![]() These numbers (p,q,r) appearing as candidates elsewhere in the same row, column, or region in unmatched cells can be deleted. These are essentially coincident contingencies. 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. 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. One of the most common is "unmatched candidate deletion". There are a number of elimination tactics. After each answer has been achieved, another scan may be performed - usually checking to see the effect of the latest number. In elimination, progress is made by successively eliminating candidate numbers from one or more cells to leave just one choice.There are two main analysis approaches - elimination and what-if. 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. The dot notation has the advantage that it can be used on the original puzzle. 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. If using the subscript notation, solvers often create a larger copy of the puzzle or employ a sharp or mechanical pencil. 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. In the subscript notation the candidate numbers are written in subscript in the cells. There are two popular notations: subscripts and dots. Many find it useful to guide this analysis by marking candidate numbers in the blank cells. From this point, it is necessary to engage in some logical analysis. Scanning comes to a halt when no further numbers can be discovered. 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. 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). 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). 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.Īdvanced solvers look for "contingencies" while scanning - that is, narrowing a number's location within a row, column, or region to two or three cells. Counting based upon the last number discovered may speed up the search. Counting 1-9 in regions, rows, and columns to identify missing numbers.It is important to perform this process systematically, checking all of the digits 1-9. For fastest results, the numbers are scanned in order of their frequency. ![]() This process is then repeated with the columns (or rows). 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.Scanning comprises two basic techniques, cross-hatching and counting, which may be used alternately: Scans may have to be performed several times in between analysis periods. Scanning is performed at the outset and periodically throughout the solution. The strategy for solving a puzzle may be regarded as comprising a combination of three processes: scanning, marking up, and analysing. This leaves only one possible cell (highlighted in green). By hatching across and up from 5s located elsewhere in the grid, the solver can eliminate all of the empty cells in the top-left corner which cannot contain a 5. The 3x3 region in the top-right corner must contain a 5.
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