## Exotic patterns a resume

Advanced methods and approaches for solving Sudoku puzzles

### Re: Exotic patterns a resume

Code: Select all
`9    8     12345 |7     234    1234    |6     125   123  7    14    1234  |5     2346   12346   |123   8     9    123  15    6     |128   2389   12389   |12357 1257  4    --------------------------------------------------------8    14567 145   |3     2467   24567   |12457 9     1267 1346 2     13459 |468   46789  456789  |13457 14567 1367 346  4567  3459  |246   1      245679  |23457 24567 8    --------------------------------------------------------5    146   124   |12468 234678 1234678 |9     12467 1267 1246 3     7     |9     246    1246    |8     1246  5    1246 9     8     |1246  5      12467   |1247  3     1267`

Guardians meets Double JExocet:

We have an AADJE (bases r7c23, r8c45; targets r9c4,r8c8; r9c1,r7c8). The two spoilers are 12r1c8. Note that the DJE must be false, as a true DJE would eliminate 1246 from r8c1, leaving no solution. So one of the spoilers (Guardians) must be true to prevent the illegal DJE. So r1c8 <> 5, and we got to the end with singles, locked candidates and a couple of kites.

Xsudo:

Hidden Text: Show

17 Truths = {1246C1 1246C4 1246C8 7N23 8N156}
25 Links = {1r3578 2r3678 4r5678 6r5678 1n8 1246b7 1246b8}
1 Elimination --> r1c8<>5.
Last edited by sultan vinegar on Sat Sep 28, 2013 2:24 am, edited 1 time in total.
sultan vinegar

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### Re: almost complementary AAHS

withdrawn not the expected action
Last edited by champagne on Sun Sep 22, 2013 5:35 pm, edited 1 time in total.
champagne
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### Re: Exotic patterns a resume

Leren wrote:I think this might be a bit tougher

98.7..6..7...6......6....5.4..3.9.....8.573.....8......2......1..9...5....7.839..;975059;GP;13_03;21;1234 ;3;34

Yes. The key is to derive r8c5=7 ... but good luck on doing so.
daj95376
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### Re: Exotic patterns a resume

daj95376 wrote:
Leren wrote:I think this might be a bit tougher

98.7..6..7...6......6....5.4..3.9.....8.573.....8......2......1..9...5....7.839.. ;975059;GP;13_03;21;1234 ;3;34

Yes. The key is to derive r8c5=7 ... but good luck on doing so.

r8c5=7 is conditionally true for JE base r1c89=1234 because [edit: there is a weak inference between target cell r3c5 and 3 digit spoiler cell r8c5 ... (1234-9)r3c5 = (9-7)r7c5 = (7-124)r8c5.] The Xsudo logic set is ...

16 Truths = {124C37 123479C5 1N89 3N12}
28 Links = {12r1346 3r13 4r1367 2n3 13678n5 26n7 124b13}
11 Eliminations --> r6c689<>4, r4c28<>1, r4c89<>2, r12c3<>3, r2c3<>5, r7c5<>4,

The targets for base r1c89 are r2c3 and r3c5 but the exclusions (r2c3<>35 and r3c5<>none) are unconventional. So far I've found nothing similar for JE base r3c12=1234.

Leren, thanks for posting this "a bit tougher" ADJE.
Last edited by ronk on Mon Sep 23, 2013 4:33 am, edited 3 times in total.
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### Re: Exotic patterns a resume

Leren wrote:I think this might be a bit tougher

98.7..6..7...6......6....5.4..3.9.....8.573.....8......2......1..9...5....7.839..;975059;GP;13_03;21;1234 ;3;34
Code: Select all
`+----------------------+---------------------+-----------------------+| 9      8       12345 | 7      1234   1245  | 6      1234    234    || 7      1345    12345 | 12459  6      12458 | 1248   123489  23489  || 123    134     6     | 1249   12349  1248  | 12478  5       234789 |+----------------------+---------------------+-----------------------+| 4      1567    125   | 3      12     9     | 1278   12678   25678  || 126    169     8     | 1246   5      7     | 3      12469   2469   || 12356  135679  1235  | 8      124    1246  | 1247   124679  245679 |+----------------------+---------------------+-----------------------+| 3568   2       345   | 4569   479    456   | 478    34678   1      || 1368   1346    9     | 1246   1247   1246  | 5      234678  234678 || 156    1456    7     | 12456  8      3     | 9      246     246    |+----------------------+---------------------+-----------------------+`
#1. Analysis of (1234)r3c12,r1c89 [Note : C(S)= Contradiction using only Singles] :

r3c1=1,r1c8=1 -> LC(1r89c2) :=> r5c4=1; 11 Singles; Chain[3] : SF(4R9C67) : r1c6=r78c6-r9c4=r9c89-r7c7=r23c7 :=> -4r1c9(=2); C(S)
r3c2=1,r1c8=1 -> LC(1r89c1) :=> r5c4=1; 11 Singles; Chain[5] : 2r5c1=(24)r5c89,r1c9-(24=6)r9c9-6r7c8=6r7c1 :=> -6r5c1(=2); C(S)
r3c1=2,r1c8=2 -> LC(2r89c9) :=> r5c4=2; C(S)
r3c1=2,r1c9=2 -> LC(2r89c8) :=> r5c4=2; 11 Singles; Chain[8] : Kraken 1r126c3 -> 4r1c36=4r23c7 :=> -4r1c8; C(S)
Code: Select all
`   1r1c3-1r1c6=*Wing[(4=*5)r1c6-(5=4)r7c6-4r7c7=4r23c7]-4r1c8   ||   1r2c3-4r2c3=*FXWing(4r17c3,r237c7)-4r1c8   ||   1r6c3-(1=6)r5c1-6r7c1=6r7c8-(6=4)r9c9-4r7c7=4r23c7-4r1c8`
r3c12,r1c89=3 -> 3C5 is empty
r3c2=4,r1c8=4 -> C(S)
r3c2=4,r1c9=4 -> 17 Singles; NP(12)r4c35 :=> LC(1r56c8) :=> -1r1c8(=2); C(S)

Conclusion : r3c12=ab->r1c89=cd where {a,b,c,d}={1,2,3,4} :=> -1234r1c3(=5),r3c79

#2. NP(12)r4c35 :=> NP(78)r34c7 :=> r7c7=4; 4 Singles; NP(12)r46c3 :=> -1r5c1(=6),r456c2; ste

Comment : apparently, there is no exocet !? IOW, 2#1+1 in this puzzle.
JC Van Hay

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### Re: Exotic patterns a resume

Champagne wrote: I have to thanks abi who pointed a new kind of pattern having some solving potential.

Hi Champagne, it was easy for me to implement abi's method. I tested it against your recent list of puzzles and got a response via either a ADJE move or an "ABI" move for all except the following 7 puzzles.

369483, 513302, 715461, 773111, 804218, 805097 and 836915."

I've made a minor extension to abi's method as illustrated for puzzle 369483.

Code: Select all
`*--------------------------------------------------------------------------------*| 2359    2378    389      | 789     6       4        | 23589   389     1        || 1569    1678    1689     | 1789    2       3        | 4589    489     4789     || 1239    12378   4        | 5       1789    1789     | 2389    6       23789    ||--------------------------+--------------------------+--------------------------|| 4       13      135      | 6       13589   2        | 1389    7       389      || 1236    1236    7        | 4       1389    189      | 1389    5       3689     || 8       9       1356     | 137     1357    157      | 1234    134     2346     ||--------------------------+--------------------------+--------------------------|| 1369    1368    2        | 1389    4       1689     | 7       1389    5        || 1369    4       13689    | 13789   135789  156789   |A1389    2      A389      || 7       5       1389     | 2      B1389   B189      | 6      c13489  c3489     |*--------------------------------------------------------------------------------*`

The potential complementary AAHs are r8c79 and r9c56. As per the ABI method I show that 189 can't be in both AAHSs. Thus 3 must be in at least 1 AAHS. So far nothing new.

There is no single cell <1389> to force 3 to be in both AAHSs. Instead we have r9c89 = <13489> with S/L on 4 in Row 9, so either r8c9 = <1389> or r9c9 = <389>.

Either would force a contradiction if the AAHSs were complementary, so r9c5 = 3 and r8c7 or r8c9 = 3 and we get eliminations as per ABI's method.

In this way I get a response for puzzles 369483, 715461, 804218 and 805097. That leaves 513302, 773111 & 836915 with no response from me at this stage.

Perhaps you could check these 3 for me. There may be a bug or some inadequacy in my code. Meanwhile I'll investigate further.

Leren
Leren

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### Re: Exotic patterns a resume

Leren wrote: it was easy for me to implement abi's method. I tested it against your recent list of puzzles and got a response via either a ADJE move or an "ABI" move for all except
...
Perhaps you could check these 3 for me. There may be a bug or some inadequacy in my code. Meanwhile I'll investigate further.

Hi Leren,

You went to fast. I did not touch the code since I posted my remark, but I agree that implementation of that will not be a problem.

It's obvious for me that we will see all the panel from an immediate collapse as in abi's example to something tougher as in the former puzzle (975059) for a full complementarity;

For the time being, I am still more on the extraction of specific families than in the solving process. Many new things came recently and I have to apply that to the data base of potential hardest.

Here, my priority (likely to-day) will be to extract all puzzles having a potential pattern for a 3 or 4 digits complementarity in AAHS.

The next step is what you already implemented, to filter all puzzles having an easy response to to feed our experts. They already can see your 3 "unsolved" puzzles.

We should be in a position to compare our results this week, and may be others will come with adequate code.
champagne
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### Re: Exotic patterns a resume

I have extracted all puzzles having a pattern fitting with a possible established complementary double AAHS or with the pattern described by "abi"

In fact nothing objects to have a similar solving potential with 4+3 digits or 3+3 digits.

This open again the field to break these puzzles where 3 or 4 digits are interleaved in such a way that the chain nets have some difficulty to solve the puzzle.

This will my next search after I have finished the analysis of the lot extracted.
champagne
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### Re: Exotic patterns a resume

Sorry to be late to the party ...

98.7..6..7..5...8...6.....48..3...9..2...........1...85.....9...379....5..8.5..3.;174173;GP;12_11
Code: Select all
` *--------------------*-----------------------*--------------------* | 9      8     12345 | 7      234    1234    | 6     125 #   123  | 12!   | 7      14    1234  | 5      2346   12346   | 123   8       9    |  | 123    15    6     | 128    2389   12389   | 12357 1257    4    | 12 *--------------------*-----------------------*--------------------* | 8      14567 145   | 3      2467   24567   | 12457 9       1267 |  | 1346   2     13459 | 468    46789  456789  | 13457 14567   1367 | 146 | 346    4567  3459  | 246    1      245679  | 23457 24567   8    | 246 *--------------------*-----------------------*--------------------* | 5      146 B 124 B | 12468  234678 1234678 | 9     12467 T 1267 | | 1246   3     7     | 9      246 b  1246 b  | 8     1246  T 5    | | 1246 t 9     8     | 1246 T 5      12467   | 1247  3       1267 | *------------------*----------------------*-----------------------*    S                     S                            S`

Almost Double JExocet [a=r7c23,r8c8,r9c4][b=r8c46,r7c8,r9c1] => r8c1 <> 1246 (false)

SV's transformation of the DJE spoilers in r1c8 to Deadly Pattern guardians is absolutely right, but I couldn't find any kite patterns without resorting to branching.

Assuming the spoilers aren't so conveniently placed, I wanted to see what other deductions can be made in the JE tier when the DJE is clearly impossible because it would produce an empty cell.

In this puzzle there can only be a single spoiler digit, (1) or (2) r1c8, which consequently ** must be true in both pairs of base cells. (If there are two potential spoilers the options would probably be too numerous to be decisive.)

Here is tier 3 repeated and then re-labelled using (a) to represent the spoiler and (bcd) the other digits to show how thy must repeat in the mini-lines as shown in < this post >:

Code: Select all
` *------------------*----------------------*------------------* | 5    146   124   | 12468 234678 1234678 | 9     12467 1267 |   | 1246 3     7     | 9     246    1246    | 8     1246  5    | | 1246 9     8     | 1246  5      12467   | 1247  3     1267 | *------------------*----------------------*------------------*  *------------------*----------------------*------------------* | 5    ab    ab    | d8    d38-7   d38-7  | 9     c7    c7   | [abcd] = [1246] | d    3     7     | 9     ac     ac      | 8     b     5    | [a]    = [1 or 2]  | c    9     8     | b     5      7-abdc  | ad-7  3     ad-7 | *------------------*----------------------*------------------*`

The non-spoiler digits (bcd) will repeat in the regular mini-line diagonal directions for the JE pattern. (a) will be true with one of (bcd) in each mini-row, while the other mini-rows will contain two non-base digits repeating in patterns forced by the givens. Here the cells seen by both sets of base cells can't contain (abc) but can contain (d). Because they will be limited to two truths in the partial fish cells, (b) and (c) must be true in at least one of the target cells, but could be true in two, and similarly for (d) in r7c4 & r8c1.

Note the targets for the r1c23 JE at r8c8,r9c4 must both contain (b) as there are no non-base digits available.
Now r9c6 must contain a non-base digit which can only be 7 which provides the eliminations shown.

Implementing these eliminations isn't enough for my methods to solve the puzzle, but adding the (5)r1c8 elimination leads to r7c2 = 6. So, as (a) = (1) or (2), (b)r7c2 = (6) and can be assigned in r8c8 & r9c4. This in turn leads to r7c4 = 1, and so (a)r7c3 must = 2 from which the solution falls out.

To avoid being labelled a hypocrite, I should register such impossible DJEs as a pattern that gives rise to the inferences I've used, but then there is the problem about how to notate the algebra involved.

** On reviewing this I found I'd missed an important logical step, namely that neither component JE can be true.
If r7c23,r8c8,r9c4 were true, r8b9 would have to hold two base digits, and if r8c46,r7c8,r9c1 were true, this would hold for r9b7, neither of which is possible.
Last edited by David P Bird on Fri Sep 27, 2013 11:39 am, edited 1 time in total.
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### Re: Exotic patterns a resume

9..8..7...8..7..6...5..4...8....53...2.1.........8...94....8..3.1.2..8......6..7.;11.10;1.20;1.20;GP;Kz1 b;15666
98.7..6..5...9..4...3..2...81......5.3...5.....5...2....8..3..1...6..9......4..7.;11.30;11.30;9.40;GP;Kz1 b;17065

these 2 puzzles have complementary AAHS;

I found no identified exotic pattern and could not prove the complementary property with my code (voluntarily limited in that situation).
champagne
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### Re: Exotic patterns a resume

[Withdrawn: realized there was an error in the logic.]
Last edited by daj95376 on Mon Sep 30, 2013 7:42 pm, edited 1 time in total.
daj95376
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### Re: Exotic patterns a resume

daj95376 wrote:Excellent observation. Since I wasn't able to fully follow champagne's or DPB's logic on this puzzle, I decided to see what made sense to me.

Hi Danny,

'abi' path is very simple and can surely be reproduced in other puzzles, so it is worth being understood.
Although the ADJE pattern is in background, the logic does not use it at all. we consider directly AAHS r7c23 and r8c56.

Just considering the floors 4 and 6 separately, digits 4 and 6 can not occupy both AAHS (this is a kind of ADJE effect)
For digit 1, you come to the same conclusion adding the "5" floor (a kind of ADJE spoiler elimination)

So you have now 3 of the 4 digits that can not occupy both AAHS

The fourth digit (2) must be in one of the AAHS, may be in both.

In our example, r8c1 sees both AAHS, so we can not have complementary digits, the minimum is that one digit must be there twice. It can only be '2'.

so we have r7c3=2 and 2r8c56
champagne
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### Re: Exotic patterns a resume

David P Bird wrote:SV's transformation of the DJE spoilers in r1c8 to Deadly Pattern guardians is absolutely right, but I couldn't find any kite patterns without resorting to branching.

I should have said grouped kites (finned mutant x-wings). The first one is near the start:

2 Truths = {6R4 6C4}
3 Links = {6r9 6c9 6b5}
1 Elimination --> r9c9<>6

And I'll let you find the second one (digit 4, right near the end).
sultan vinegar

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### Re: Exotic patterns a resume

champagne wrote:9..8..7...8..7..6...5..4...8....53...2.1.........8...94....8..3.1.2..8......6..7.;11.10;1.20;1.20;GP;Kz1 b;15666
98.7..6..5...9..4...3..2...81......5.3...5.....5...2....8..3..1...6..9......4..7.;11.30;11.30;9.40;GP;Kz1 b;17065

these 2 puzzles have complementary AAHS;

I found no identified exotic pattern and could not prove the complementary property with my code (voluntarily limited in that situation).

Hi Champagne, any chance of listing the cells with the complementary AAHS so that I can get to work on them (or is that what the code at the end of the grid is?).

For what it's worth, I think that the complementary AAHS is more fundamental than the Double Exocet, as you can derive the Double Exocet results from the complementary AAHS.

Hidden Text: Show
Here is a DJE, with traditional explanation:

1......8...71....609.....5...56....7..17.4..57.....34.57.2......1......2..2.61.7.

16 Truths = {3489C3 3489C4 3489C9 7N56 9N12}
36 Links = {3r13789 4r13789 8r36789 9r16789 8n3 8n4 79n9 3489b7 3489b8 3489b9}
39 Eliminations --> r7c3<>3489, r9c4<>3489, r1c256<>3, r1c257<>4, r1c567<>9, r3c156<>3,
r3c157<>4, r3c156<>8, r6c256<>8, r8c7<>489, r6c56<>9, r8c8<>39, r7c9<>1,
r8c4<>5, r8c3<>6.

An alternative view:

Step 1. Show the 4 base cells are a virtual locked set. I'm going to use the complementary finned-fish to show this:

19 Truths = {3R2458 4R248 8R2458 9R2458 7N56 9N12}
35 Links = {3489r7 3489r9 3c12568 4c1257 8c12567 9c15678 3489b7 3489b8}
8 Eliminations --> r7c3<>3489, r9c4<>3489.

The four base cells share two columns, so at most 2 instances of any digit can occur. Note that for digits 3,8,9, the fish structure has 4 row truths and 5 column links. For digit 4, the fish structure has 3 row truths and column links. So, let's assume that two of the four base cells are digit 3. This leaves us with 4 row truths and 3 column links remaining which is an illegal fish. Hence, the assumption that two of the four base cells are digit 3 is false. Likewise for digits 8,9, and the same logic for candidate 4 except that there is one less row and column. All up, there must be exactly one of digits 3,4,8,9 in the base cells, i.e. a virtual locked set.

Step 2. Account for the eliminations due to the four target cells also being a locked set. Note that I don't need to prove that the target cells are a locked set to show this. There are finned fish and a multi-fish left over once we perform the eliminations from the first step.

Finned fish:

12 Truths = {3489C3 3489C4 3489C9}
16 Links = {349r1 348r3 89r6 3489r8 3489b9}
5 Eliminations --> r8c7<>489, r8c8<>39.

Multi-Fish:

12 Truths = {3489C3 3489C4 3489C9}
12 Links = {349r1 348r3 89r6 8n3 8n4 79n9}
26 Eliminations --> r1c256<>3, r1c257<>4, r1c567<>9, r3c156<>3, r3c157<>4, r3c156<>8,
r6c256<>8, r6c56<>9, r7c9<>1, r8c4<>5, r8c3<>6.

Finally, has anyone looked for exocet logic based on Franken or Mutant fish? Here is a concept for a Sashimi Franken Swordfish based exocet. There are three base cells and three target cells, but the logic is the same.

Hidden Text: Show

15 Truths = {1234R1 1234R6 45N3 9N5 1234B7}
23 Links = {1234r9 1234c1 1234c3 1234c5 17n2 6n8 1234b4}
6 Eliminations --> r17c2<>5, r17c2<>6, r6c8<>56.
Last edited by sultan vinegar on Sat Sep 28, 2013 2:21 am, edited 2 times in total.
sultan vinegar

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### Re: Exotic patterns a resume

Hi "sultan vinegar",

Congratulations first for the wording of your start. At the end, it's the same as 'abi's path, but I like it.

sultan vinegar wrote:any chance of listing the cells with the complementary AAHS so that I can get to work on them (or is that what the code at the end of the grid is?).

Hidden Text: Show
Code: Select all
`9     346   12346 |8    1235 1236 |7     12345 1245  123   8     1234  |359  7    1239 |12459 6     1245  12367 367   5     |369  1239 4    |129   12389 128   ----------------------------------------------------8     4679  14679 |4679 249  5    |3     124   12467 3567  2     34679 |1    349  3679 |456   458   45678 13567 34567 13467 |3467 8    2367 |12456 1245  9     ----------------------------------------------------4     5679  2679  |579  159  8    |12569 1259  3     3567  1     3679  |2    3459 379  |8     459   456   235   359   8     |3459 6    139  |12459 7     1245  r4c8r6c8 r1c9r2c99     8      124   |7     135   14    |6    1235  23     5     267    1267  |138   9     168   |1378 4     2378   1467  467    3     |1458  1568  2     |1578 1589  789    --------------------------------------------------------8     1      24679 |2349  2367  4679  |347  369   5      2467  3      24679 |12489 12678 5     |1478 1689  46789  467   4679   5     |13489 13678 16789 |2    13689 346789 --------------------------------------------------------2467  245679 8     |259   257   3     |45   256   1      12347 2457   1247  |6     12578 178   |9    2358  2348   1236  2569   1269  |12589 4     189   |358  7     2368   r5c1r6c1 r2c2r3c2`

these are my 2 starting positions with the complementary AAHS
but I have doubts that in that situation this is the best start.

sultan vinegar wrote:For what it's worth, I think that the complementary AAHS is more fundamental than the Double Exocet,
as you can derive the Double Exocet results from the complementary AAHS.

IMO, if it is a classical DJE, a player will immediately see it,
but if it is an extended exocet, than the direct attack of the double AAHS will pay

BTW, in that specific case, my solver had seen 2 exocets in extended mode,

r7c2r7c3 r8c5 r8c8
r8c5r8c6 r9c1 r9c7

That's the reason why that puzzle was posted
This is not a double exocet, the minimum condition to have a double exocet is that one of the base sees the target of the other one (outside of it).

sultan vinegar wrote:Finally, has anyone looked for exocet logic based on Franken or Mutant fish? Here is a concept for a Sashimi Franken Swordfish based exocet. There are three base cells and three target cells, but the logic is the same.

May be we will see such an example in puzzles not solved through identified exotic patterns.
My conjecture remains that a limited number of such patterns should cover nearly all (if not all) the field of potential hardest
champagne
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