Solving tips

Solution Methods
The strategy for solving a puzzle may be regarded as comprising a combination of three processes: scanning, marking up, and analyzing.

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:
• Cross-hatching: 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.

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. A popular method is the subscript notation. In the subscript notation the candidate numerals are written in subscript in the cells.

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 (also called "guess-and-check", "bifurcation", and "Ariadne's thread"), 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 must be the solution. In logic, this is known as reductio ad absurdum. Nishio is a limited form of this approach: for each cell's candidate, 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