Solve Sudoku - Coloring Technique with Examples

Get out the Crayola! Try and stay within the lines.

Despite the childish name, this technique is VERY powerful.

This technique is called by a few names. Amongst them are simple coloring, multi-coloring, colors. I prefer just coloring. I also prefer to extend this idea to include any technique that is performed on just one type of candidate - or just one number, if you prefer.

Xwings, Swordfish, Jellyfish, Finned Xwings, Finned Swordfish, Locked Candidates, and many other techniques involving just one type of candidate can be placed under the tag, coloring.

Simple Coloring Example

First Coloring Example

In this example, all the possible locations for 3's are highlighted by some sort of color. In column g, the 3's are limited to two locations, g4 and g8. If g8=3, then clearly e8 cannot be 3. If however g4=3, then h5<>3, thus e5=3. In this case, again e8 cannot be 3. The term, coloring is derived from the common illustration of this technique using colors to deduce the possible eliminations. In a proof, I would present this idea as:

  • fc on 3's: g8 == g4 -- h5 == e5 forbids e8=3
Whoa! What did I just write?!

Here is a brief key to terminology above:
forbidding chain
A == B
A OR B, equivalent to A Union B
A -- B
(not A) OR (not B), equivalent to A forbids B
shorthand for g8=number, in this case: g8=3
The reason that I introduce forbidding chains at this time is quite simple:

All the sudoku solving techniques, except perhaps those involving uniqueness of solution, fit 
under one umbrella,  Forbidding Chains and Forbidding Matrices.

Blog pages currently under construction will discuss forbidding chains in some detail.

More Complex Coloring

Second Coloring Example

This example is from the same puzzle as the first, after preforming the previously prescribed exclusion. The logic is colorfully highlighted, but here is an explanation:

  • If c6=3, then clearly d6 cannot be 3
  • Since 3's in column c are limited to c2,c6 - If c6<>3 then c2=3
  • If c2=3, then without even looking at the grid, e2 cannot be 3
  • Since 3's in column e are limited to e2,e4,e5 - If e2<>3 then 3's must exist within e45
  • Either one of e4, e5 being 3 prevents d6 from being 3
Conclude therefor that in every event, d6=3 is impossible and therefor forbidden. In a proof, this step could look like this:
  • fc on 3's: c6 == c2 -- e2 == e45 forbids d6=3
Again, do not be disheartened by the forbidding chain language. It is not required to understand this idea - it will help though, both in clear communication and thought once fully explained.

Technique names are almost as much fun as the puzzles. The example above is a Finned X wing. However, if one generalizes the idea of coloring, then learning each specific technique involving just one number is superfluous. The elimination above is missed by programs such as the solver in Simple Sudoku merely because of the grouping of e45=3. In my humble opinion, such a grouping adds little complexity.

Coloring Example From Tough Puzzle of 12/23/06

Third Coloring Example

This example is one step deeper then the previous two. Here we have:

  • 1's in box e8 limited to two locations: d8,f9
  • 1's in column g limited to two locations: g9,g3
  • 1's in row 1 limited to two locations:i1,a1
  • Clearly: d8=1 or f9=1
  • If f9=1, then g9<>1 thus g3=1
  • If g3=1, then i1<>1 thus a1=1
  • Thus, sans further grid examination, we know that a8<>1
This step could be written as:
 fc on 1's: d8 == f9 -- g9 == g3 -- i1 == a1 forbids a8=1

Interesting coloring example

Complex Coloring Example

In this example, think of the blue cell at c6 as the starting point.

  • If c6=1, then c3<>1
  • If c6<>1, then both a4=1 and h6=1
  • If a4=1 and h6=1, then neither a8=1 nor h8=1
  • If neither a8=1 nor h8=1, then e8=1
  • If e8=1, then e2<>1
  • If e2<>1, then d3=1
  • Conclude: If c6=1, then c3<>1. If c6<>1, then still c3<>1. Thus, c3=1 is not possible
This complex coloring idea could be presented in a proof as:
fc on 1's: d3 == e2 -- e8 =={fc on 1's: c6 == a4 -- a8 == h8 -- h6 == c6} forbids c3=1
 Do not be discouraged - most coloring that is needed to solve puzzles is not nearly
as difficult. Nevertheless, understanding an idea such as this one is certainly needed
to tame the monster puzzles without guessing.

Many solvers would use multi-coloring for the elimination above. In most cases, two colors is quite enough. In fact, one color is enough for the elimination above. Thinking clearly about coloring involves thoughtful partioning of native strong sets.

 Native strong set: Any grouping of puzzle possibilities that contains
at least one item that  must be true.

Diabolic Coloring Example

Diabolic coloring on 5's

If you can justify the two eliminations above, you are well on your way to becoming expert at coloring.

 fc on 5's: e7== f8 -- ai8,e7 =={swordfish on 5's at a237,e23,i23} forbids bg7=5

Symmetrical presentation

Diabolic coloring on 5's alternate conditional swordfish

The following puzzle links are provided for practice in coloring:

Indicate which comments you would like to be able to see

Hopefully, all of you had a wonderful !
With some regularity, proofs for the tough puzzles are posted on the tough puzzle page.

Sadly, I tend to be grandiose and disparage some of the less difficult puzzles. Please forbear my arrogance.

I searched my portfolio of such proofs, and have compiled the More...
I just made it through the first example - is there a typo in the proof? the text refers to 'e8 cannot be 3', but the proof says 'forbids 'g8=3' (i think s/b 'forbids e8=3').

Thanks for the great explanations, though - very enlightening!
This is totally awesome. I love learning the techniques. Thanks a million.
Thanks Susan!
You are correct, I did commit one of my frequent typo's. The page will be corrected.
The tough puzzle of March 16, 2006 is one of my all time favorite examples of coloring.
The proof on that page was written in my sudoku infancy - and not as well presented as later proofs. I think the suggestions for proof discussed on that day on that page helped significantly to open my eyes on how to solve these puzzles well - especially the help from Andrei.
Don’t quite follow you terminology.

For g8 == g4 -- h5 == e5
Does it mean (g8 == g4) -- (h5 == e5)
or g8 == (g4 -- h5) == e5
or something else?

Any precedence for the operators == and --
Hi pChu!
It means:
g8 == g4, g4 -- h5, h5 == e5.
No precedence is required.
Note: -- links are provided as a courtesy. They are obvious without grid consideration. The list:
{g8,g4}, {h5,e5} is all that is really required.
Eventually, the blog will give a full explanation of forbidding chains.
Thank you.
It makes sense now.
Thank you, Steve!

I had been somewhat discouraged from the study
of fc:s - until now. Coloring (together with practice) will undoubtedly make spotting these chains 'easier'. May be I should say 'possible'?

Hope you all enjoyed your Xmas. Gotta go and install the broadband! More...
I wonder which calender I consulted when I wrote June 31?
Steve, is there a way to put use Simple Sudoku to put in one's own puzzle so that you can color them? I am able to put in numbers, but then can't eliminate or color.
Hi Susan!
Yes - I do it all the time.
After entering in the numbers, under the file heading, you should see 'start'. After 'start', you can now manipulate/solve the puzzle as you wish. You can also save the puzzle. I do that all the time. Using Simple Sudoku, I have a sizable archive of More...
Thanks Steve,
I knew I must be missing something.
Steve, I am still working my way through this stuff - it provides a rationale for some of my intuitions/guesses. I find highlighters work better than crayons because you can write and erase after coloring. Thanks for taking the time to write all this out.
What is 'simple sudoku' that you refer to?? I thought I would have to print the puzzle out and then colour it which seemed a little tedious. Is there a better way to learn this?
Try this link: (for Simple Sudoku)

If you paste the link, you may have to remove spaces - spaces seem to creep into links posted in these comments....
Steve, you rock!
Some of these things I have worked out through your proofs posted throughout last year, but it is awesome that you've taken the time to put it in a blog. Thankyou.
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