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Sudoku Guide

The pleasure of solving sudoku puzzles comes from the different types of logic required. Some are more advanced than others although - even for the most difficult puzzles - much of the challenge is derived from careful observation of the grid. Elimination of candidates from a particular square or squares requires logic concepts which are actually rather straightforward; it's all down to spotting the clues.

Before we begin, a quick note on terminology:

Seed
A seed is any digit which appears in the original puzzle. The seeds are the starting point. Creators of hand-made sudoku puzzles often set themselves interesting challenges when providing these seeds. This can take the form of interesting, attractive patterns or - perhaps more commonly - providing the fewest possible seeds. An average (especially computer-generated) sudoku will have 26-30 seeds. Many hand-made puzzle designers won't be satisfied until they have reduced this to 24, and the majority look for less. A designer will generally be very happy with 20 seeds, which is surprisingly difficult to achieve. Symmetrical puzzles with 19 seeds or fewer are notoriously tough to make. 17 seeds is almost regarded as a "jackpot" achievement, and although we have found many examples none are symmetrical and, oddly, it appears that the majority of these are actually based on one particular arrangement of numbers in the finished sudoku. As for 16 seeds - at the moment, this seems to exist as rumour only.

Number
This is simply a number which the solver has committed to the grid as the only possible candidate for a particular square. For our purposes, as we describe the various types of logic needed to complete sudoku puzzles, we will use the word "number" to identify any digit, whether it is one of the puzzle's original seeds or one placed by the solver.

Candidate
A candidate is any number which might be the correct one for a particular square. For all but the simplest of puzzles, it is wise to write - in pencil and in small type - all possible candidates for a square. Solvers have their own preferences. Some use dots arranged in a 3x3 pattern, while others write the actual numbers in the same pattern. This latter tactic is probably the better one, but it's all down to personal preference.

Row
A row is a horizontal line of nine squares in which the digits 1-9 need to appear.
Column
A column is a vertical line of nine squares in which the digits 1-9 need to appear.
Region
A region is one of the nine 3x3 divisions of the sudoku grid. Again, this must contain the digits 1-9.

OK, it's time to work through the types of logic used in solving a sudoku puzzle, starting with the simplest. Different solvers and setters use different terminology and we can't cover all versions, but the aim here is merely to help with identifying these logical processes.

Naked Single
This is the give-away of the sudoku world. A naked single is a square which has only one valid candidate. In its simplest form, if we have a row in which the first eight squares have the digits 1-8 placed, then 9 is the naked single for the ninth square because no other number can be used there. That does not mean all naked singles are simple to spot. It may be, for example, that you have a row containing the numbers 1,6,7 and 9 intersecting a column containing 2,4,5 and 8. If the intersecting square is blank, its only possible number will be 3. Furthermore, if the row or column intersects a region (both containing those same numbers) this can be a little harder to spot. The principle is identical in all three cases - how obvious it is depends on how carefully you are watching the grid.

Hidden Single
This is slightly different to the naked single. Instead of looking for a square in which only one candidate is possible, we are looking for a candidate which is unique to one square in a row, column or region. That square may contain other candidates, but if our "target" is unique to that square we can eliminate them all.

Naked Pairs
We are now embarking on the first of the main elimination processes. Identifying naked pairs does not necessarily mean the immediate conversion of a candidate into a placed number. Very often, it simply means that we can remove candidates from some squares because they must appear elsewhere. Here's an example:

In the left region we have two squares containing the candidates 1 and 3. The highlighted squares each contain one or both of these candidates. We don't yet know which way round the 1 and 3 will be placed in the left region, but we do know that they cannot survive as candidates in the highlighted squares, so we can safely eliminate them.

Hidden Pairs
The difference between naked and hidden pairs is the same as for naked and hidden singles. Imagine some or all of the numbers removed from the left region, and some of them placed as candidates in addition to the 1 and 3. Then imagine the candidates 1 and 3 removed from all of the highlighted squares. The 1 and 3 would thus become hidden pairs, and all other candidates in those two squares can be removed, since one square must contain 1 and the other must contain 3.

Beyond pairs, we also encounter naked/hidden triplets, quads, even quints. The principle leading to candidate elimination is exactly the same but, of course, when you have 3, 4 or 5 candidates in play they can be harder to spot.

Locked Candidates
The visual example above actually demonstrates the principle of locked candidates as well as naked pairs. One reason for removing the 1 and 3 from the highlighted squares is that they are restricted to that row in the left region. Here's another example:

Are all of the candidates in the highlighted squares valid? The answer is No. Looking at the middle region, the candidate 4s only appear in its top row. Since one of these must be correct, we can safely remove 4 as a candidate for both of the highlighted squares. Here's a harder example:

The top and bottom regions hold the clues here. What they have in common is their candidate 7s. In both cases these are restricted to either the middle or right column. Eventually, when the puzzle is solved, the top region will will have a 7 in the middle square and the bottom region will have a 7 in the lower right square - or it may be the other way around.

The important thing, though, is that however it turns out a 7 placed in either of the highlighted squares of the middle region would clash. So we can remove them.

This touches very gently on what is sometimes, rather grandly, referred to conjugate relationships. In plain English, a conjugate relationship is a yes/no argument.

Welcome to X-Wing!

The blue squares all contain candidate 3s - we just don't know which way round. But we do know that on the left one square must be 3 and the other must be 9. On the right, one square must be 3 and the other must be 7. To avoid a clash, that means one 3 will be in the upper row and the other in the lower one. That means we can eliminate 3 as a candidate from the three highlighted squares in the middle (which, happily, will confirm 5 as the number to go in the top left of these three squares).

And the rest...
Seasoned sudokulists will be familiar with the exotic names given to the "shapes" formed by these conjugate, yes/no relationships. Swordfish, Skyscraper, 2-String Kite, the gloriously named Squirmbag. They are all based on the same principle, namely that candidates which are restricted to (usually) two squares in a row, column or region necessitate their removal from other squares through which their connecting lines pass. There are too many variations to describe here, and refraining from doing so is probably a good idea anyway. The difficulty in providing examples is that solvers will be tempted to look for types based on precisely the same shape or layout as any given example, and in the real world of sudoku it doesn't work that way. Our X-Wing illustration is based on one version of a rectangle, and there are hundreds of potential rectangles in a sudoku grid.

The fundamental methods of logic are quite easy, so it isn't really a case of "learning" - it's all about looking.