The New Sudoku Players' Forum

[tarek]

the upshot is that this method is brute force
there are no hints for how a human solver should proceed
(other than to pick a cell with the least # candidates)
in the worst case you have to make lucky random cell guesses to make progress

Hi gsf,

have a look at the start of this linked thread, the member solving the puzzle spotted the forcing implications.

Now it really depends on the person if he can spot a pattern or not. some people may not spot a simple colouring elimination at once. but when they systematically go through the grid, they will find it.

The other thing is that you can the can get to forcing implications result you want without seeing the implications right to the end (not spotting the contradiction). A human solver will spot one cell, as the member in mentioned thread has, & act accordingly.

As for the Hinting, well it will not be straight forward I must say, but most of the very advanced techniques suffer the same at this department. If you are interested in how a hint would come out the I will tell you about the RAINBOW cell.

[ronk]

I swept through ~225M canonical order puzzles and canonicalized the top1465 to eliminate duplicates (because the 225M contain random generated puzzles along with puzzles scraped from the forums)

Would you consider making the canonicalized form of the top1465 available on the web?

If "yes", thank you in advance, Ron

[tso]

the upshot is that this method is brute force
there are no hints for how a human solver should proceed
(other than to pick a cell with the least # candidates)
in the worst case you have to make lucky random cell guesses to make progress

Hi gsf,

have a look at the start of this linked thread, the member solving the puzzle spotted the forcing implications.

You are grossly misrepresenting that thread! That thread was about valid FORCING CHAINS (NOT what you are calling forcing implications) in which the state of each individual cell forces the value of one other individual cell. This is nothing like bifurcation (or trifurcation) or what you are using in your proofs.
In my opinion, you should [edit] not use the name "forcing implications" for what is only a slight variation of a tactic that already has a name -- plus, the name is misleading. The casual reader will mistake this for much more explicity defined "forcing chains".

Michael Memphan called it "Ariadne's thread" or bifurcation. The only distinction you are making is that after your random placement, you restrict your solving methods arbitrarily to finding singles. Mempham would have bifucated the puzzle, creating essentially two separate puzzles, then use whatever logical methods possible to advance them both until reaching the solution, a contradiction or another brick wall requiring a second bifurcation. The method can also be used to find cells that have the same value regardless of the value placed in in the starting cell.

The issue of whether or not a contradiction is found is a red herring. Forcing chains can be used to find a contradiction just as well as to find dual implications leading to the same placement.

"If A implies B which implies C which implies Not A" is a valid forcing chain that proves "Not A".

"If A implies B which implies C" AND "Not A implies D which implies C" together are dual forcing chains that prove "C".

Both of these ideas are encompassed under forcing chains.

[tso]

I really DO understand what you are driving at, but I must confess that my feeling is that the use of the word HIGHLIGHTING is just a clever way of saying ASSIGNING. the same is the use of the words LOOK FOR.

Indeed there is a pattern, but what is the math behind the pattern, They all ASSIGN or ASSUME a certain chain of events.

I think that is why, all of the long debates about T&E started & that is why The line which everybody seem to recognise for sure is when there is a contradiction elimination.

I strongly disagree. Coloring does not require or imply any assumptions. Further, contradiction is another red herring. For example:

Code: Select all

  3     26    1     | 4     9     8     | 5     7     26   
  9     248   7     | 6     5     3     | 248   248   1     
  46    5     48    | 7     2     1     | 3     4689  4689 
 -------------------+-------------------+-------------------
  248   3     6     | 5     1     7     | 9     248   248   
 +18    7     9     | 2     6     4     |-18    3     5     
  124   14    5     | 3     8     9     | 7    +1246  246   
 -------------------+-------------------+-------------------
  7     19    3     | 8     4     6     | 12    5     29   
  5     4689  48    | 1     7     2     | 468   489   3     
 x146   1468  2     | 9     3     5     | 1468 -148   7     


Code: Select all

  .     .     .     | .     .     .     | .     .     .     
  .     .     .     | .     .     .     | .     .     .     
  .     .     .     | .     .     .     | .     .     .     
 -------------------+-------------------+-------------------
  .     .     .     | .     .     .     | .     .     .     
 +1     .     .     | .     .     .     |-1     .     .     
  1     1     .     | .     .     .     | .    +1     .     
 -------------------+-------------------+-------------------
  .     1     .     | .     .     .     | 1     .     .     
  .     .     .     | .     .     .     | .     .     .     
 x1     1     .     | .     .     .     | 1    -1.    .     

I've colored 1s in four cells based these three descrete truths:
r5c1=1 IF AND ONLY IF r5c7<>1
r5c7=1 IF AND ONLY IF r6c8<>1
r6c8=1 IF AND ONLY IF r9c8<>1

This coloring shows that r5c1=1 IF AND ONLY IF r9c8<>1. Since r9c1 can *see* both of these cells, r9c1<>1.

This can described as finding a contradiction if you wish:

if r9c1=1, then r5c1<>1, then r5c7=1, then r6c8<>1, then r9c8=1, then r9c1<>1, contradiction


This is no trick to avoid bifucation. No values were ever assinged to the first four cells -- in fact, we don't know what they are. It differs only be degree from this deduction:

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12 . . |12 . . |23 . .
 . . . | . . . | . . .
 . . . | . . . | . . .
 ------+-------+------
 . . . | . . . | . . .
 . . . | . . . | . . .
 . . . | . . . | . . .
 ------+-------+------
 . . . | . . . | . . .
 . . . | . . . | . . .
 . . . | . . . | . . .

r1c1=1 IF AND ONLY IF r1c4=2
r1c1=2 IF AND ONLY IF r1c4=1
r1c7 can see them both, therefore, r1c7<>2

This can described as finding a contradiction if you wish:

if r1c7=2, then r1c1=1, then r1c4=2, then r1c7<>2, contradiction

[tarek]

My intentions to start the thread was not to herald the invention of a new technique, but to find a suitable name for something which I thought to be Forcing chains then discovered that it wasn't (or is it ??!!).

returning to the example in that thread. My solver with modifications solves it as follows

Code: Select all

*-----------------------------------------------------------------*
| 18     19     3     | 7      189    5     | 4      2      6     |
| 126    14569  249   | 3      129    29    | 7      59     8     |
| 7      59     28    | 28     4      6     | 59     3      1     |
|---------------------+---------------------+---------------------|
| 5      2      89    | 1      7      89    | 6      4      3     |
| 68     369    7     | 4      269    2389  | 1      589    259   |
| 4      369    1     | 5      269    2389  | 89     7      29    |
|---------------------+---------------------+---------------------|
| 12     14     24    | 6      3      7     | 589    589    59    |
| 9      7      5     | 28     28     1     | 3      6      4     |
| 3      8      6     | 9      5      4     | 2      1      7     |
*-----------------------------------------------------------------*
Any Candidate in r1c1 forces r1c2 to have only 9 as valid Candidates (Forcing Implications)
Any Candidate in r1c1 forces r2c8 to have only 5 as valid Candidates (Forcing Implications)
Any Candidate in r1c1 forces r7c1 to have only 2 as valid Candidates (Forcing Implications)
Any Candidate in r1c1 forces r7c2 to have only 1 as valid Candidates (Forcing Implications)
Any Candidate in r1c1 forces r7c3 to have only 4 as valid Candidates (Forcing Implications)

Now if you used forcing chains to get to your conclusion then I must have done the same thing.

I strongly disagree. Coloring does not require or imply any assumptions. Further, contradiction is another red herring. For example:

Code: Select all

  3     26    1     | 4     9     8     | 5     7     26   
  9     248   7     | 6     5     3     | 248   248   1     
  46    5     48    | 7     2     1     | 3     4689  4689 
 -------------------+-------------------+-------------------
  248   3     6     | 5     1     7     | 9     248   248   
 +18    7     9     | 2     6     4     |-18    3     5     
  124   14    5     | 3     8     9     | 7    +1246  246   
 -------------------+-------------------+-------------------
  7     19    3     | 8     4     6     | 12    5     29   
  5     4689  48    | 1     7     2     | 468   489   3     
 x146   1468  2     | 9     3     5     | 1468 -148   7     


Code: Select all

  .     .     .     | .     .     .     | .     .     .     
  .     .     .     | .     .     .     | .     .     .     
  .     .     .     | .     .     .     | .     .     .     
 -------------------+-------------------+-------------------
  .     .     .     | .     .     .     | .     .     .     
 +1     .     .     | .     .     .     |-1     .     .     
  1     1     .     | .     .     .     | .    +1     .     
 -------------------+-------------------+-------------------
  .     1     .     | .     .     .     | 1     .     .     
  .     .     .     | .     .     .     | .     .     .     
 x1     1     .     | .     .     .     | 1    -1.    .     

I've colored 1s in four cells based these three descrete truths:
r5c1=1 IF AND ONLY IF r5c7<>1
r5c7=1 IF AND ONLY IF r6c8<>1
r6c8=1 IF AND ONLY IF r9c8<>1

"If" is an Assumption, I can't see how you label it as Truth.

Is the problem then that too many elimination cycles are above the human level or is it that the same cell might feature in the same chain twice ??

I could potentially make the function more human by only following limited elimination cycles (which also means that I could present it easier here as a series of eliminations reaching the target cell). Selecting only bivalued cells for elimination is one way of preventing the a cell to be featured in the chain twice (which ends up also with easy nice presentable chain which can feature in a helper program more easily).

By the way, why do I get this feeling that I'm alone on this ???

[tso]

My intentions to start the thread was not to herald the invention of a new technique, but to find a suitable name for something which I

thought to be Forcing chains then discovered that it wasn't (or is it ??!!).

Bifurcation, (or trifurcation, etc), Ariadne's thread, etc. "Forcing implications" is inappropriate and confusing.

returning to the example in that thread. My solver with modifications solves it as follows

Code: Select all

*-----------------------------------------------------------------*
| 18     19     3     | 7      189    5     | 4      2      6     |
| 126    14569  249   | 3      129    29    | 7      59     8     |
| 7      59     28    | 28     4      6     | 59     3      1     |
|---------------------+---------------------+---------------------|
| 5      2      89    | 1      7      89    | 6      4      3     |
| 68     369    7     | 4      269    2389  | 1      589    259   |
| 4      369    1     | 5      269    2389  | 89     7      29    |
|---------------------+---------------------+---------------------|
| 12     14     24    | 6      3      7     | 589    589    59    |
| 9      7      5     | 28     28     1     | 3      6      4     |
| 3      8      6     | 9      5      4     | 2      1      7     |
*-----------------------------------------------------------------*
Any Candidate in r1c1 forces r1c2 to have only 9 as valid Candidates (Forcing Implications)

But none of us reading this know what this means without solving ourselves! No one simply says A leads to M by forcing chain without giving the chain. If not, its simpler just to say "Place a 9 in r1c2. Trust me."

This first deduction can be expressed by forcing chains aka dual implication chains:

-- r1c1=1 => r1c2=9
-- r1c1=8 => (r5c1=6 AND r3c3=2) => r2c1=1 => r1c2=9
-- therefore, r1c2=9

Each link leads to the next with no 'memory', though one of the links is two cells combined.


But the fact that your solver is giving these series of convoluted and opaque 'steps' might hint at the problem. There is a simple xy-wing (4 cell forcing chain):

-- r1c1=1 => r7c1=2 => r7c3=4
-- r1c1=8 => r3c3=8 => r7c3=4
therefore, r7c3=4 and the rest of the puzzle is trivial.

Now if you used forcing chains to get to your conclusion then I must have done the same thing.

Whoa now ... that's not true. Just because we both started in LA and we're now in London doesn't mean we were on the same plane. Or that we both flew.

I strongly disagree. Coloring does not require or imply any assumptions. Further, contradiction is another red herring. For example:

...snip...

I've colored 1s in four cells based these three descrete truths:
r5c1=1 IF AND ONLY IF r5c7<>1
r5c7=1 IF AND ONLY IF r6c8<>1
r6c8=1 IF AND ONLY IF r9c8<>1

"If" is an Assumption, I can't see how you label it as Truth.

No it isn't. "if/then" and "if and only if" are relationships. "If I drop this ball, it will fall." assumes only the existance of gravity, not that I will drop the ball. If you prefer, I can state the same thing without the ifs:

r5c1 and r5c7 have the opposite values (1 or not 1)
r5c7 and r6c8 have the opposite values
Therefore, r5c1 and r6c8 have the same value

r6c8 and r9c8 have the opposite values
Therefore r5c1 and r9c8 have the opposite values

Another way to say it is that r5c1-r5c7-r6c8-r9c8 form a chain of conjugate connections. Chains with an even number of links will have the opposite state on the ends.

Is the problem then that too many elimination cycles are above the human level ...

No. Length has nothing to do with it. In some cases, very long forcing chains can be easily followed by humans as long as no eliminations are required along the way, no memory other than where the chain started.

... or is it that the same cell might feature in the same chain twice ??

No. The problem is you aren't actually giving an answer that we can follow along ourselves. No matter how complex an actual forcing chain might be or how difficult it is to find -- I could draw it on the grid (or list the sequence of implications) and anyone will be able to follow it from cell to cell. You have not done this. You have chosen a cell at random, filled in each of the candidates it can contain sequentially, solved the resulting 2 or 3 puzzles as far along as you can go using an arbitrary subset of tactics, then looked for cells that contained the same number in the same spot. In most cases, you could not map them out on the grid because some of the connections *do not exist yet*.

Selecting only bivalued cells for elimination is one way of preventing the a cell to be featured in the

chain twice (which ends up also with easy nice presentable chain which can feature in a helper program more easily).

And then the result *would* be a forcing chain.

[tarek]

The problem is you aren't actually giving an answer that we can follow along ourselves. No matter how complex an actual forcing chain might be or how difficult it is to find -- I could draw it on the grid (or list the sequence of implications) and anyone will be able to follow it from cell to cell. You have not done this. You have chosen a cell at random, filled in each of the candidates it can contain sequentially, solved the resulting 2 or 3 puzzles as far along as you can go using an arbitrary subset of tactics, then looked for cells that contained the same number in the same spot. In most cases, you could not map them out on the grid because some of the connections *do not exist yet*.

Well, I thoght that when i said eliminations would follow then it would be trivial. Back to the same example, here I modified the function so that no elimination can occur in the same cell twice (& added the word "Simple" ):

Code: Select all

*-----------------------------------------------------------------*
| 18     19     3     | 7      189    5     | 4      2      6     |
| 126    14569  249   | 3      129    29    | 7      59     8     |
| 7      59     28    | 28     4      6     | 59     3      1     |
|---------------------+---------------------+---------------------|
| 5      2      89    | 1      7      89    | 6      4      3     |
| 68     369    7     | 4      269    2389  | 1      589    259   |
| 4      369    1     | 5      269    2389  | 89     7      29    |
|---------------------+---------------------+---------------------|
| 12     14     24    | 6      3      7     | 589    589    59    |
| 9      7      5     | 28     28     1     | 3      6      4     |
| 3      8      6     | 9      5      4     | 2      1      7     |
*-----------------------------------------------------------------*
Any Candidate in r1c1 forces r1c2 to have only 9 as valid Candidates (Simple Forcing Implications)
Any Candidate in r1c1 forces r2c2 to have only 4569 as valid Candidates (Simple Forcing Implications)
Any Candidate in r1c1 forces r2c8 to have only 5 as valid Candidates (Simple Forcing Implications)
Any Candidate in r1c1 forces r3c2 to have only 5 as valid Candidates (Simple Forcing Implications)
Any Candidate in r1c1 forces r3c7 to have only 9 as valid Candidates (Simple Forcing Implications)
Any Candidate in r1c1 forces r5c8 to have only 59 as valid Candidates (Simple Forcing Implications)
Any Candidate in r1c1 forces r6c6 to have only 239 as valid Candidates (Simple Forcing Implications)
Any Candidate in r1c1 forces r6c7 to have only 8 as valid Candidates (Simple Forcing Implications)
Any Candidate in r1c1 forces r7c1 to have only 2 as valid Candidates (Simple Forcing Implications)
Any Candidate in r1c1 forces r7c2 to have only 1 as valid Candidates (Simple Forcing Implications)
Any Candidate in r1c1 forces r7c3 to have only 4 as valid Candidates (Simple Forcing Implications)
Any Candidate in r1c1 forces r7c7 to have only 58 as valid Candidates (Simple Forcing Implications)
Any Candidate in r1c1 forces r7c8 to have only 89 as valid Candidates (Simple Forcing Implications)


For example regarding the first line (r1c2)
r1c1=1 => r1c2<>1 => r1c2=9
r1c1=8 => r3c3<>8 => r7c3<>2 => r7c2<>4 => r1C2<>1 => r1c2=9

is that valid ? Now if u insist i could do the same with the rest.

"If A implies B which implies C which implies Not A" is a valid forcing chain that proves "Not A".

"If A implies B which implies C" AND "Not A implies D which implies C" together are dual forcing chains that prove "C".

Both of these ideas are incompassed under forcing chains.

Well that is just what I did & didn't at the same time .

[tarek]

But the fact that your solver is giving these series of convoluted and opaque 'steps' might hint at the problem. There is a simple xy-wing (4 cell forcing chain):

-- r1c1=1 => r7c1=2 => r7c3=4
-- r1c1=8 => r3c3=8 => r7c3=4
therefore, r7c3=4 and the rest of the puzzle is trivial.

I know about the xy-wing, I had to disable it from the solver to reach to the grid which you posted. was the xy wing technique present in June when u posted the grid ???

Again, I thought that by stating that single eliminations would follow, the 'steps' would be clear

[tarek]

But none of us reading this know what this means without solving ourselves! No one simply says A leads to M by forcing chain without giving the chain. If not, its simpler just to say "Place a 9 in r1c2. Trust me."

Well I think you need to trust me on this. I could add an option to show the chain if that makes you happier though

[tarek]

Regarding the name.... The tecnique as I see it at the moment can fall within (Bifurcation, ForwardChecking & Forcing chains).

This comes from another thread, gsf describes how Forward checking works.....

move() assigns a value to a cell and iteratively propagates the constraints to all the other cells

forward_check() for each unsolved cell it calls move() for each candidate value on a copy of the grid; if only one of the candidates succeeds then the cells is assigned that value; otherwise the union of all changed values for the assignments for this cell are used to update the candidate lists for the other unsolved cells (which may result in more assignments) -- frisch mentions this for his ocaml solver

backtrack() when forward check fails to progress a "good" cell and candidate value is chosen and assigned via move() using standard DFS backtrack logic; a "good" cell is one which has the minmax # unsolved cells for all cells examined by forward_check()

Now the bit that says "otherwise the union of all changed values for the assignments for this cell are used to update the candidate lists for the other unsolved cells (which may result in more assignments) -- frisch mentions this for his ocaml solver " seems to describe what the technique does.

Which again brings it back closer to T&E. the technique however does not CHECK anything, it basically STOPS. so does it STOPS short of being T&E?

The idea is that although it is part of other general techniques that go into T&E & therefore not add much to a non-helper solver. This part of technique can be used to help solving in the non T&E environment

[Jeff]

If another bi- or poly-valued cell turns out to have exactly the same candidates with each candidate from the original cell then we can safely say that the the 2nd cell must have those same candidates

This is not new of course. I just don't know what is its name. There is no Contradiction involved so it shouldn't fit the GENERAL term of T&E.

Hi Tarek,

Excuse me for being frank. Your statement "There is no Contradiction involved so it shouldn't fit the GENERAL term of T&E" is not correct for the following reason:

T&E stands for trial & error which can be described as:

1. 'Trial' could mean testing each cell in a grid in turn
2. 'Trial' could mean testing each candidate in a cell in turn
3. 'Error' could mean backtracking when a contradiction is identified
4. 'Error' could mean a consistent implication result is not found and the process reiterates for other cells

The technique you described involves items 1, 2 and 4 above, so it is a T&E process.

[tarek]

Excuse me for being frank. Your statement "There is no Contradiction involved so it shouldn't fit the GENERAL term of T&E" is not correct for the following reason:

T&E stands for trial & error which can be described as:

1. 'Trial' could mean testing each cell in a grid in turn
2. 'Trial' could mean testing each candidate in a cell in turn
3. 'Error' could mean backtracking when a contradiction is identified
4. 'Error' could mean a consistent implication result is not found and the process reiterates for other cells

The technique you described involves items 1, 2 and 4 above, so it is a T&E process.

Thanx Jeff,

I don't mind a frank discussion if it helps clearing out issues.

I do concede that it is "Trial" but you have to do a bit more convinicing regarding the "ERROR" part as the process in my opinion was not looking for something in particular. (No progrees, Contradiction are STOP signals of my choice - or the human solver using a pencil or paper- & the process does NOT take a a specific action when it reaches the specific predefined STOP signal, it just keeps just looking looking & looking & reports what it registered in its field of vision at the end). I can change the stop signals to whatever I want.

Having to read a bit more about something I knew little about, it seems that this specific technique as it is (although being a cog in the wheel of forward checking) looks like a double/poly implication chain.

Now I will concede that if ALL double/triple/... implication chains are T&E then this is definitely a T&E technique. That is how I see it (at the moment of course ), I'm not trying to be difficult (although i can be on other occasions), but others' arguments until now failed to convince me (and apparantly my arguments failed to convince anyone ).

[ronk]

Code: Select all

*-----------------------------------------------------------------*
| 18     19     3     | 7      189    5     | 4      2      6     |
| 126    14569  249   | 3      129    29    | 7      59     8     |
| 7      59     28    | 28     4      6     | 59     3      1     |
|---------------------+---------------------+---------------------|
| 5      2      89    | 1      7      89    | 6      4      3     |
| 68     369    7     | 4      269    2389  | 1      589    259   |
| 4      369    1     | 5      269    2389  | 89     7      29    |
|---------------------+---------------------+---------------------|
| 12     14     24    | 6      3      7     | 589    589    59    |
| 9      7      5     | 28     28     1     | 3      6      4     |
| 3      8      6     | 9      5      4     | 2      1      7     |
*-----------------------------------------------------------------*
Any Candidate in r1c1 forces r1c2 to have only 9 as valid Candidates (Simple Forcing Implications)
Any Candidate in r1c1 forces r2c2 to have only 4569 as valid Candidates (Simple Forcing Implications)
Any Candidate in r1c1 forces r2c8 to have only 5 as valid Candidates (Simple Forcing Implications)
Any Candidate in r1c1 forces r3c2 to have only 5 as valid Candidates (Simple Forcing Implications)
Any Candidate in r1c1 forces r3c7 to have only 9 as valid Candidates (Simple Forcing Implications)
Any Candidate in r1c1 forces r5c8 to have only 59 as valid Candidates (Simple Forcing Implications)
Any Candidate in r1c1 forces r6c6 to have only 239 as valid Candidates (Simple Forcing Implications)
Any Candidate in r1c1 forces r6c7 to have only 8 as valid Candidates (Simple Forcing Implications)
Any Candidate in r1c1 forces r7c1 to have only 2 as valid Candidates (Simple Forcing Implications)
Any Candidate in r1c1 forces r7c2 to have only 1 as valid Candidates (Simple Forcing Implications)
Any Candidate in r1c1 forces r7c3 to have only 4 as valid Candidates (Simple Forcing Implications)
Any Candidate in r1c1 forces r7c7 to have only 58 as valid Candidates (Simple Forcing Implications)
Any Candidate in r1c1 forces r7c8 to have only 89 as valid Candidates (Simple Forcing Implications)


For example regarding the first line (r1c2)
r1c1=1 => r1c2<>1 => r1c2=9
r1c1=8 => r3c3<>8 => r7c3<>2 => r7c2<>4 => r1C2<>1 => r1c2=9

is that valid ? Now if u insist i could do the same with the rest.

I think your "solver log" should not tie all those placements and eliminations back to "Any Candidate in r1c1 ...".

After the placement r1c2=9, the puzzle can be solved with only naked and hidden singles, so reporting the placements as such would greatly enhance understandability IMO. For example ...
Naked single: r3c2=5
Naked single: r3c7=9
Naked single: r2c8=5
Naked single: r6c7=8
Naked single: r7c7=5
Naked single: r5c8=9
---
---

BTW Jeff and Carcul would (I think) apply a discontinuous Y-cycle nice loop to your double implication chains:

[r1c2]-1-[r1c1]-8-[r3c3]-2-[r7c3]-4-[r7c2]-1-[r1c2] which implies r1c2<>1 and r1c2=9

Ron

[Carcul]

BTW Jeff and Carcul would (I think) apply a discontinuous Y-cycle nice loop to your double implication chains:

[r1c2]-1-[r1c1]-8-[r3c3]-2-[r7c3]-4-[r7c2]-1-[r1c2] which implies r1c2<>1 and r1c2=9

Yes I would. Very well spoted Ronk.

Regards, Carcul

[Jeff]

BTW Jeff and Carcul would (I think) apply a discontinuous Y-cycle nice loop to your double implication chains:

[r1c2]-1-[r1c1]-8-[r3c3]-2-[r7c3]-4-[r7c2]-1-[r1c2] which implies r1c2<>1 and r1c2=9

Hi Ronk, With so many names to choose from, why did you call it a y-cycle. It's is more meaningful to call it a simple pure bivalue nice loop or more elegant to call it an xy-chain since all cells involved are bivalued which is what 'xy' implies. BTW, the nice loop notation is spot on.