Comments  Talk
... by: Jackg
In the third example down J9 is linked into box 6 containing more than two 5's G3 is not linked up into box 4 containing more than two 5's resulting in E3 seeing two colors. Any reason (besides giving an example of rule 5) that G3 couldn't be linked to E3 resulting in two colors (blue) in a unit therefore allowing for the removal of all blues (rule 2)?
... by: Steve Thier
I think rule 2 should include the condition that they be on the one chain.
Andrew Stuart writes: Good idea, I've added a line at the top before the first rule.
... by: td
andrew, great website. I have a suggestion and a question. Suggest the use of a tracing paper grid identical to the sudoku grid for coloring . The lines are easily erased leaving the sudoku uncluttered.. question: how do you go about choosing the candidate most likely to yeild results? Greatest number of candidates? If one has to check all of the possible candidates, it would be a very lengthy process. Any hints?
... by: fry
poderia me explicar melhor a cadeia de singles, eu nao entendi.......por favor....
obrigado
... by: Ian
Andrew, I'm puzzled by one aspect of the chaining and networking type strategies inasmuch as what sort of criteria do you use when choosing the starting tuple? In the example diagram 1 above, you start by saying; "Ive clicked on the 5 at C8 to highlight all the 5's in the puzzle...". But why 5? What determined that you looked at 5 to start with in that specific case? I'm trying to tie down  to define precisely  the differences between 'trial and error' strategies and 'logical' strategies  or what I like to think of as 'pattern recognition' strategies. At the time of asking this question, I put these single tuple type networking strategies in the 'trial and error' box but I desperately want for them to be moved to the other camp. However, for this to happen I need a 'pattern' type of reason. I hope this question makes sense. Thanks in advance.
Great site! Fascinating! Ian
... by: Mike Kerstetter
So just to clarify, am I correct in thinking that while the top diagram shows two unconnected 5 chains, I should not try to apply these coloring rules to multiple chains at the same time? I should apply them only to a SINGLE connected chain at a time?
Andrew Stuart writes: You can mix two isolated sets of chains but it's quite complicated. Have a look at: MultiColoring StrategyBut if you have two isolated chains you can use them individually to use Single Chains strategy  they shouldn't inhibit each other. Just don't reach across.
... by: Can I use singles chains in Windoku puzzles?
I am trying to get through a series on fiendish HyperSudoku puzzles (from an Android app  Super Sudoku). I recently found they are also called Windoku. I find that I can use singles chains when I have exhausted easier strategies. I can check my board and the app will tell where i have made mistakes and let me back up, so I know when I have made a mistake. HyperSudoku has a set of four shaded square areas overlaying the board (B2D4, B6D8, F2H4, F6H8) and they must be also filled in accordance with the normal rules.
Anyway, I find that sometimes singles chaining will indicate a number can be eliminated from a cell and the program indicates that it cannot. I'm thinking that either I am missing something, but cannot use your solver to find what I've missed. Maybe I can use the killer solver, but I think it works differently. I just want to know if I should be using singles chaining for this type of Sudoku puzzle and if your solvers can be used to solve them. I really like your solvers and they are very helpful in learning the various strategies.
Andrew Stuart writes: Hi Dan It is very dangerous to use the normal Sudoku solver for Windoku. Windoku has extra constraints (the four 3x3 blocks) that mean you can't assume certain things. All the normal Sudoku strategies apply but the normal Sudoku solver is not aware of the extra constraints so it will go down the wrong path eventually. It is also likely the solver will find multiple solutions as well. I intend to provide a solver for Windoku just as I have a seperate solver for Sudoku X and Jigsaw.
... by: Ludek Frybort
I found a sudoku, which can demonstrate what I find an interesting twist to the colouring strategies:
The interesting part comes if you run it through the solver to the point where it uses simple colouring on 9s. You get this: CLICK (but if you load this in the solver, you'll have to go through some pointing pairs and an Xwing to eliminate some candidates, which were already eliminated in the original process, to get to the point I'm talking about)
At this point (actually right before this point but the order doesn't matter) I tried to use simple colouring on 2s  unsuccessfully, nothing was eliminated. But since I did it on paper, the "colors" remained on the sheet. Now I correctly guessed it was the time to look for Ywings. Before I could spot the D2 E3 E7 Ywing, which the solver uses to solve the puzzle, I spotted another one at E4 F5 F9. The latter Ywing is useless itself, it doesn't eliminate anything, but I noticed that the 2s on both its ends have the same color from the previous attempt at colouring. So the logic of Ywing says one of them must be the solution and the logic of simple colouring says that if one of them is, the other must be, too. So I could conclude both must be the solution, quickly solve all the remaining 2s and the rest was trivial.
I'm posting this firstly to just share what I find an interesting puzzle, and secondly in the hope that maybe somebody with deeper understanding of the strategies might find a way to generalize it and maybe extend the power of the colouring strategies.
... by: John Riddle
OK I get it...E3 was eliminated by rule 5 on the previous turn leaving E1 and D3 as the only cantidates in box 4.
Andrew Stuart writes: Yes but I used a screen shot of the previous step incorrectly. I have updated the diagram with 5 removed from E3.
... by: John Riddle
There are three 5's cantidates in box 4. So why is E1 and D3 listed as a valid link? Please advise  I don't understand.
... by: Luke
How do you choose which colour to start a particular chain with?
Andrew Stuart writes: It shouldn't matter which colour your start with or where you start. You should more or less get the same result.
... by: Peter Foster
It appears to me that simple colouring and the 3D Medusa techniques are just another name for 'trial and error'. We colour a possibility and 'see where that takes us'. Maybe during the colouring it indicates that the number we started with can't be right or the same number in other parts of the grid can't be there. But essentially it seems to me that we are 'trying' a number and finding out whether it is valid or not. The only difference is that should we 'try' the number and it 'not' result in eliminations in other parts of the grid, then it can't be assumed that that number was correct. It's a negative rather than positive result. Am I misunderstanding something ? I use the 2 extensively, but it does seem like I'm getting shortchanged when compared to Xwing etc.
Andrew Stuart writes: I know where you are coming from. And there are degrees of "trail and error". I distinguish between the pattern found that leads to an elimination and searching for a pattern. The later can seem arbitrary long but the cut off between a logical strategy and trial and error is  is the logic based on a static pattern that exists on the board now  or have you modified the board by inserting and removing numbers? I try and be more clear about this subject on this page Crook's AlgorithmI hope that helps a bit
... by: Joe Badillo
I need help with example 2: Why is the 5 in A2 eliminated and not the 5 in J2 Why is the 5 kept in D3 and not the 5 in D6 Why is the 5 in A1 not colored at all and will eventually kept Simple coloring is not so simple for me Please help
Andrew Stuart writes: The 5 in A1 is critical to upsetting the neat array of pairs of 5 in rows, columns and boxes. It means there are three 5s in row A and column 1. That means no links can be made as part of the 'simple chains' web. But 5's across the board can be connected. It so happens that when they are connected with alternating colours (starting from any 5) the ringed 5s have the same colour. Since ALL of one colour is true and ALL of the other colour is false, the colour found twice in the row, column or box will be the false one, and all yellow can be eliminated.
... by: Michael Wallis
I was working on the this puzzle: Your sudoku solver returned the following when I clicked 'Take Step': "SIMPLE COLORING (RULE 4): Two different colors for 4 appear in Box 3 with more than two occurences for 4. The uncolored candidates can be eliminated from B8 C7". However, when I clicked on the explanation for 'Simple Colouring', I only saw references to Rules 2 and 5. What is 'Rule 4' and why isn't it in the explanation for 'Simple Colouring'. You did mention a Rule 4 for 3DMedusa, but that seems to be a different strategy than simple colouring.
Andrew Stuart writes: Correct, I hadn't provided an example for this Rule for Simple Colouring, and your example is very good, so I've poached it. Rule 4 now added to the page above. Thanks for sharing and pointing it out.
... by: Guy Renauldon
For Glisten and Donald Waddell
The candidates 5 are not shown in E3 in the introduction and in the second diagram because these diagrams represent a step after the third diagram (the third diagram is the first step if you prefer) First and second grid are identical.
To solve this sudoku two steps are necessary:
First step (third diagram): rule 5 allows you to remove the 5 in E3.
Second step (second diagram): the 5 being removed from E3 give a strong link between E1 and D3. Then you can use Rule 2 which allows you to remove all yellow candidates 5 in the second diagram.
Very good example of Single's chains: rule 5 followed by rule 2
... by: glisten
There seems to be a problem with the first three stages of the example sudoku. In the first two stages, E3 is shown only with candidates 1,2. But in the third stage, the candidate 5 appears. I believe candidate 5 should appear at stage 1 and 2. And if so, that would invalidate the links to the left and down from D3. You are way better at this than I am. What did I miss?
... by: Donald Waddell
Love the Sudoku site and increasing my knowledge of its logic. In following the example for Single's Chain, is there a reason why the candidate 5 in cell E3 is not shown on the introduction, but is shown on a subsequent diagram for the same puzzle for "Rule 5"?
Andrew Stuart writes: I believe the example in Rule 5 happened before the examples above it. I have to search quite hard to find examples, and they are not always in the order of the documentation. Only that really :)
... by: Zophuko
Rule 5, two colors elsewhere , appears logically quite transparent to me but I am having trouble understanding Rule 2, twice in a unit, regarding removing all instances of the number that occurs twice in one unit. Would appreciate a more detailed explanation of the logic underlying Rule 2.
Thanks for a wonderful wevbsite
Andrew Stuart writes: In the example for Rule 2, you will notice that A1 contains a 5 as well. If the eliminations of all one colour left no number X in some rows, columns or boxes, something very wrong would be going on. Infact, your chains would be very different in the first place. Rule 2 works because there are 3 or more candidates of X in such units. I suggest tracing the chain network from any starting point  you can only make chains along a unit with 2 candidates. You should get the same result every time.
... by: John
Thanks for the wonderful instruction. My question has to do with practicality/tactics for when one is doing a paper (not online) puzzle. Do you draw pencil lines between the numbers? Could get extraordinarily messy. What is the best way of applying this on in a pencil/paper mode?
... by: Zophuko
Thank you Andrew. The linking rules now appear very clear.
Another observation and question regarding your example next to the Rule 5 discussion above. It seems to me that, under the rules, the following connections could also be made: A4 to B5 to B1; B5 to H5 to H6 to D6 to D3; D6 to F4 to A4; F4 to F7 to G7. What is the reason you do not show those links?
Andrew Stuart writes: Several chains might coexist at the same time but with no link between them. In this example the solver has found one chain and highlighted those numbers which form that chain, but it is not required to highlight all links in all chains. I don’t believe any of the links you found can be connected to that chain. The solver only returns the first chain that actually does something  there might be other chains and there might be other eliminations on other chains at this point. I am planning a version of the solver which will allow one to cycle through every elimination at a given point, which might show shorter solve routes over all
... by: Zophuko
Thank you Andrew. It is now very clear. Also, it appears to me that if intrabox links are omitted, one obtains only possible Xwing or swordfish patterns.
... by: Zophuko
RE: constructing single chains.
Referring to the example, I do not perceive clear and restrictive rules for not drawing links between the following: A2 and C3, D6 and E7, H6 and G7, D6 and F7, A4 and D3, A1 and E1; or B1 to E1, or A1 to D3 leaving B1 unlinked.
Someone please explain the rules that prevent these kinds of links but allow those shown.
Andrew Stuart writes: The connections you list are between cells that do not see each other, that is they share no common row, column or box. Not every cell can 'see' every other cell. A link is always along one common 'unit', sometimes two (same row and box for example).
... by: Ed
I'm fairly new to Sudoku puzzles and have enjoyed reading through the explanations for the various strategies presented on this website. The step by step solver with the detailed explanations that you provide seems like an excellent way to approach figuring out the most difficult puzzles. Thanks for all of the hours you've put in to this website.
From reading through the posts below it looks like there may be some confusion regarding single chains and the explanation above. Some of the user comments seem to prove the logic or point out alternative methods to solving the example cells in question, but it still looks like there may be a bit of confusion on, for example, where to start, what color to start with, etc.
Basically restating what you've presented above...
1) Pick a number and link all of the single chains: A single chain occurs in a row, column or box (9 cells) when there are only two occurrences of that number in the given row, column or box.
From the first example puzzle above you chose the number 5. Notice that box 1 has three 5s so there are no single chains (both starting AND ending) in that box. Column 2, however, has only two 5s in it and they are connected as a single chain (A2>J9: both starting AND ending in that column). The 5s in Row D get connected (D3>D6) because there are only two in that row. Notice that the 5s in Row E don't get connected because there are more than two in that row. This process is repeated until all of the single chains are completed and you end up with what you've shown in the first example puzzle above.
2) Add the alternating colors: Start anywhere in the puzzle with any the chains you've just laid out and color one of the cells that is pointed to by one of the chains. It doesn't matter where you start. From there, branch out, making sure to alternate colors as you move along the various connections.
3) Eliminate the numbers in the cells (if possible) using the rules provided above:
Rule 2 (Twice in a Unit): If you see a box, row or column with two cells that have the same color, remove all instances of the number that share that color.
In the example above, there are three instances of units that contain more than one 5 with the same color: Box 1: (A2 & B1), Row A (A2 & A4) and Column 1 (B1 & E1). In addition to the four cells (A2, A4, B1 and E1) that are colored the same in these three units, all of the cells of the same color can be removed.
Rule 5 (Two Colors Elsewhere): If there is a cell that can "see" two other chained cells with alternate colors, it can be eliminated.
The third example puzzle above shows the single chains drawn in blue (I think the G7>J9 link is supposed to be blue to show it is also a chain). Cell E3 sees a blue 5 (E9) and green 5 (G3) and therefore the 5 in cell E3 can be eliminated. Notice that the 5 in cell E3 was not part of the original network of single chains.
Basically, Rule 2 helps eliminate candidates that are part of the network of chains and Rule 5 helps eliminate candidates that the network of chains can "see".
Reasonable?
... by: John
I believe the easiest way to state this rule is
We have a chain, between PAIRS of a DIGIT, which has an ODD number of links. Any cell which can see BOTH ends of the chain cannot have that DIGIT.
Why? See the diagram where we have a chain between pairs of 9. One end is C2, the other end is G7. Case 1: Try C2=9, follow the chain, G7="not9". Case 2: Try C2="not9", follow the chain, G7=9. One of these cases must be right but we don't know which. If Case 1 is true, and C2=9, then C7 cannot be 9 because it can see the 9 at C2. If Case 2 is true, and C2="not9", then C7 cannot be 9 because it can see the 9 at G7. So C7 cannot be 9 in either case  because it can see both ends of a chain with an odd number of links.
It does not matter "where we start".
If we have a chain of pairs of digits with more than 3 links, we can look at all the subchains which have an odd number of links. So, with 4 links pq, qr, rs, st, we first look at pq, qr and rs. Any cell which can see both p and s cannot have the digit. We also look at qr, rs and st. Any cell which can see both q and t cannot have the digit.
... by: Lenni Nero
Best site I've found yet. Might I please ask, for those of us who are redgreen colourblind (about 10% of males), would please avoid using those colours together whenever you do a rewrite of the examples. TIA.
... by: ozzy
I get stuck often at "SIMPLE COLORING (RULE 2)" where the only colour is green and (and yellow) not combined green & blue as in example above (where it is shown the combination beteween those two colours, plus yellow for the "unwanted candidates"). In my solutions are only green and yellow. Am I missing something? The second rule is very hard for me to understand. Thank you for a wonderful site!
... by: Robin
Hi, Many thanks for the simple colouring explanation; that helped with a puzzle I couldn't solve; however the "simple colouring Rule 5 example" appears to be a very simple puzzle that does not require anything beyond very basic skill to solve. This must be an error ?
... by: Mike
X and A form a conjugate pair, so why can't X be one end of the chain and D the exlusionary square?
... by: Marge Falconer
Agree that this explanation remains confusing. However, using the basic premise of the chain, I've just tried to replace at any point in the chain the number I'm dealing with and one answer soon becomes apparent. Unfortunately, I need more practice with this strategy and only come across it rarely. I would love a site that explained a strategy and then gave you three or four puzzles which would require you to use it.
... by: Lea Hayes
Thank you for your interesting descriptions on various Sudoku strategies.
What is the difference between single's chains and xcycles? Do xcycles always cover the same ground, or are they completely different?
... by: Ben Wearn
The 11link chain in the last example may be shortened to a 5link chain by linking B5 to A6.
... by: Steve
Colouring example 1 in fact shows rule 2, not rule 1: A and X are a conjugate pair, and there is no justification for arbitrarily ending the chain at A instead of continuing it to X. If you add 5 as a candidate in F3, then you will have a valid example of rule 1.
Interestingly, though, example 1 also has an Xcycle with a weak link discontinuity at X, which provides a different reason for eliminating 5 at X.
... by: Mike Wallis
"Another way of looking at this is the popular technique of Colouring". What does "this" refer to?
The explanation for Rule 1 states that 5 can be removed from X (G3) because it's outside the chain and points to A (D3) and D (G8).
However, the explanation for Rule 2 states that X (G3) and D (G8) are the "false' color because they're in the chain, both are blue, and both in the same row.
How can a square be both in a chain and outside of it at the same time? Moreover, Example 1 shows X (G3) and D (G8) in two separate chains  not in "a' chain.
There is another chain that runs from F to C, C to D, D to E, E to B, B to A, and A to X. However the example for Rule 2 doesn't show that.
In fact, none of the examples on this page show Rule 2 in action. They only show Rule 1.
The disconnect between Rule 2 and the examples has been very confusing to me and has caused me difficulty in solving puzzles using this technique. For this reason, I think the article for Singles Chains needs to be revised with new examples that more accurately reflect Rule 2.
... by: Don
Cell X cannot be green because cells D and X do not form a conjugate pair. That's because two other cells (G1 and G2) in row G contain 5 as a candidate. In column 3, however, cells A and X are the only two that contain 5 as a candidate; hence they do form a conjugate pair, making cell X blue.
... by: Clark
If you start with E=green, then B=blue, C=green, D=blue, X=green. 'A' now looks like the exclusion. Can you explain this better? Why is X the exclusion and not A?
... by: Semax
Every part of the chain links two cells which share one or two units, but these and only these shared units have to fulfill the condition that there can't be any other cells containing the candidate.
So in the first example you can link D7 to E8 because in their shared unit (box 6) there is no other cell containing a 5. In other words, if D7 is 5 then E8 can't be 5, and vice versa. Therefore it doesn't matter what happens in rows D and E or in columns 7 and 8.

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