Rules for Sudoku

Sudoku is a logic-based number placement puzzle. The objective is to fill a 9×9 grid so that each column, each row, and each of the nine 3×3 boxes (also called blocks or regions) contains the digits from 1 to 9, only one time each (that is, exclusively). The puzzle setter provides a partially completed grid.

1. Fill the grid so that the numbers 1 through 9 appear in each row.

2. Fill the grid so that the numbers 1 through 9 appear in each column.

3. Fill the grid so that the numbers 1 through 9 appear in each 3x3 box.

4. A complete Sudoku puzzle contains the numbers 1 through 9 in every row, column, and 3x3 box.

Strategies

The strategy for solving a puzzle may be regarded as comprising a combination of three processes: scanning, marking up, and analyzing. The approach to analysis may vary according to the concepts and the representations on which it is based.

The top right region must contain a 5. By hatching across and up from 5s elsewhere, the solver can eliminate all the empty cells in the region which cannot contain a 5. This leaves only one possibility (shaded green).

Scanning

Scanning is performed at the outset and throughout the solution. Scans need be performed only once between analyses. Scanning consists of two techniques:

• Cross-hatching: The scanning of rows to identify which line in a region may contain a certain numeral by a process of elimination. The process is repeated with the columns. 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, particularly in tougher puzzles, that the best way to ascertain the value of a cell is to count in reverse—that is, by scanning the cell's region, row, and column for values it cannot be, in order to see what remains.

Advanced solvers look for "contingencies" while scanning, narrowing a numeral's location within a row, column, or region to two or three cells. When those cells lie within the same row and region, they can be used for elimination during cross-hatching and counting. Puzzles solved by scanning alone without requiring the detection of contingencies are classified as "easy"; more difficult puzzles are not readily solved by basic scanning alone.

Logically, every sudoku puzzle, regardless of difficulty, is solved via scanning heuristics. In a true sudoku puzzle, every number has a necessary position in each part of the grid which can be deduced from the description or if you prefer definition of what a "true" sudoku is. The only difference between solving advanced puzzles and simpler puzzles is not the techniques used to solve the puzzle but recognizing the logical implications of the scanning heuristic. One such implication would be recognizing logical "contingencies" which just basically means narrowing down the possibilities of a given square via the relations between every other square.

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.

The partially filled sub-square determines that 3,5, and 6 must go in the top row. These create a contingency for the far right hatched cell based on the complete row across. It must be a 4.

Marking up

Scanning stops when no further numerals can be discovered, making it necessary to engage in logical analysis. One method to guide the analysis is to mark candidate numerals in the blank cells.

Subscript notation

In subscript notation, the candidate numerals are written in subscript in the cells. Because puzzles printed in a newspaper are too small to accommodate more than a few subscript digits of normal handwriting, solvers may create a larger copy of the puzzle. Using two colors, or mixing pencil and pen marks can be helpful.

Dot notation

The dot notation uses a pattern of dots in each square, where the dot position indicates a number from 1 to 9. The dot notation 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 easily erased.

An alternative technique is to mark the numerals that a cell cannot be. The cell starts empty and as more constraints become known, it slowly fills until only one mark is missing. 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.

An analysis in Sudoku, done in superscript notation, with all possible values for the squares written in. There are three squares which contain only three values: 4, 6, and 8. If 4, 6, or 8 were written in any square where they're red, it would be impossible to complete the squares where they're blue. Therefore, the numbers in red can be erased. This logic works with rows, columns, sections, and diagonals. (if applicable)

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