The New Sudoku Players' Forum

[champagne]

Champagne, you post made me think I'd slipped up for a moment! However my deduction is good because:

Hi david,

My view has nothing against yours and BTW, both are using the same hidden pair.

It's just that in that pattern, I want to be prepared to process puzzles being relatively far from a double exocet.

I checked meantime, we go to the end easily just using the fact that both base have complementary digits.

I am sure that in some cases showing that we have an ADE will help

[daj95376]

[Withdrawn: I completely missed the objective.]

[David P Bird]

_
Observation from an outsider to the current discussions (in general) and DPB's latest puzzle selection (in particular).

-- the conjugate pattern needs to be established for <1267> in r7c12 and r8c46
-- no one is arguing that the conjugate pattern holds through testing <167>
-- this leaves testing <2> and determining/resolving spoiler candidate cells

I find 2r6c8 to be the only spoiler cell for the conjugate pattern.

-- 2r7c12 & -2r6c8; Sashimi Swordfish 2c358\r148 w/fin cell r9c5 => -2r8c4

That leaves showing 2r6c8 => -2r8c4. This step is why I'm an outsider. I don't like it one bit.

-- 2r7c12 & (2-3)r6c8 = 3r6c1; LNT 249r4c123; -2r4c5; 2r9c5 => -2r8c4

DAJ, Firstly remember that we are considering only Almost Double JEs with 4 target cells.

For a digit to be common to both base sets, its partial fish cells must hold 3 truths. (Take any example case and eliminate the same digit from the cells seen by the two base sets and it becomes very obvious.) Therefore any digit that complies with requirement 3) for the partial fish CANNOT be common to both base sets – end of story – no need to examine cases.

In the puzzle just discussed the only way for the partial fish cells to hold 3 truths for (2) is for both (2)r4c5 and (2)r6c8 to be true. (This is obvious as without (2)r6c8, only two rows in the partial fish hold (2)s, and without (2)r4c5 only two columns do.)
But in the chain I gave:
(2)r4c5 – (2)r4c123 = (267-3)r5c12,r6c1 = (3-2)r6c8
shows that these two spoilers are weakly linked

1. they can never both be true,
2. the partial fish cells can therefore only hold 2 truths at most,
3. (2) cannot be common to both pairs of base cells,
4. the double JE is true.

I'm afraid I just don't understand the basis for your approach stemming from the sashimi fish.

Now a bit of a confession. For the examples I've tried in the limited time I've had, I haven't had problems proving the DJE is true when (against my principles) I allowed myself to use forcing nets. But my aim has been to find linear chains to prove them. However in some cases I've assumed a conjugate link between a spoiler and a DJE which now I realise is unsound. Certainly if a spoiler is false the DJE must be true, but if the DJE is true it may be possible for the spoiler and just one other instance of the digit in the partial fish to be true, so the link is not bi-directional but forcing. I'm going to have to review that.

DPB

[sultan vinegar]

Thanks DPB, I see my misunderstanding now! Its the parallel lines (II) case, which I didn't see the significance of when I first read it, but now I think it is a nifty way to deal with that particular (sort of degenerate) case, bringing it all under the one umbrella so to speak.

A couple of eliminations that I think are missing from the guide (unless I missed them):

1) JE, the sort of degenerate case from above:

Code: Select all

  *-------*-------*-------*
  | B B . | . . . | . . . |  B = Base Cells 
  | . . . | Q . . | R . . |   
  | . . . | Q . . | R . . |  Q = 1st Object Pair
  *-------*-------*-------*  R = 2nd Object Pair
  | . . S | / . . | S . . |       
  | . . S | / . . | S . . |  S = Cross-line Cells     
  | . . S | / . . | S . . |  / = Candidate missing from potential Cross-line Cells 
  *-------*-------*-------*  . = Any candidates
  | . . S | / . . | S . . | 
  | . . S | / . . | S . . |     
  | . . S | / . . | S . . |   
  *-------*-------*-------*

If the candidate is true in the base, then in column 4, it must be true in the first target (Q), so it may be eliminated from the second target (R).

2) DJE:



As is already known, the cannibal eliminations (red) are explained by the virtual locked set of the 4 base cells (r9c12, r7c56), which allows you to eliminate those candidates that can see all the base cells. Also of course the orange eliminations in the 4 target cells (r8c34, r79c9) are already known.

What I think is not already known is the 'covering line' eliminations (orange). These can be explained by the virtual locked set of the 4 target cells. Whichever of the target cells holds a particular candidate will eliminate that candidate from one of the 3 'S' columns, leaving 2 'S' columns, which are covered by the 2 covering lines, making the two covering rows rank 0 logic.

To make it really clear, consider the cases for candidate 3: If either r79c9 is a 3, then we are left with the X-wing 3:c34r13 (note that r8c34 <> 3 in this case as the 4 target cells are a virtual locked set, r7c3,r9c4 <> 3 as the 4 base cells are a virtual locked set, and r9c3, r7c4 must be empty for the DJE pattern anyway). If r8c3 = 3, then we are left with the X-wing 3:c49r13, and if r8c4 = 3, then we are left with the X-wing 3: c39r13. So no matter which target cells holds digit 3, we get the eliminations in r13 for digit 3. Likewise for the other candidates.

This is particularly useful to get eliminations with ADJE, e.g. DPBs example:



r6c8 <> 2 => DJE => r4c9 <> 2.
r6c8 = 2 => r4c9 <> 2, so either way it goes.

Of course, DPB showed that the DJE is true in this example anyway, but that won't be the case for every ADJE.

By the way, I'm not sure about the two cannibal eliminations yet.

[David P Bird]

SV, Thanks very much for your comprehensive contribution!

I was aware of your first elimination when the partial fish is confined to two cross lines but must have missed including it in the list. It follows from the general principle that in a JE pattern the division of truths for base digits in the 3 cross lines must be 1 in the JE band and 2 in the partial fish. If one of the partial fish cross lines can't contain a digit, it must therefore be true in the JE band if it turns out to be true in the base cells.

Your second elimination isn't the only one that is available - for example:
If r6c8 is false there is this logic chain:

1. r12c358 must contain 6 base digits
2. In c8 r12c8 must contain 2 base digits and the targets in r8c78 must contain the other two
3. r456c8 therefore can't contain a base digit, and r4c8 <> 2

If r6c8 is true r4c8 <> 2 anyway.

I'm no expert on XSudo constructs and would appreciate a list of the truth and link sets in your second grid including the format of the virtual sets you use. It would help me see what rank it turns out to be.

Note the logic I used for that (2)r4c8 elimination amounts to a truth counting method. Can XSudo replicate it?

For every scenario there will be similar specific eliminations, and I feel that we can't extend the elimination lists to consider every possibility. That said, a note about checking any cells in the partial fish patterns that must contain a base digit might be useful.

DPB

[sultan vinegar]

I'm no expert on XSudo constructs and would appreciate a list of the truth and link sets in your second grid including the format of the virtual sets you use. It would help me see what rank it turns out to be.

No virtual sets, all native. Note the extra box cover for candidate 2 in box 6 to take care of the "almost" part of the ADJE.

16 Truths = {1267C3 1267C5 1267C8 7N12 8N46}
29 Links = {1r2478 2r1478 6r1278 7r1278 9n3 9n5 78n8 2b6 1267b7 1267b8}
3 Eliminations --> r4c9<>2, r7c5<>2, r8c3<>2

For the grid with the DJE (again, all native):

16 Truths = {3489C3 3489C4 3489C9 7N56 9N12}
28 Links = {3r1379 4r1379 8r3679 9r1679 8n3 8n4 79n9 3489b7 3489b8}
34 Eliminations --> r7c3<>3489, r9c4<>3489, r1c256<>3, r1c257<>4, r1c567<>9, r3c156<>3,
r3c157<>4, r3c156<>8, r6c256<>8, r6c56<>9, r7c9<>1, r8c4<>5, r8c3<>6,

Note the logic I used for that (2)r4c8 elimination amounts to a truth counting method. Can XSudo replicate it?

First try, remove 2r6c8, and add cell links in the six cells 12n358. Indeed, those six cells are shown by Xsudo to have to contain a base candidate (if 2r6c8 is false).



Second try, add the 2r6c8 back in. We get a cannibal elimination for 2r4c8.



But, third try, we still get the cannibal elimination for 2r4c8 even with the cell links in cells 12n58 removed???

For every scenario there will be similar specific eliminations, and I feel that we can't extend the elimination lists to consider every possibility. That said, a note about checking any cells in the partial fish patterns that must contain a base digit might be useful.

I think the extra eliminations for the DJE in the covering lines are generally true for DJEs, and don't require any special missing candidates or extensions. They should still work for the "degenerate" cases too. The "true killer" (in the words of Champagne) just got even deadlier!

[ronk]

Second try, add the 2r6c8 back in. We get a cannibal elimination for 2r6c8.
...
But, third try, we still get the cannibal elimination for 2r6c8 even with the cell links in cells 12n58 removed???

Do you mean r4c8? And why the focus on this exclusion?

In the puzzle just discussed the only way for the partial fish cells to hold 3 truths for (2) is for both (2)r4c5 and (2)r6c8 to be true. (This is obvious as without (2)r6c8, only two rows in the partial fish hold (2)s, and without (2)r4c5 only two columns do.)
But in the chain I gave:
(2)r4c5 – (2)r4c123 = (267-3)r5c12,r6c1 = (3-2)r6c8
shows that these two spoilers are weakly linked

1. they can never both be true,
2. the partial fish cells can therefore only hold 2 truths at most,
3. (2) cannot be common to both pairs of base cells,
4. the double JE is true.

The complete (minimal) logic set for your "ADJE":

20 Truths = {3R6 1267C358 7N12 8N46 267B4}
32 Links = {1r2478 2r1478 67r1278 56n1 5n2 9n35 678n8 1267b78}
30 Eliminations --> r1c1267<>6, r1c1246<>7, r2c1249<>7, r7c5<>1267, r8c3<>1267, r2c127<>6,
r2c14<>1, r4c6<>1, r4c9<>2, r8c8<>8, r9c3<>4, r9c5<>5

Including your derived weak link (2r4c5 - 2r6c8) makes for a very nice deduction.

sultan vinegar, note that r4c8<>2 is not an exclusion for this logic set.

[daj95376]

Firstly remember that we are considering only Almost Double JEs with 4 target cells.

Okay.

For a digit to be common to both base sets, its partial fish cells must hold 3 truths. (Take any example case and eliminate the same digit from the cells seen by the two base sets and it becomes very obvious.) Therefore any digit that complies with requirement 3) for the partial fish CANNOT be common to both base sets – end of story – no need to examine cases.

In the puzzle just discussed the only way for the partial fish cells to hold 3 truths for (2) is for both (2)r4c5 and (2)r6c8 to be true. (This is obvious as without (2)r6c8, only two rows in the partial fish hold (2)s, and without (2)r4c5 only two columns do.)
But in the chain I gave:
(2)r4c5 – (2)r4c123 = (267-3)r5c12,r6c1 = (3-2)r6c8
shows that these two spoilers are weakly linked

1. they can never both be true,
2. the partial fish cells can therefore only hold 2 truths at most,
3. (2) cannot be common to both pairs of base cells,
4. the double JE is true.

I'm afraid I just don't understand the basis for your approach stemming from the sashimi fish.

And I don't understand how it is that you don't recognize the Sashimi Swordfish stemming from your JE definition. For example:

Code: Select all

    JExocet chute in [band 1]
    - with -
    r1c12=k; Sashimi Swordfish c347\r582 w/fin cell R=r3c7  =>  -(k)r2c89
    r1c12=k; Sashimi Swordfish c347\r583 w/fin cell Q=r2c4  =>  -(k)r3c56
    *-------*-------*-------*
    | B B ~ | ~ ~ ~ | ~ ~ ~ |
    | ~ ~ ~ | Q . . | y . . |  <-- line containing Q=r2c4
    | ~ ~ ~ | x . . | R . . |  <-- line containing R=r3c7
    *-------*-------*-------*
    | . . . | . . . | . . . |
    | . . S | S . . | S . . |  <-- first  cover sector intersecting the cross-lines
    | . . . | . . . | . . . |
    *-------*-------*-------*
    | . . . | . . . | . . . |
    | . . S | S . . | S . . |  <-- second cover sector intersecting the cross-lines
    | . . . | . . . | . . . |
    *-------*-------*-------*


For your grid, this translates into:

Code: Select all

 r7c1=1; Sashimi Swordfish c358\r248 w/fin cell R=r9c5  =>  -1r8c46
 +-----------------------------------+
 |  .  .  .  |  .  .  .  |  .  .  1  |
 | ~1  . *1  |  1 *1  .  |  .  .  .  |  <-- first  cover sector intersecting the cross-lines
 | ~1  .  .  |  1  .  1  |  .  .  .  |
 |-----------+-----------+-----------|
 |  .  .  .  |  . *1  1  |  . *1  .  |  <-- second cover sector intersecting the cross-lines
 |  .  .  .  |  1  .  1  |  1  .  .  |
 |  .  1  .  |  .  .  .  |  .  .  .  |
 |-----------+-----------+-----------|
 | =1  .  .  |  . ~1  .  | ~1 ~1  .  |
 |  .  . ~1  | -1  . -1  |  1 *1  .  |  <-- line containing Q=r8c8
 |  .  . ~1  |  1 #1  1  |  1  .  .  |  <-- line containing R=r9c5
 +-----------------------------------+


Code: Select all

 r7c12=6; Sashimi Swordfish c358\r128 w/fin cell R=r9c5  =>  -6r8c6
 +-----------------------------------+
 |  6  6 *6  |  . *6  6  |  6 *6  .  |  <-- first  cover sector intersecting the cross-lines
 |  6  6 *6  |  . *6  .  |  6 *6  .  |  <-- second cover sector intersecting the cross-lines
 |  6  6  .  |  .  .  6  |  6  .  .  |
 |-----------+-----------+-----------|
 |  .  .  .  |  6  .  .  |  .  .  .  |
 |  6  6  .  |  .  .  .  |  .  .  .  |
 |  .  .  .  |  .  .  .  |  .  .  6  |
 |-----------+-----------+-----------|
 | =6 =6  .  |  . ~6  .  | ~6 ~6  .  |
 |  .  . ~6  |  .  . -6  |  6 *6  .  |  <-- line containing Q=r8c8
 |  . ~6 ~6  |  . #6  6  |  6  .  .  |  <-- line containing R=r9c5
 +-----------------------------------+


Code: Select all

 r7c12=7; Sashimi Swordfish c358\r128 w/fin cell R=r9c5  =>  -7r8c46
 +-----------------------------------+
 |  7  7 *7  |  7 *7  7  |  . *7  .  |  <-- first  cover sector intersecting the cross-lines
 |  7  7 *7  |  7 *7  .  |  . *7  7  |  <-- second cover sector intersecting the cross-lines
 |  7  7  .  |  7  .  7  |  .  .  7  |
 |-----------+-----------+-----------|
 |  .  .  .  |  .  .  .  |  7  .  .  |
 |  7  7  .  |  7  .  7  |  .  .  .  |
 |  7  .  .  |  7  .  7  |  .  .  .  |
 |-----------+-----------+-----------|
 | =7 =7  .  |  . ~7  .  |  . ~7  .  |
 |  .  . ~7  | -7  . -7  |  . *7  7  |  <-- line containing Q=r8c8
 |  . ~7 ~7  |  7 #7  7  |  .  .  7  |  <-- line containing R=r9c5
 +-----------------------------------+


Code: Select all

 r7c12=2; Sashimi Swordfish c358\r148 w/fin cell R=r9c5                =>  -2r8c4
 -or-
 indirect fin cell: (2-3)r6c8 = 3r6c1 - (3=249)r4c123 - 2r4c5 = 2r9c5  =>  -2r8c4
 +-----------------------------------+
 |  2  2 *2  |  .  .  .  |  2 *2  .  |  <-- first  cover sector intersecting the cross-lines
 |  .  .  .  |  .  .  2  |  .  .  .  |
 |  2  2  .  |  .  .  .  |  2  .  2  |
 |-----------+-----------+-----------|
 |  2  2 *2  |  . *2  .  |  . *2  2  |  <-- second cover sector intersecting the cross-lines
 |  2  2  .  |  2  .  .  |  2  .  2  |
 |  2  .  .  |  2  .  .  |  . @2  .  |  <-- indirect fin cell r6c8
 |-----------+-----------+-----------|
 | =2 =2  .  |  . ~2  .  | ~2 ~2  .  |
 |  .  . ~2  | -2  .  .  |  2 *2  2  |  <-- line containing Q=r8c8
 |  . ~2 ~2  |  2 #2  .  |  2  .  2  |  <-- line containing R=r9c5
 +-----------------------------------+


Since r4c5 is part of the Sashimi Swordfish for <2>, the only anomaly is the indirect fin cell r6c8 ... and it's resolved by an ALS chain forcing r9c5=2.

Thus, any values in r7c12 result in those values not being in r8c46. This forces the conjugate values in r7c12 to be true in r8c46.

[Edit: properly identifies cover sectors outside the base chute]

[champagne]

For the exocet basics, that example is very interesting.

If any digit fails to comply with the exocet definition due to a "spoiler" to use David's terms, and if the PM context shows that the spoiler can be erased when the digit is part of the base, this remains an exocet digit. I don't know what is the appropriate name for that extended definition of the exocet, but it is surely an important concept.

I have to check again, but this was also the case in a recent example given by JC Van Hay

[sultan vinegar]

Do you mean r4c8?

Oops, I do indeed mean r4c8. I'll fix that.

And why the focus on this exclusion?

Because that's what I was asked to do, here:

Note the logic I used for that (2)r4c8 elimination amounts to a truth counting method. Can XSudo replicate it?

Finally,

I don't know what is the appropriate name for that extended definition of the exocet, but it is surely an important concept.

How about a "Phantom Menace"? I would class it as a "get out clause" (similar to the "box singles extension") rather than a new type of exocet.

[champagne]

I don't know what is the appropriate name for that extended definition of the exocet, but it is surely an important concept.

How about a "Phantom Menace"? I would class it as a "get out clause" (similar to the "box singles extension") rather than a new type of exocet.

Surely, this is not a new type of exocet, rather an extension of the way to prove the digit per digit property.

This fit better with my lay-out that the box extension, still in my to do list, but not with a high priority.

I'll surely keep it separate from the basic case, may-be with a kind of "rating" of the proof

. basic if the proof is within a floor
. a rating depending on the tool used for others
_ chain elimination of the spoiler
_ dynamic expansion to get rid of it (in the sense of serate)
_ extended dynamic expansion
_ ...

but for a player stepping in that field, a split in 2 groups can be enough. Then, a dynamic extension using all objects till nested kites ... should be appropriate.

It should be relatively easy to include that in my code

EDIT I am wondering whether the "box singles extension" will not be seen in such a process;

[David P Bird]

In a previous post I confessed that I realised that the link between a spoiler candidate in the partial fish and the DJE was not conjugate as I had been assuming. If one of these spoilers is false the DJE must be true, but if it's true the DJE may be either true or false depending on how many other partial fish candidates are true. The link between a spoiler and the DJE is therefore an example of a strong-only link where the nodes can't both be false, but could both be true. However it's a rather strange version of this link type as it's not symmetrical in the way it works.

Here's puzzle no 5 in the list as another example of a base digit (8) with multiple spoiler cells in its partial fish:
..............1.23..2.345.....6.......3.27.5..8...37.....9.54....4.7...51....627.;759669;dob;12_12_19;156;1689 ;169

Code: Select all

  *-------------------------*-------------------------*-------------------------*
  | 3456789 1345679 156789  | 2578    5689    289     | 1689B   14689   146789  |
  | 456789  45679   56789   | 578     5689    <1>     | 689B    <2>     <3>     |
> | 6789  # 1679    <2>     | 78   #  <3>     <4>     | <5>     1689T   16789   |
  *-------------------------*-------------------------*-------------------------*
  | 24579   124579  1579    | <6>     14589   89      | 1389    13489   12489   |
> | 469     1469    <3>     | 148  #  <2>     <7>     | 1689T   <5>     14689T  |
  | 24569   <8>     1569    | 145     1459    <3>     | <7>     1469    12469   |
  *-------------------------*-------------------------*-------------------------*
  | 23678   2367    678     | <9>     18      <5>     | <4>     1368    168 B   |
> | 23689 # 2369    <4>     | 1238 #  <7>     28 #    | 13689   13689T  <5>     |
  | <1>     359     589     | 348     48      <6>     | <2>     <7>     89  B   |
  *-------------------------*-------------------------*-------------------------*
    689     169               18               8! 

ADJE (1689)[a=r12c7,r5c9,r8c8][b=r79c9,r3c8,r5c7], spoilers A=(8)r3c1, B=(8)r5c4, C=(8)r8c6
The partial fish cells for (8) are flagged with # symbols. If any of the spoilers false, the partial fish cells for (8) would be restricted to two houses, so all three must be true to stop the DJE.

It now takes strong links between each of the spoilers and the DJE to eliminate nine (8)s to prove the DJE is true.
(8)SpoilerA:r3c1 = DJE - (1689=7)r3c9 - (7=8)r3c4 – Loop => r3c12 <>7, r3c89 <> 8
(8)SpoilerB:r5c4 = DJE - (1689=7)r3c9 - (7=8)r3c4 => r1289c4 <> 8
(96)r12c5 = (9)r1c6 - (9=8) - (8):SpoilerB:5c4 = DJE - (1689=7)r3c9 = (7=8)r3c4 => r12c5 <> 8
(8)SpoilerC:r8c6 = DJE - (169=7)r3c9 - (7=8)r3c4 => r1c6 <> 8
(8)r3c4 Single, r3c1,r5c4 <> 8 Hence DJE = True.

SV, Ronk, Your various XSudo constructs (thank you) are above my head at this time. I assume the difference between counts for Link and Truth Sets indicates that there must be multiple triplets involved, but it doesn't give any sort of processing order for them to produce the eliminations shown. I had hoped that defining virtual sets would help simplify the picture, but as it stands, XSudo just appears to be working like a black box for these complex patterns.

DAJ, I've read, but haven't had time to study, your post. When I said I didn't understand the basis of you approach it was because your fish included cells in the DJE band, when my focus was on the partial fish in the other two bands which I considered to be the crux of the matter.

As a minor point I would point out that that, as defined, the cross lines are the rows or columns that intersect the JE band, not those that run parallel to it.

When I have more time I'll work through your post grid by grid.

DPB

[daj95376]

DAJ, I've read, but haven't had time to study, your post. When I said I didn't understand the basis of you approach it was because your fish included cells in the DJE band, when my focus was on the partial fish in the other two bands which I considered to be the crux of the matter.

As a minor point I would point out that that, as defined, the cross lines are the rows or columns that intersect the JE band, not those that run parallel to it.

When I have more time I'll work through your post grid by grid.

Sorry about the mix-up on properly identifying the cross-lines. I tend to quickly identify the base sectors and concentrate on the cover sectors for the Sashimi fish. The location of the three sectors for the base set are fixed based on the location of the base and target cells. What needs to be identified is the cover sectors for the Sashimi Swordfish. I often mistake the two cover sectors outside the JE band as "cross-lines". In actuality, I don't believe that you ever gave these cover sectors a "name".

[daj95376]

Here's puzzle no 5 in the list as another example of a base digit (8) with multiple spoiler cells in its partial fish:
..............1.23..2.345.....6.......3.27.5..8...37.....9.54....4.7...51....627.;759669;dob;12_12_19;156;1689 ;169

Code: Select all

  *-------------------------*-------------------------*-------------------------*
  | 3456789 1345679 156789  | 2578    5689    289     | 1689B   14689   146789  |
  | 456789  45679   56789   | 578     5689    <1>     | 689B    <2>     <3>     |
> | 6789  # 1679    <2>     | 78   #  <3>     <4>     | <5>     1689T   16789   |
  *-------------------------*-------------------------*-------------------------*
  | 24579   124579  1579    | <6>     14589   89      | 1389    13489   12489   |
> | 469     1469    <3>     | 148  #  <2>     <7>     | 1689T   <5>     14689T  |
  | 24569   <8>     1569    | 145     1459    <3>     | <7>     1469    12469   |
  *-------------------------*-------------------------*-------------------------*
  | 23678   2367    678     | <9>     18      <5>     | <4>     1368    168 B   |
> | 23689 # 2369    <4>     | 1238 #  <7>     28 #    | 13689   13689T  <5>     |
  | <1>     359     589     | 348     48      <6>     | <2>     <7>     89  B   |
  *-------------------------*-------------------------*-------------------------*
    689     169               18               8! 

ADJE (1689)[a=r12c7,r5c9,r8c8][b=r79c9,r3c8,r5c7], spoilers A=(8)r3c1, B=(8)r5c4, C=(8)r8c6

When viewed as a Sashimi Swordfish, then [c1] and [c4] are [edit: the logical sectors to be] in the cover set ... leaving C=(8)r8c6 as the only spoiler cell for <8>.

[Leren]

The DJE properties of puzzles such as Puzzle no 5 can be established from a different POV as follows.

Suppose you have shown that;

1. The puzzle has ADJE properties except that a single "Rogue" digit (in this case 8) has "Spoiler" cells;

and, via a separate process

2. The conjugate properties of the 2 AJE Base cell AAHSs.

2. For Puzzle 5, 2 shows that r13c9, r8c7 <> 1689 or, in Champagne's parlance, r1c9 = 4, r3c9 = 7, r8c7 = 3. This is well known.

However further eliminations can also immediately be made without any further consideration of the puzzle's properties.

Because the Rogue digit 1 will not be true in one of the Base AAHs, that AAHs and its Target cells will form a 169 Jexocet (of course at this stage we don't know which one it is but that doesn't matter in what follows).

Thus 2 of the 3 "Good" digits 169 will appear in 2 of the 4 Target cells. Suppose those digits are 1 and 6. We also know that the third "Good" digit 9 satisfies the Jexocet row counts (by 1 above). Thus 9 must be true in the Exocet stack
in Rows 3, 5 and 8. Noting the AAHS eliminations above and the companion cells in those rows it is clear that 9 must occupy one of the 2 other Target cells.

In general then, the 3 ""Good" digits 169 must occupy 3 of the 4 Target cells. Thus the 3"Good" digits 169 can be eliminated from any cells that can see all 4 Target cells.

For Puzzle 5 this leads to r4c8 <> 89 r6c8 <> 169.

I've tested this property against all 2214 puzzles in Champagne's Conjugate AAHS file with no failures, so I am reasonably confident that it is generally true.

A corollary of all this is that, if all 4 Target cells only contain Base digits then the "Rogue" digit must also occupy 1 Target cell. In puzzle 5 this means that if r5c9 <> 4 and r8c8 <> 3
then the 4 Target cells must contain the 4 Base digits in exactly the configuration for a DJE solution. In puzzle 5 since r1c9 = 4 and r8c7 = 3 this property is established and the DJE for this puzzle is confirmed.

<Edit> I forgot to mention that if you establish the Double Exocet property in this way you can make all Double Exocet eliminations except for the fin cell eliminations for the "Rogue" digit.

None of this adds great value to actually solving the puzzle, as it solves pretty easily with just the well known AAHs eliminations, but it may prove of interest to those who may wish to establish DJE properties for aesthetic reasons, after a complementary AAHs has been established.

Leren