# How To Play SuDoku Puzzles

This page will outline the various solution methods for solving SuDoku puzzles. We will discuss scanning, marking up, and analysing in that order.

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# How To Play SuDoku Puzzles

## Rules

The puzzle is most often 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). Some cells already contain numbers, known as "givens" (or "clues"). The goal is to fill in the empty cells, one number in each, so that each column, row, and region contains the numbers 1 – 9 exactly once. Each number in the solution therefore occurs only once in each of three "directions".

## Solving the puzzle

#### Scanning

Scanning is performed at the outset and periodically throughout the solution. Scans will have to be performed several times in between analysis periods. There are two basic techniques for scanning:

• 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. This process is then repeated with the columns (or rows). For fastest results, the numbers 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 numbers. Counting based upon the last number discovered may speed up the search. 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.

Advanced solvers look for "contingencies" while scanning — that is, narrowing a number'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. Particularly challenging puzzles may require multiple contingencies to be recognized, perhaps in multiple directions or even intersecting — relegating most solvers to marking up (please see below). Sudoku 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 scanning alone.

#### Marking Up

Scanning comes to a halt when no further numbers can be discovered. From this point, it is necessary to engage in some logical analysis. You may find it useful to guide this analysis by marking candidate numbers in the blank cells. There are two popular notations: subscripts and dots.

• In the subscript notation the candidate numbers 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 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. 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 pencil would then be recommended.

Another technique that you may find easier is to mark up those numbers that a cell cannot be. Thus a cell will start empty and as more constraints become known it will slowly fill. When only one marking is missing, that has to be the value of the cell.

#### Analysing

There are two approaches to analysis: "candidate elimination" and "what-if".

1. In elimination, progress is made by successively eliminating candidate numbers 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 latest number. There are a number of elimination tactics, all of which are based on the simple rules given above, which have important and useful corollaries, including:

1. A given set of n cells in any particular block, row, or column can only accommodate n different numbers. This is the basis for the "unmatched candidate deletion" technique, discussed below.

2. Each set of candidate numbers, 1–9, must ultimately be in an independently self-consistent pattern. This is the basis for advanced analysis techniques that require inspection of the entire set of possibilities for a given candidate number. Only certain "closed circuit" or "n×n grid" possibilities exist (which have acquired peculiar names such as "X-wing" and "Swordfish", among others). If these patterns can be identified, elimination of candidate possibilities external to the grid framework can sometimes be achieved.

2. One of the most common elimination tactics is "unmatched candidate deletion". 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; essentially, these are perfectly coincident contingencies. For example, cells are said to be matched within a particular row, column, or region (scope) if two cells contain the same pair of candidate numbers (p,q) and no others, or if three cells contain the same triplet of candidate numbers (p,q,r) and no others. The placement of these numbers anywhere else in the matching scope would make a solution for the matched cells impossible; thus, the candidate numbers (p,q,r) appearing in unmatched cells in the row, column or region scope can be deleted. This principle also works with candidate number subsets—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 in the scope can be deleted. The principle is true for all quantities of candidate numbers.

3. A second related principle is also true — if the number of cells - in a row, column or region scope - where a set of candidate numbers only appear is equal to the quantity of candidate numbers, the cells and numbers are matched and only those numbers can appear in matched cells. Other candidates in the matched cells can be eliminated. For example, if (p,q) can only appear in 2 cells (within a specific row, column, region scope), other candidates in the 2 cells can be eliminated.

4. The first principle is based on cells where only matched numbers appear. The second is based on numbers that appear only in matched cells. The validity of either principle is demonstrated by posing the question 'Would entering the eliminated number prevent completion of the other necessary placements?

5. In the what-if approach, a cell with only two candidate numbers 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 number prevent completion of the other placements of that number? 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 you need to find a combination of techniques which will avoid some of the drawbacks of the above elements. The counting of regions, rows, and columns can be very boring. Writing candidate numbers into empty cells can be very time-consuming. The what-if approach can be confusing. We suggest you find a technique which will minimize counting, marking up, and rubbing out.

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