Colors Redefined: Multicolors no longer useful.

TheSpaniard

Good day to anyone. This post will be long, but its importance deserves it.

I will redefine the technique "Colors" in such a way that the standard technique will be much more efficient, and, it has a bonus: You can forget Multicolors forever: it will be no longer useful.

The story:
Several months ago, I accidentally discovered one very powerful way to extend and probably complete "Colors", in such a way that it makes "Multicolors" obsolete. And, the better, was very simple to spot.

I must say at this point that I'm interested in solving Sudokus exclusively by human means. Then, a lot of complicate and exotic techniques that computers can implement are completely unuseful for me, due to the fact that they are almost impossible to spot by humans. During some weeks I have made some investigation and theoretical work, and I am now sure that the following is right.

This new technique (better: Extension of the standard Colors technique) is called "X-Ray", because it works in a very similar way than the also known X-Ray technique in chess. In this technique, you "look behind" one or several pieces that are in front of your tower, bishop or queen, imagining the effect on the row, column or diagonal if this piece were removed. The principle is similar, because you watch the peers (row, column or box) of one cell marked with a color (let say green) after applying standard Colors, using X-Rays to find what happens to other cells, columns or boxes in order to colour them.

Here is the NEW DEFINITION of Colors:

1) Once selected one digit candidates, select one pair of conjugate cells (only two possibilities in a row, column or box), and colour them with two alternate colors (let say green-blue).
2) If one of these two cells are conjugate of another cell, colour this new cell with the opposite colour. This process is recursive, and ends when all the chain of conjugates are alternatively coloured green-blue (or whatever colors).

... Until here, it is the Standard "Colors". Here comes the new stuff:

3) If ALL BUT ONE (called the Exception) of all the candidates in a house (row, column or box) are peers of cells coloured with the same color (eg: blue), then COLOUR THE EXCEPTION WITH THAT COLOR (blue, in the example). This process is recursive, and ends when you cannot find any more cells to colour.
(IMPORTANT: If as result of this process you colour blue one cell that has a conjugate not been coloured blue or green until yet, YOU CAN NOT COLOUR THIS NEW CELL WITH THE OPPOSITE COLOR. This is made in steps 1 and 2 of the technique, but we are now in step 3).

4) Once marked green-blue all the possible cells:
4.1) If one cell has two or more peers coloured with both colors (gree AND blue), you can safely exclude this cell as a candidate for the digit. (Same than in "Colors").
4.2) If two or more cells of the same house (row, column or box) are coloured with the same color , then you can conclude that THE OTHER COLOR IS TRUE. Note that is very common to find definitions for this case in which you can exclude the candidates coloured with these "autodestroying color". In "Colors" is actually the same, because it only deals with conjugate pairs of cells, and then "excluding one color" is the same that "making true the opposite color". Now, it is different, and a "Color Wrap" in SudoCue terminology determines not that this color is false, but that the other is true.

5) As a consequence, you can throw all the Multicolors, Ultracolors and XXX-Colors to the Trash Bin. All situations in which Multicolors can be applied are solved by this Newly-Defined-Colors technique. Even more, you can find now some exclusions that Multicolors cannot.

And that's all. As a last proof of the theory, I have tested all of AngusJ Multicolors**.ss Sudokus. All of them were solved with simple Colors as defined here. No need of Multicolors.

A human can easily spot this situation. Only needs TWO colors. Is quite easy of understand and apply. After this post, everyone knows this finding. Congratulations to all Sudoku players, from now on, it will be more easy and funny to solve hard, extreme, evil, unfear, etc Sudokus.

And now, some examples:

1)

angusj · Site Admin

Unfortunately you've pushed the logic of your 'X-ray' technique too far and so it's not always valid. Your final solution above is a good example. You say you can colour r2c9 the same colour as r1c6, which if you complete the puzzle is WRONG. However, if there wasn't a candidate 3 in r1c5 then you'd be safe to extend the colours as you suggest.

Always keep in mind BOTH of the following questions:
1. if this cell is TRUE what does this tell me about another cell?
2. If this cell is FALSE what does this tell me about another cell?

Ruud · Site Admin

This technique does the same type of coloring as weak coloring described by David Bird. It is essentially a forcing net restricted to single digits.

Like in David's method, Pedro extends a basic coloring pattern with forcing nets for one or both colors. Only the core coloring pattern is fully conjugate. The extensions are not conjugate, but because they are forced by the core colors, one of these 2 sets must be completely true, which allows us to do the same eliminations as with SC/MC.

The only issue I challenge is the "freshness" of this technique. I already pointed Pedro to David's post. The only differences between these techniques are the name and the details of their implementation. Weak coloring is actually more powerful because it allows grouped candidates in the extensions, where X-Ray is restricted to single candidates.

The "backtrack" proof for this technique is a Nishio test for the eliminated candidates.

Ruud

Jean-Christophe · Posted: Fri Sep 01, 2006 12:46 am Post subject:

I don't see anything new here : simple colouring, strong links, conjugate pairs are all the same for me.

For the first puzzle :
R3, C5, N9 and R9, C8, N2 forms a Generalized Swordfish (222) on 3 -> not elsewhere in R9, C8, N2 -> R9C2, R12C8 <> 3
But after that it requires an XY Wing to solve (for me)

For the second puzzle (the simplest for me) :
Turbot Fish on 3 with 3 links : R9C3 == C3 == R1C3 .. R1 .. R1C9 == C9 == R7C9 -> R7C2, R9C8 <> 3

(the "==" are for strong links / conjugate pairs, the ".." are for weak links / peers / buddies)

For the 3rd puzzle :
Either a Strong chain with 5 links (6 cells) :
Turbot Fish on 3 with 5 links R7C3 == R7 == R7C9 .. C9 .. R2C9 == R2 == R2C5 .. C5 .. R9C5 == N8 == R8C6 -> R8C1 <> 3

Or a grouped Turbot fish (similar to finned X Wing) :
Grouped Turbot Fish on 3 with 3 links R2C5 == R2 == R2C9 .. C9 .. R79C9 == N9 == R9C8 -> R9C5 <> 3

For the latter : either R2C5 = 3, or else R2C5 <> 3 -> R2C9 = 3 -> R79C9 <> 3 -> R9C8 = 3. Thus either R2C5 = 3 or R9C8 = 3
_________________
Jean-Christophe
"When you have eliminated the impossible, whatever remains, however improbable, must be the truth." Sherlock Holmes.

TheSpaniard · Posted: Fri Sep 01, 2006 8:34 am Post subject:

Good day to everybody, and thanks for your comments.

Angus: I didn't said that "AT THE END" R2C9 and R1C6 MUST contain the same DIGIT. I said that, in this position they HAVE THE SAME COLOR. In fact, I have stated that in the case 4.2 (which happens in example 3), you cannot exclude the blue cells, but trueing the Green cells. This leaves some blue cells misconnected, that are not useful from now on and must be uncoloured (Perhaps I should include that statement in the definition, I thought it was obvious, but you made me clear that it is not).

Maybe the better is to rewrite the 4.2 rule, adding at the end:

"Two or more cells having the same color means that, in this situation, if one of them is true, then the rest of the cells are true too. But two or more cells having the same color DOES NOT MEAN that if the other color is true, they are then false."
The rule must be applied as I stated in my previous post, with this addition to make more clear the rule.
What I do assure you is that it ALWAYS WORK.
If it is "mathematically correct" or not, I ignore it. Please remember my aim: Solving Sudokus by human means. And the Colors rule as stated in my previous post works, and its very simple to spot.
Again, Angus, thanks a lot for the feedback.

Ruud: Thank you again. Yes I saw the weak coloring described by David Bird. Surely it is completely correct; I am sure it is better founded than this X-Ray technique and probably it is a generalisation of X-Ray. But I hardly could understand it, and I am sure that I could never use it solving sudokus by hand.
If people think this X-Ray technique is useful, here is. If not, Ok. I posted it only to share my discover with the community.

Jean Christophe: Thank you for your evaluation.
Yes, you can apply several methods to solve the third example.... if you are a computer. Not me, I couldn't.
I am afraid my limitations as human player avoids me to spot conjugate pattern or Nishio or whatever else to solve this sudoku (Sorry, but I don't want to spend hours to solve one single Sudoku. My limit is exactly the same than Simple Sudoku: techniques more complicated than XY-Wing or maybe XYZ-Wing in some cases, are completely outside my limits).
I solve sudokus by hand, and that's my only aim. Again: I say that the technique ALWAYS WORK, at least I am using it several months and it always worked in hundreds of Sudokus, and tested against many sudokus from different sources. I you prefer other techniques, Ok, I understand. Everyone finds funny different things, thats why life is so funny...

Anyway, thanks to you all.

angusj · Site Admin

Ruud · Site Admin

TheSpaniard · Posted: Fri Sep 01, 2006 11:22 am Post subject:

You're right. But you are applying the axiomatic definition of the Standard Color technique, that only works with conjugates, to a New one technique that perhaps is different, and needs different axiomas.
In this new X-Ray Colors technique (expanded Color, if you like), it is no longer needed that condition. The constraint is, as I explained above that "Once coloured the cells using steps 1,2,3, If one cell coloured with color "A" is true, then every cell coloured with color "A" is true too." Nothing more ahead of that. The "B" cells can be true or not, it doesn't matter.

I agree that the technique may be flawed, but it is due to the fact that I have forgotten the Logic I studied at the University thirty years ago. What I only say is: X-Ray Technique Always Works. Never Fail. Maybe there is the need to redefine the fundamentals, I don't know, but
* It is easy to find by humans.
* It overlaps both Colors and "Multicolors" techniques, that may be removed from the arsenal of techniques.
* And in a few cases, it even find some exclusions Multicolors cannot.

Using it or not, it's up to everybody. I am personally happy to contribute with that to the repository of information about such a funny puzzle.

Angus, thanks a lot again, discussing with you is a pleasure. Gracias.

Buenos dias a todos.

TheSpaniard · Posted: Fri Sep 01, 2006 11:34 am Post subject:

Ruud, thank you both for answering and for putting the schema in a way it is easily understandable. I confess I am dumb in html and that.

Again: In this schema, the ONLY thing you can assure is that the green cell (R8C3) is true. Once you set 3 in this cell, the cells in darker blue disappear (they are conjugates). The cells in lighter blue do not disappear. They remain misconnected. They are unuseful from now on. You must uncolour them and find other schemas (hidden singles, and so on). They have served to state that all Blue cells together cannot be true, and therefore green is correct now. Its utility after that is none.

Again, I am sorry not to have the mathematical skills to better found this. The only important thing is that it always works and is very easy to spot by mere humans. That's all.

Gracias, again, to you all.
Buenos dias.

Jean-Christophe · Posted: Fri Sep 01, 2006 12:28 pm Post subject:

Angus,

This is a weak link (one way, not both ways)
R1C6 = blue -> R2C9 = blue.
But it does not work the other way :
if R2C9 = blue we cannot conclude that R1C6 = blue.
But we can conclude is that either R1C5 or R1C6 is blue (the solution is both R1C5 & R2C9 = 3)

TheSpaniard made it clear at its step 3 : you cannot make the other cell green (which is the equivalent of a weak link).

TheSpaniard,

I just wanted to inform you that this isn't really new. You are just applying Grouped Turbot Fish AKA finned XWing without even noticing it Wink

I think there are already too many names for just the SAME logic.

I'll translate the grouped Turbot Fish terminology into your XRay. For the 3rd puzzle, we have a Grouped Turbot Fish : a strong chain from R2C5 to R9C8 going through R2 (strong link), C9 (weak link), box 9 (grouped strong link). Using the true/false logic :
Either R2C5 = true or else R2C5 = false -> R2C9 = true -> R79C9 = false -> R9C8 = true
Thus either R2C5 = true or R9C8 = true -> R9C5 = false (ie <> 3)

Translating this into your XRay terminology :
R2C5 = G, R2C9 = B -> R9C8 = B
And since R9C5 is both under control of a green cell (R2C5) and a blue cell (R9C8), it cannot be 3.

This is exactly the SAME logic.

Kind regards,
_________________
Jean-Christophe
"When you have eliminated the impossible, whatever remains, however improbable, must be the truth." Sherlock Holmes.

angusj · Site Admin

TheSpaniard · Posted: Fri Sep 01, 2006 2:06 pm Post subject:

Jean Christophe:
Thanks for the acclaration.

I am sure you're right. I don't claim to be the first finding the technique. I said it is very easy to understand by humans. Actually, I was very surprised when I haven't found any similar, when it seems to be much simpler than Aligned Pairs, for instance. And people have formalised Aligned Pairs, so I couldn't believe I was the first. You say it is the same technique than turbot-fish AKA Finned X-Wing, at least in this example it is. I don't know if it is possible to generalise that X-Ray will always have the same results than Turbot Fish and viceversa. You say it is so, and I cannot discuss about it... I cannot do that!, my skills does`'t permit it.

I only wonder: Having read many articles and posts about weak links, turbot fish, nishio, etc, as I have read trying to find something similar... How is it possible that I didn't realized that Finned X-Wing is the same logic?. And, even knowing that there is a turbot fish here, why is so dificult for me to spot it?

Of course I assume that's my fault, my lack of knowledge, even my bad english, or maybe I have not read the exact post where it is described.

Anyway, my excuses if I seemed to try to "be the father" of the Hyper-Mega Technique... Nothing of that. I am only interested in solving Sudokus by humans, and I thought my post could help doing that.

Again, is a pleasure to discuss with you all.

Hasta la vista.

P.S: INCREDIBLE, Spanish Basketball team will play the World Championship Final against Greece... I cannot believe my eyes. :D

TheSpaniard · Posted: Fri Sep 01, 2006 8:56 pm Post subject:

Jean Christophe:
I have had some time to think and I am now sure that X-Ray and Turbot Fish are completely different techniques.
The thing that has confused you (and me) is that in example 3 of my first post it can be solved with both techniques.

Try to do that with example 2 and you will see that Turbot does not work.

Hasta la vista.

Jean-Christophe · Posted: Fri Sep 01, 2006 9:46 pm Post subject:

Yes it does. Here goes, using the same links as you used :
The strong chain is R1C7 == R9C7 ... R9C46 == R8C6
That is either R1C7 = 7 or else R1C7 <> 7 -> R9C7 = 7 -> R9C46 <> 7 -> R8C6 = 7. Thus either R1C7 or R8C6 = 7 -> R1C6 <> 7
The links goes through C7 (strong), R9 (weak), box 8 (grouped strong)

Now there are other ways to "break" this :
R9C7 == R1C7 .. R1C46 == R2C4
Either R9C7 = 7 or else R9C7 <> 7 -> R1C7 = 7 -> R1C46 <>7 -> R2C4 = 7. Thus either R9C7 or R2C4 = 7 -> R9C4 <> 7
And then : 7 of box 8 locked in R89C6 -> R1C6 <> 7
Or 7 of C4 locked in R12C4 -> R1C6 <> 7

Whichever way, you'll get the same result : both R1C6 & R9C4 <> 7

But there is indeed an even simpler way, using simple coloring (after some basic moves in box 9) :
R1C7 = G, R9C7 = B, R8C9 = G, R8C6 = B -> R1C6 <> 7
This is going through C7, box 9, R8

Or else, for the same grid, you can unlock the puzzle looking at the 3. The pattern is in R1, R9, C3, C9.

Also, sometimes your technique will lead you to a conflicting color. eg for some color, there are 2 cells in the same row/column/box -> this color cannot be the solution. This is because, your are going one "link" furter compared to turbot. But you should also get the same result removing the candidate form the cell(s) which are "under control" of 2 cells of different colors, which "can see" both the 2 different colors.
_________________
Jean-Christophe
"When you have eliminated the impossible, whatever remains, however improbable, must be the truth." Sherlock Holmes.

TheSpaniard · Posted: Sat Sep 02, 2006 10:41 am Post subject:

Jean Christophe. Thank you for that. I didn't realised that in puzzle number two. I know that in this puzzle there are some more ways to continue solving: I used its configuration because I think it is very easy here to spot the thing.
Anyway, I am not sure to understand all your steps. That's my fault, of course. I only am interested in solving Sudokus by human means, and I confess XY-Wing is the limit to my comprension.
You said it is the same techique than Turbot Fish: That's Excellent. Because I am preparing a Formal Definition that (luckily) I will publish tomorrow, generalising both Colors and Multicolors and maybe (although I am not very confident of that) more techniques.

On monday I begin working again after holidays, and I am almost sure that I will be offline for a while.

Thank you again