You can also use this technique if you have more than two candidates. For example, let us say the pairs circled in red were instead triple candidates of the numbers 1, 4, 7. This would mean those three numbers would have to be placed in either rows 1, 2 or 5. We could remove these three numbers as candidates in any of the remaining cells in the column. Hidden subset This is similar to Naked subset, but it affects the cells holding the candidates.

In this example, we see that the numbers 5, 6, 7 can only be placed in cells 5 or 6 in the first column marked in a red circle , and that the number 5 can only be inserted in cell number 8. Since 6 and 7 must be placed in one of the cells with a red circle, it follows that the number 5 has to be placed in cell number 8, and thus we can remove any other candidates for this cell; in this case, 2 and 3.

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X-Wing This method can work when you look at cells comprising a rectangle, such as the cells marked in red. In this example, let's say that the red and blue cells all have the number 5 as candidate numbers. Now, imagine if the red cells are the only cells in column 2 and 8 in which you can put 5. In this case you obviously need to put a 5 in two of the red cells, and you also know they cannot both be in the same row. Well, now, this means you can eliminate 5 as the candidate for all the blue cells. This is because in the top row, either the first or the second red cell must have a 5, and the same can be said about the lower row.

Swordfish Swordfish is a more complicated version of X-Wing. In most cases, the technique might seem like much work for very little pay, but some puzzles can only be solved with it. So if you want to be a sudoku-solving master, read on!

Example A In example A, we've plotted in some candidate cells for the number 3. Now, assume that in column 2, 4, 7 and 9, the only cells that can contain the number 3 are the ones marked in red. You know that each column must contain a 3. Example B Look at example B.

We can eliminate 3 as candidate in every cell marked in blue. The reason for this is that if we consider the possible placements of the number 3 in the red cells, we get two alternatives: either you must put 3s in the green cells, or in the purple cells, as example C shows. In any case, each of the columns 2, 4, 7 and 9, must contain a 3 in one of the colored cells, so no other cell in those rows can contain a 3.

Example C How do you recognize a swordfish pattern? You also managed to get the two diagonals to add up to 87, as well as the four numbers in each of the corner quadrants …. Unfortunately, the two side quadrants will not add to 87 but the four corners of the large grid will! Save that for the magic square that you make for your guests. Because 87 is an odd number, we had a remainder that we needed to use for the boxes with 13, 14, 15 and 16 in them. And then you divide 20 by 4 to get 5 with no remainder! All you have to do is add 5 to each of the 16 numbers in your new grid and it will work.

Incidentally, if your target number is even, then those two side quadrants will also add up to the target number. If you want to make this look tougher, you can memorise your original magic square that adds up to Also, instead of working straight across the rows one at a time, you can ask your spectator to point to any empty box and you can add your special number or special number plus remainder to whatever number belongs in that box.

This article appeared in Cosmos 66 - Dec-Jan under the headline "Magic by the numbers". Astronomers have solved the mystery at the heart of the beautiful Rosette nebula. Digital Issues Buy a back issue. Renew my subscription Give a Gift Manage my subscription. Features Mathematics 23 November How to solve a magic square. Learn the secret of the magic square, then you too can dazzle onlookers with feats of mathematics.

Jason England is a magician based in Las Vegas and a renowned authority on casino gambling and card handling.

Looking for more science? Click here to see our subscription options. Click here to see our gift options. Our Sudoku Dragon software has a free tutorial that explains twins in more detail with an animated guide.

### SOLVING SUDOKU

This rule is named the hidden twin rule as the twins are only found by considering other squares in the group. Discovering the twins is the challenge. Another way to exclude possibilities in a group is with naked twins. In this case the twin squares are evident on their own and so they are termed 'naked' to distinguish them from the previous 'hidden' case and these are used to exclude possibilities in other squares in the same group.

Here's how it works. This 4x4 Sudoku has the region Ca highlighted in green. Because these two squares have no other possibilities we can deduce that a 2 must go in Ca and 3 in Cb or else 3 in Ca and 2 in Cb , there are no other alternatives for these two squares. So looking at square Da the naked twin rule excludes 2 from occurring here because we have just shown that region Ca must have a 2 in either Ca or Cb. As Da is now left with one possibility a 1 can be safely allocated there.

For the same reason the naked twin also eliminates 2 from square Cc and a 4 must go there. The two 'twin' rules are particular examples of the general Sudoku logic. It is all down to permutations. Each Sudoku group is a permutation of the numbers 1 to 9 for a 9x9 grid.

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## Mathematics of Sudoku

If you can identify a group within this permutation that is restricted to the same number of squares then you have a Sudoku permutation rule. Note : In fact the 'only square'; 'single possibility' and 'only choice' are just special cases of this general rule - only one square is involved in this case. This general rule has more exotic applications. The twin, triplet, quadruplet rules just reflect different number of possibilities 2,3, However there are also chains. Detecting this chain lets you safely exclude a possible 1, 4 and 7 elsewhere in the same group. So the logic applies equally for chains as it does for twins, there are 'naked chains' and 'hidden chains'.

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One of the more complex Sudoku strategies is the 'X-Wing' and its cousin the 'Swordfish'. These rules are useful for solving the really difficult Sudoku puzzles when all else has been tried and failed. In looking for twins and permutations we restricted ourselves to looking at possibilities within a single group. The shared sub-group rule is the simplest example of a rule where two groups are looked at to eliminate possibilities.

The X-Wing requires looking at multiple groups as well. A better name for this strategy might be 'Box' as you are looking for four squares forming the corners of a box. These squares must be the only permitted squares for that number in that row or column for one particular symbol.

## The Sudoku solution | Science News

This box arrangement forms a two dimensional link. If the symbol spotted occurs in the top left corner of the box it must then also occur in the bottom right corner of the box. The only other alternative is that it occurs in the top right corner in which case it must then occur in the bottom left corner. No other option is possible for these four squares and this number.

Just as with the sub-group rule, this can knock out possibilities somewhere else in the Sudoku grid. Here's an example and good X-Wings are hard to find. Sudoku Dragon has highlighted all the squares where a 4 is allocated or looks like it can be allocated. The rows C and G are crucial. They both have only two squares that can take a 4 : Ca, Cf, Ga and Gf - highlighted in blue - this is the vital starting point. Moreover these 4 s form the corners of a rectangular box highlighted in orange. How is this useful? Well, because there must the 4 in column a must either go in Ca or Ga and nowhere else in that column.

Similarly in column f the 4 must be in either Cf or Gf and so we can exclude all the other 4 s from these two columns. So all the yellow highlighted squares Aa, Ba, Bf and Ha can have the possibility of 4 safely discounted. If you are lucky then eliminating the 4 s will mean you can allocate one of these excluded squares or at least a square in the same group as them. Believe it or not the complexity does not end at the X-Wing, the Swordfish is a further refinement of the X-Wing. Instead of four squares forming a box of possible allocations the Swordfish rule uses six squares.

In the example puzzle there are not just two pairs of squares for 9 but three pairs: in columns b ; e and h. They are linked by rows to form a box with an extension 'sword' jutting out on one side : hence the term Swordfish. The other squares forming the swordfish are highlighted in orange and yellow. Because all these three columns have coinciding end squares the rule applies again.

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Any 9 s that we find in rows that link the columns can be safely excluded because we know that a 9 must occur in one of the two highlighted squares in that row; they can't be allocated elsewhere. These excluded squares are highlighted in yellow Fc; Gf and Fg. Of course, the Swordfish is not the end of the matter we can extend the logic to four interlinking pairs of possibilities and then five etc.. You'll feel a real sense of achievement if you locate a Swordfish and use it to solve a Sudoku puzzle.

Further complex strategies are available for fiendishly difficult puzzles. They require a lot more thought and analysis to learn about and use correctly. When all else fails, there is one technique that is guaranteed to always work, indeed you can solve any Sudoku puzzle just using just this one strategy alone. You just work logically through all the possible alternatives in every square in order until you find the allocations that work out. If you choose a wrong option at some stage later you will find a logical inconsistency and have to go back, undoing all allocations and then trying another option.

Because there are so many alternatives billions you won't want to use it too often. You start with a square and choose one number from the available possibilities. This is a completely different type of strategy as it uses 'brute force' rather than 'logic'.