Using Almost Continuous Networks Study 02

The Tough Sudoku of August 11, 2010 contains another reasonable teaching moment on how one might combine smaller almost continuous networks or loops to form deductions. A continouous network or continuous loop is not only a powerful Sudoku technique, trick or tip, it also is my personal favorite type of sudoku deduction..

If this is your first visit to this blog, welcome! Unfortunately, if you are a first time visitor, this page may seem like it is written in a different language. Well, it is!! Previous blog pages may be helpful. Links to these pages are found to the right, under Sudoku Techniques. The earliest posts are at the bottom, and if you have never perused the intricacies of our special coded language here, you may wish to start close to beginning. The list is rather large, so below find a list of links that may be pertinent to this particular puzzle.

The Puzzle

Puzzle at Start

Some easy moves are available here:

  • (3)a1% Box (Hidden Single (3), only (3) in Box)
  • (5)h4% Box & Column
  • (9)b6% Box & Column
  • (8)i9% Box
  • => UP 27 (Unique Possibilities to 27 filled and given cells)
  • LC(6)a46 => a89≠6 (Locked Candidate)
  • (6)b9% Row
  • (4)b7% Column
  • (8)c7% Row, Box
  • (8)a4% Row, Box
  • (8)g2% Row
  • (6)a6% Column, Box
  • => UP 33
  • LC(1) ac8 & ac2 => ef8,dehi2 ≠ 1
  • LC(1)i46 => i1 ≠ 1
  • LC(7)g45 => g13 ≠ 7

Simple Sudoku can find no further deductions to make at this point. The puzzle in this state is shown below.

A Y Wing Style: YWS

Note that although much of the stuff we will analyze is available without this next step, it is hard to pass on one that is this easy. It is quite likely that one can find a path to solve this puzzle without the next step.

YWS After SSTS is done

Above, one can find the following rather easy Y Wing Style (YWS)

  • (3)i56 = (3-6)i8 = (6)i2 - (6=3)g3
  • => g56 ≠ 3
Alternatively, one could find an ALS xz with restricted common candidate (6). Since (6) cannot exist at both g3 and i2, one has
  • (3)g3 = Naked Five-agon (12349)i12456
  • => g56 ≠ 3
IMO, the former is easier to find. Some, however, have a fondness for the latter. In such a puzzle, the two must co-exist. Thus, one might say they are equivalent.

In either case, one can continue with two rather simple steps: (HP means Hidden Pair)

  • LC(3)i56 => i8 ≠3
  • HP(36)g38 => g8 ≠ 29
Please note that what follows is available without the YWS. However, I do not believe the puzzle solves without some similar additional step.

Almost Continuous Loops A'Plenty

Below, looking at AIC's which do not eliminate anything but are instead Almost Continuous Loops (ACLs)can unlock the puzzle once one finds how to combine them.

I shall include three rough diagrams of some of the ACL's at the end of this blog page.


The following AIC's which are also ACL's are available. Note additionally not only the logical symmetry of some of these, but also the actual puzzle symmetry. IMO, these symmetries make the following relatively easy to locate.

  • (2): i2 = [h2 = ac2 - b1 = b8 - i8 = h79 Loop]
  • The next two items are Almost WWing Rings. They should be self-explanatory as two halves of Almost WWings - which are a type of YWS.
    • (9)h2 = [(2=7)h2 - ac2 = b13 - (7=2)b8 - b1 = ac2 Loop]
    • (9)h2 = [(2=7)h2 - ac2 = b13 - (7=2)b8 - i8 = h79 Loop]
  • Note: The next item is included as a curiosity, and for those who wish to one can use the information for a nice alternative solution
    • (4)i2 = de2 - (4=NP18)f13 - (1)f6 = HP(13-35)ei6 = (5)f56 - (5=2)f8 - LOOPs that target (2)f8
  • The following item is a less complex way to see a relationship between the loops:
    • (27=9)h2 - (9=NP24)gi1 - (2)i2
    • (27=9)h2 - (9=NP24)gi1 - (2)b1 = (2)b8
One can see from the above fragments that some (2)'s in row 8 are easily eliminated. The hardest thing to do from this point forward now is to choose a presentation to continue. Although I prefer AIC's for their generality, they are not the only way to proceed from here. Since the item I noted above as a curiosity overlaps with the almost chain of only candidate (2) and with one of the Almost WWing Rings, it is relatively easy to see that simply finding a way to knock off (2)f8 will make some progress in the puzzle. IMO, this is not selecting a target. Rather, it is letting the puzzle choose a target. Not only do I know that (2)f8 is a reasonably effective target, I also know that it is a reasonably easy item to eliminate. I also know of more than one way to eliminate it (from the information above)! Below is a relatively easy AAIC
  • [WWing Ring: (2)h79 = i8 - (2=7)b8 - b13 = (7)ac2 - (7=2)h8]
  • = (9)h2 - (9=NP24)gi1 - (2)b1 = (2)b8
  • => h1,acef8 ≠ 2
  • => Cascade of Singles until finished; UP 81
Some may prefer a diagram. This type of diagram is championed by the great Sudoku solver, ttt:
  • (9)h2 - (9=NP24)gi1 - (2)b1 = (2)b8
  • ||
  • (2)h2 - h79 = (2)i8
  • || Note the Almost WWing Ring Loop
  • (7)h2 - ac2 = b13 - (7=2)b8
Regardless of the preference in presentation, IMO finding the step is served well by looking at ACL's.

Here is yet another way to knock off the puzzle with the given information:

  • [WWing Ring: (2)h79 = i8 - (2=7)b8 - b13 = (7)ac2 - (7=2)h8]
  • = (9)h2 - (9=NP24)gi1 - (2)i2
  • = (2): [h2 = ac2 - b1 = b8 - i8 = h79 LOOP]
  • => h1,acef8,i2 ≠ 2
  • => Cascade of Singles until finished; UP 81
Although this particular manner is more complex in expression, I find the symmetry that it exploits to be especially pleasing to the eye and brain. It also serves to emphasize the process of finding the elimination group, since it maintains the ACL's that one can locate and exploit.

In summary, I believe strongly that ACL's are extremely well used as ALS's and some more typical forms. However, I believe that their utility in other forms such as those shown above are not as well utilized. They deserve some greater study, IMO.

Almost Continuous Loops Diagrams

Below, find what I think to be the primary Almost Continuous Loops that can unlock this puzzle. Since one of the WWing ACLs is shown, I trust that the other need not be shown.

ACL candidate 2

ACL WWing Ring

ACL ALS plus links

The ALS above includes some link information.

Indicate which comments you would like to be able to see

Hi Steve,
You lost me on the first step. I can't make see the logic in the YWS technique either here or where it is introduced in this blog. I also have no idea what an argon is other than a gas.
04/Sep/10 11:07 PM
Correction, now I see the ALS-XZ elimination. I misread agon as argon. Sorry.

04/Sep/10 11:12 PM
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