# Cycles of two symbols to solve Sudoku puzzles

### Extending the excluded twin rule to cycles of symbols.

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**Cycles and Twins**[Game solving Strategies forum]

Many Sudoku puzzle books describe the 'twin' rule and I see that SudokuDragon has a tutorial on the 'twin exclusion' rule. I know that this can be extended to triplets, quadruplets, quintruplets etc. - although I have yet to find an actual real use for it with five symbols involved.

However, what I have not seen discussed is the extension to chains of symbols. Let me try to explain. The 'twin' rule says that if you have two squares containing two numbers and those are the only occurrences of the number in the region then you can safely deduce that any other possibilities for those two squares are excluded. Not very useful in itself but often this exclusion helps you to work out that the excluded number is therefore forced to go in some other square. Now if instead of twins you have a 'chain' of symbols, say in three squares you have all the possible occurrences of the symbols 3;4 and 5 say. The possibilities are '3';'4' in one square '4';'5' in another and '5';'3' in the third then there is a chain of three squares and just as in the case of 'Twins' all other possibilities are excluded in these squares.

Has anyone else comments on using this technique. It does not seem to be supported by SudokuDragon at present.

**Cycles coming soon!**(re: Cycles and Twins) contributed by

**SudokuDragon**

The support for spotting exclusions for chains of symbols not just twins or triplets is under development and should be available soon. It is quite a tricky bit of implementation to get right

**Cycles arrived at last!**(re: Cycles coming soon!) contributed by

**SudokuDragon**

At long last the support for detecting chains (or cycles) of possibilities has been added to release 15.

It should detect 2;3 and 4 membered chains of indefinite length, for example squares with possibilities 1 5; 5 2; 2 6; 6 1 would constitute a chain of 4 squares with 4 symbols.

**Extra cycles**(re: Cycles arrived at last!) contributed by

**Huw**

The new sudoku cycle analysis is very useful but I am still left with lots of puzzles that it doesn't work out a solution for. Quite often there are groups of the type where there are three squares with related possibilities say: 1;3, 1;3;5 and 3;5. Is it possible to do anything with these?

**No extra Cycles**(re: Extra cycles) contributed by

**Alexander**

No in general you can't make anything of the three squares 1;3, 1;3;5 and 3;5. This is because all the possibilities in the three squares are genuine, it could be 1;3;5 or 3;1;5 or 1;5;3. However all other possibilities for 1;3;5 can be discounted in the remaiing squares in the group, I am not sure offhand whether SudokuDragon checks that.

**General Permutations**(re: Cycles and Twins) contributed by

**John Wood**

The chains are just a special case of the general logical rule about subgroups of permutations.

If you have **n** squares with only **n** possible values to go in them within one group you can use this to exclude possibilities elsewhere in the group. This general rule covers twins (hidden and naked), triplets, quadruplets... and chains as well.

For example a simple twin could be two squares with just two possibilities say {4,7} so 4 and 7 can't go elsewhere. A chain could have 3 squares A possible {2,6}; B possible {6,8} and C possible {2,8}. The three squares A;B;C have only 3 possibilities {2,6,8} between them so the same exclusion logic can be used.

Chains are however much more difficult to spot and reason about than twins and so I think that is why you won't see them mentioned so much.

**Naked adjacency**(re: Cycles and Twins) contributed by

**Paul Kimball**

As a novice newcomer please forgive what may be an ignorant question.

Does a naked twin need to be adjacent to one and other? Or may it be separated and still be a valid exclusion instrument?

**Adjacent in groups**(re: Naked adjacency) contributed by

**John Wood**

Hello there and welcome.

The 'naked' and 'hiden' twins and triplets etc. only have to be somewhere in the same group (a row or a column or a region).

So that they could be at opposite ends of a row or column but they can be adjacent too - That makes them easier to spot!

**Naked and hidden**(re: Cycles and Twins) contributed by

**Nivlem**

In the Sudoku Dragon puzzles I have looked at the software only seems to detect and highlight 'hidden twins' and does not spot the more easy naked twins is this true?

NOTE:A hidden twin is where you have two numbers that only occur twice in the same two squares. You can use this to exclude the other possibilities in either of the squares. So a sudoku square with 1;3;4 as possibilities and one with 3;4;8 AND where 3;4 can occur nowhere else in the group then {3;4} is a hidden twin and '1' can be excluded from the first square and '8' from the second one.

NOTE:A naked twin is where you have two numbers that form a pair that occurs again in the same group. So if one square has 2;5 then a pair square must also have 2;5 as the ONLY possibilities. This stops '2' and '5' occurring elsewhere in the group - it must be in the two squares. So it is a different sort of rule to the 'hidden twin' rule.

**Triple cycles**(re: Cycles and Twins) contributed by

**John Wood**

You may be alarmed to know that the example of chains in groups of squares is just the beginning. There do not have to be two symbols in squares to form a Sudoku chain. So a chain of {1;7} {7;2} and {2;1} in three squares is just the very simplest case.

In four squares you can have (once again as the simplest case) {1;4;7} {4;7;2} {7;2;5} {2;5;7} and {5;7;1} (with these as the only places that 1;4;7;2;5 can occur in the group.

Looking closely you'll there are no pairs or triplets at all and would quickly miss any further analysis but these five squares only have five values and so form a chain. So you can be sure that all other possibilities that might seem possible on these squares can be safely excluded.

These are very hard to spot!

**Using cycles**(re: Cycles and Twins) contributed by

**Alexander**

Yes, I've used the cycles (chains) for solving the 'hard' puzzles from time to time. They do not crop up all that often though. They are to my mind a method of last resort when all my other strategies have failed.

**Cycles solve all**(re: Using cycles) contributed by

**Lynne**

It's my view that you can solve any Sudoku puzzle using the cycle rule and you never need the backtrack rule to solve a puzzle. Perhaps I haven't come across the right puzzle yet that would prove otherwise. Have you an example to clinch it?

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