (the fact that every number in each row, each column, and each 3×3 region must be unique, and the provided "givens"); a finite domain solver applies the constraints successively to narrow down the solution space until a solution is found. Backtracking may be applied when alternate values cannot otherwise be excluded.
Although for standard Sudoku problems highly optimized and sophisticated backtracking programs are the most efficient, another popular way of solving such constraint problems is Donald Knuth's Dancing Links Algorithm for solving the exact matrix cover problem, of which the Sudoku problems are a special case. Knuth's algorithm can be applied by converting the Sudoku puzzle to a matrix cover problem, solve this problem instead, and convert the solution obtained back to a completed Sudoku grid. This method is now preferred by many Sudoku programmers, by virtue of its execution speed, simplicity and ease of implementation, and the availability of documentation and reference source code.
Rapid solvers are preferred for trial-and-error puzzle-creation algorithms, which allow for testing large numbers of partial problems for validity in a short time; human-style solvers can be employed by hand-crafting puzzlesmiths for their ability to rate the challenge of a created puzzle and show the actual solving process their target audience can be expected to follow.
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