Maybe I was unfair as I wasn't specific:

George wrote:Guys

We're not all "guys", thank you very much.

George wrote:Here is an example that got everyone stuck for a few weeks.

Who is "everyone"? "A few weeks?" What are you talking about? This is an easy one.

George wrote:9XX/5XX/362

253/9XX/81X

6X1/2X3/95X

X69/X32/X85

1XX/6X8/X39

3XX/49X/X76

53X/XX4/698

89X/3XX/741

X1X/8X9/523

George sweetie, wouldn't you rather try to make sense of a diagram that looks like this? It's easier for us 6 year olds to make sense of:

- Code: Select all
` 9 . . | 5 . . | 3 6 2 `

2 5 3 | 9 . . | 8 1 .

6 . 1 | 2 . 3 | 9 5 .

-------+-------+------

. 6 9 | . 3 2 | . 8 5

1 . . | 6 . 8 | . 3 9

3 . . | 4 9 . | . 7 6

-------+-------+------

5 3 . | . . 4 | 6 9 8

8 9 . | 3 . . | 7 4 1

. 1 . | 8 . 9 | 5 2 3

George wrote:Applying the "OPEN CHAIN OF SUDO" technique, I am able to identify 2 open chains of sudo "7". Don't forget we are propogating along "7".

How could I forget? I don't know what your talking about. How many undefined terms can you put in a floramate snofflebit before it beclorinates from yal'nrk and cannot be parsed? It's probably me -- my memory isn't that great at 6.

George wrote:These chains are:

Chain 1: (9,1)(4,1)(4,4)(7,4), these are all conjugate links and (9,1)(7,4) are the open ends which are also conjugates.

Chain 2: (9,1)(4,1)(4,4)(5,5), these are all conjugate links and (9,1)(5,5) are the open ends which are also conjugates.

Since "conjugates" means that one of them must be a true condidate of "7", it follows that other "7" cells that lies on the intersection of the conjugate links or open conjugate ends must be a false condidate of "7" and can be safely excluded.

From chain 1, it can be seen that (7,3) lies on the intersection of the chain ends (9,1) connected by a box and (7,4) connected by a row. Therefore, (7,3) must not be "7" because either (9,1) or (7,4) must be "7". It follows that (7,3) must be "2".

Likewise, (9,5) lies on the intersection of the chain ends (9,1) connected by a row and (5,5) connected by a cloumn. Therefore, (9,5) must not be "7" because either(9,1) or (5,5) must be "7". It follows that (9,5) must be "6".

Although one more open chain of sudo "7", can be identified, no other "7" cells can be eliminated. This chain is listed below for observation purposes:

(9,1)(4,1)(4,4)(1,6)(2,6)(2,9)(3,9), where all of these are conjugate links except for (4,4)and(1,6) which is a like link.

Don't forget, conjugate link must be unique on a row, a column or a box, whereas like link could be anywhere in the 9x9 grid.

This has been described half a dozen times or more in this forum -- though I had to read your post half a dozen times to realize this, as you've invented your own language and assume we know it. Honestly, I only know what you are trying to say because I KNOW what you are trying to show. "Propagating along "7"? Row 7? Candidate 7? False condidate (sic) of 7?

Setting aside my obnixious way of expressing this, you might want to use a diagram -- maybe call row 5 column 3 "r5c3 instead of the ambiguous (5,3) or (3,5). (If you click on QUOTE, you can examine the way diagrams are bracketed by [ c o d e ] and [ / c o d e ] for legibility.) One of many simpler solutions to this puzzle is:

A diagram is much easier for me to understand -- of course, us 6 year olds are easily distracted, so I've removed the extraneous information:

- Code: Select all
` . . . | . . . | . . . `

. . . | . . . | . . .

. . . | . . . | . . .

-------+-------+------

47 . . |17 . . | . . .

. . . | . . . | . . .

. . . | . . . | . . .

-------+-------+------

. . . |17 . . | . . .

. . . | . . . | . . .

47 . . | .67 . | . . .

Could anything be easier?

1) r4c1=4 => r9c1=7 => r9c5=6

2) r4c1=7 => r4c4=1 => r7c4=7 => r9c5=6

Therefore, r9c5=6, and the rest solves quickly.

This type of proof has been called

"forcing chains" or "dual implication chains" or usefully discriptive titles that can be parsed from context and possibly an English dictionary. (Mine doesn't list "sudo".) It generally isn't written in all caps with quotes and theme music.

This specific pattern of 5 cells, which is a subset of all forcing chains which may have more cells and even more chains, has been recently dubbed

"turbot fish" by Nick70 who described it in clear detail here.To answer my own question, yes, something could be easier:

- Code: Select all
` . . . | . . . | . . . `

. . . | . . . | . . .

. . . | . . . | . . .

-------+-------+------

. . . | . . . | . . .

. . . | .57 . | . . .

. . . | . . 15| . . .

-------+-------+------

. . . | . . . | . . .

. . . | . . 56| . . .

. . . | .67 . | . . .

1)r9c5=6 => r8c6=5 => r6c6=1

1)r9c5=7 => r5c5=5 => r6c6=1

Therefore, r6c6=1 and the rest follows.

This is the simplest possible forcing chain. (It is not an example of an x-wing.)

Now tell me Georgie, honey, wasn't that a lot clearer to you than what you wrote?

MadOverlord wrote:A forcing chain is a chain of assumptions about the puzzle. Let's say R1C1 has possibilities 1,2 in some puzzle. Then you can make a chain of implications, ie something like this:

If R1C1=1 then that means R1C9=5 which means R9C9=3... and so on.

It is sometimes possible to find a chain that starts with R1C1=1 and ends with R1C1=2. Since this is impossible, you know that the original guess, R1C1=1, must be false, and it can be eliminated.

This technique can be elaborated, with the chains splitting off from each other, and so on. It gets horribly complicated and, IMHO, except at the simplest levels, is not a practical technique for human solving. It is effectively brute-force recursion.

The real interesting stuff, to me, in Sudoku algorithm research, is coming up with algorithms that are easy for humans to "execute" and take advantage of our skills in pattern recognition. Simple forcing chains is right on the edge of what I think is reasonable.

One approach I am currently working on is recasting some of the more complex algorithms in such a way that they play to human strengths and are reasonably executable; that's how I came up with Bowman Bingo (an extended Nishio) a few weeks back. I've got another in the works that I'll be announcing soon.

MadOverlord's description of "forcing chains" is somewhat inaccurate. No guesses are made, no assumptions are made. Further, he mixes opinion with definition. Any tactic can be complicated in one situation and simple in another. Some of the easiest tactics we take for granted might be beyond an individual's capabilities -- if say, you decided to solve without using pencil marks because the grid in the paper was very small an all you had was a pen -- the underlying concepts do not change from complex to easy and then to beyond human ability. They remain. Forcing chains is a more *general* description that includes many *specific* formations. Some of us would rather learn to cook rather than have to carry around a pile of recipes. The "forcing chains" philosophy is that it's more fun to *think* than it is to *recall", more fun to "invent" than it is to "look-up". I respect his opinion about Sudoku algorithm research and I support it and look forward to new findings. But remember, if someone were to find THE solution, the one tactic that could be used to solve ALL Sudoku's -- and if that tactic was simple enough so that once you knew the secret, you couldn't pretend you didn't -- this would not be a boon for solvers -- it would be the end of Sudoku altogether. Fortunately, this is (out on a limb) not possible. My opinion is there is a big difference between trying to solve an individual Sudoku and trying to solve Sudoku in general. Both are plenty interesting. Of *course* brute force is riduculous to use when trying to solve the general case, but if you can use brute force in your head on the bus with a pen -- you are the MASTER, not the cheater! I bow to you. If you can -- as I do -- find by examination, forcing chains or proofs by contradiction that are 4, 6, 8, 10 cells long, this in no way negates the search for general solutions to Sudoku. General case knowlege improves specific case solving but tells us nothing about the specific case in front of me. (Men are taller than women -- my three sisters are all taller than my brother.)

George wrote:So, the forcing chain technique is just another form of bifurcation, because it involves guessing a number to propogate and return when a contradiction is identified.

I am not into bifurcation at all. I like to solve all my puzzles by logic which consider bolean, not number branching.

A common myth among 8 year olds -- it is ridiculous on two levels. First, forcing chains are NOT proof by contradiction. There IS no contradiction identified in using a forcing chain. (This in no way disparages Proof by Contradiction, and equally useful tactic.)

Take a look at this diagram. There are two proofs, one above and one below the line of equal signs, both proving that the "34" cells must be "4":

- Code: Select all
`12 . . |23 .13 | . .34 `

. . . | . . . | . . .

. . . | . . . | . . .

=======+=======+======

12 . . |13 . . | . . .

. . . | . . . | . . .

23 . . |34 . . | . . .

-------+-------+------

. . . | . . . | . . .

. . . | . . . | . . .

. . . | . . . | . . .

To call one bifurcation and the other logic -- is illogical. We you disagree, please point to or construct a strict general definition that descrimates between these structures. In the lower case, I am no more or less "guessing" what r4c1 is or "bifurcating" it than I am in the upper case "guessing" and "bifurcating" that r1c146 is 1, 2, 3 or 2, 3, 1.

(Also, if you wanted to, you could start your examination by look at the two "34" cells and "guess" they were 4's. In both cases, it would lead to contradiction. If we must toss out PBC as being beneath us, simply renaming the upper structure "triples" doesn't change anything -- it's the same four cells, the same logic from a different vantage point. I can turn ALL proofs in to PBC -- and vice versa.)

If you agree that those two trivial structures are logical, then look below:

- Code: Select all
`12 . 23|34 .56 |16 .67 `

. . . | . . . | . . .

. . . | . . . | . . .

=======+=======+======

12 . . | . . . | . . .

. .13 | . .46 |34 . .

46 . . |67 . . | . . .

-------+-------+------

34 . . | . . . | . . .

. . . | . . . | . . .

23 . . | . . . | . . .

Again, both above and below the line, on must use the information from ALL the cells, taken together, to prove that the "67" cells must each be 7. There is no guessing, no trial and error, no looking into the future. Either you can recognize the patterns in your head, or you can't. If you can, you've proved it.

The fact that I might coin the word "quints" for the pattern above the line or "quinty fish" for the *exact* configuration below the line, does not magically change they underlying logic. It just changes what the solver does from *thinking* to *recalling*. Computers MUST recall as they cannot think -- yet. It may vary well be more interesting to be able to apply dozens of known tactics and avoid looking free-form for various implications that might never be seen twice -- but it is doing the *latter* that will more likely lead to the discovery of more of the *former*.