## exocet pattern in hardest puzzles

Advanced methods and approaches for solving Sudoku puzzles

### Re: exocet pattern in hardest puzzles

When I was struggling to follow the last truth and link diagram I was looking for a proof that I could put into words. Being rather sleep-deprived recently, this took me some time, but I believe I eventually got there.

We need to show a) that one of the object cell pair can't reduce to a base digit while b) the other one must.
a) is satisfied as we know that one must contain a digit locked in the hidden set
b) is satisfied if condition 3 is met
– job done!

I was wondering if a base digit could exist in the AHS outside the object cell pair and from this analysis the answer is yes. I see ronk also posed this question in his edited response.

This required a re-write of my previous post so I've taken the opportunity of completely reworking the previous definition which appears in the following post for peer review. I'll withdraw my previous effort in a couple of days.

This has been a team effort and every contributor should be able to recognise his influence somewhere it.

I suggest that we open an Exocet/JExocet Summary thread starting with the agreed definition and followed with links to the examples different contributors have found.
David P Bird
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### Re: exocet pattern in hardest puzzles

Junior Exocet Definition

An Exocet exists when it can be shown that a base cell pair and two target cells must reduce to holding the same two digits out of a set of 3 or 4 base cell candidates. This then allows any non-base candidates to be eliminated from the target cells.

As defined by champagne, the way this base-target cell equivalence is demonstrated is unrestricted and may for example use multiple templates. However analysis has shown that in most cases there are identifiable pattern elements that exist.

The Junior Exocet (JE) pattern restricts the base and target cells to a single band of boxes and the digits to those in the base cells.
The Junior Exocet Plus (JE+) extends this by also checking for other candidates that are locked in the cells of interest.

Pattern Element Map
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 . . | S . . |  S = Cross-line Cells       | . . S | S . . | S . . |      *-------*-------*-------*  . = Any candidates  | . . S | S . . | S . . |    | . . S | S . . | S . . |        | . . S | S . . | S . . |     *-------*-------*-------*`

The different cell pairs occur in different boxes in the same band (the JE band).
In each object pair one cell will be a target cell and the other will be a companion cell that will reduce to a non-base candidate.
The three cross-lines intersect this band as shown, passing through the object cell pairs but not the base cell pair.

Requirements

1) The base cells must be restricted to a set of three or four digits (the base candidates)

2) Each object cell pair must only be capable of accommodating one base digit. This requires that they
a) have one cell that contains at least one base candidate (the target cell) and the other that contains none of them (the companion cell)
or
b) (JE+) must overlap an Almost Hidden Set that restricts the object cells to holding one base digit.
The simplest and most frequent situation will be when the AHS is an Almost Hidden Pair with a single extra digit locked in the object cells.

3) The two target cells must be forced to reduce to different base digits. This is satisfied when all occurrences of a digit (solved or not) in the "S", cross-line, cells are contained by two houses. A way to check this is to consider how many lines would be needed to cover all of them.

These diagrams show examples of the maximum number of occurrences of a digit in the "S" cells that can be contained by different combinations of two houses.
Code: Select all
`                                    v           v                         v    r4   . . \ | \ . . | \ . .      . . O | \ . . | O . .     . . \ | \ . . | O . .   r5   . . O | O . . | O . . <    . . O | \ . . | O . .     . . O | O . . | O . . <   r6   . . \ | \ . . | \ . .      . . O | \ . . | O . .     . . \ | \ . . | O . .   r7   . . \ | \ . . | \ . .      . . O | \ . . | O . .     . . \ | \ . . | O . .   r8   . . O | O . . | O . . <    . . O | \ . . | O . .     . . \ | \ . . | O . .    r9   . . \ | \ . . | \ . .      . . O | \ . . | O . .     . . \ | \ . . | O . .      2 Parallel Lines(I)        2 Parallel Lines(II)      2 Orthogonal Lines         rows 5 & 8                 columns 3 & 7             row 5 & column 7`

Eliminations

In JE patterns any non-base candidates can be eliminated from the target cells
In JE+ patterns (relying on condition 2b) the identity of the target cell won't be known. One object cell must eventually hold a base digit, and the other a candidate locked in the AHS, so any candidates that aren't in either set can be eliminated from both the object cells.

After a JE has been found

When the givens in the JE band are suitably placed, the inference that the two target cells must contain different base digits will often allow eliminations to be made in the target cell mini-lines.

When a base digit is known it will often form one of two Swordfish patterns and it will be possible to eliminate it from the fin cells common to both.

Short Proof

The columns of interest are c347 in the pattern map.

In the JE band the digits that are true in the base cells will be excluded from c3r123, c4r1 and c7r1.

From condition 2, the objects cells are limited to holding 2 truths at most for this digit pair. With no other openings in the JE band there therefore must be at least 4 truths in the 'S' cells to satisfy the three columns.

But condition 3 limits the number of truths that can be held in these cells to 4 at most, (2 for each digit).

Consequently each digit must be true twice in the 'S' cells and once in the object cells.
Last edited by David P Bird on Sat Jul 28, 2012 6:10 am, edited 4 times in total.
David P Bird
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### Re: exocet pattern in hardest puzzles

daj95376, would you please identify a handful of useful "JExocets" with 4 candidates in both base cells?

David P Bird, your proposed definition looks good to me, but I can't help but think that champagne, daj95376, and I make lousy judges. We're all so close to this topic that we unknowingly fill in important points that might be missing. Perhaps Lerner, a relative newcomer, will comment. In any case, I recommend waiting for champagne to weigh in before proceeding.
ronk
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### Re: exocet pattern in hardest puzzles

ronk wrote:daj95376, would you please identify a handful of useful "JExocets" with 4 candidates in both base cells?

I (now) only identify QExocet/qExocet puzzles.

There are so many "similar" puzzles in champagne's collection, but maybe these will qualify as unique.

Code: Select all
`98.7.....76....5....5.......9..4..3...85..6.......2..1..98..7......3..2......1..4;160 ### -1234- QExocet   Base = r3c12   Target = r1c7,r2c498.7..6......98.......6....5..4...3..3..7.4....2.....11.......2.6...78....3....5.;1887 ### -1235- QExocet   Base = r23c4   Target = r8c5,r5c698.7..6....5.9........8..4.83..6.7.....3....6.....7...67...98....2.........6....1;2715 ### -1245- QExocet   Base = r56c5   Target = r1c6==r8c4,r7c498.7..6..5..9..7......84...6..5..9....3..........29...1......76.59...1.....1...9.;2859 ### -2348- QExocet   Base = r8c89   Target = r9c1==r7c6,r7c498.7..6....5.4..3......9...7....2....2.8.......6.5..4...3...5.1....1.4.3.......6.;4302 ### -2789- QExocet   Base = r9c79   Target = r7c5,r8c3`
daj95376
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### Re: exocet pattern in hardest puzzles

ronk suggested that I comment on David P Bird's Junior Exocet Definition, so I hope that the following adds value.

The definition covers what is required but needs to be accompanied by one or more examples to make it more "real world" for a beginner.

I find the cross line diagrams difficult to interpret. I think what you are saying is that the cross row count is increased by
one for:

(1) each line (that runs at right angles to the three Exocet cross lines) that intersects one or more candidates for the digit; and

(2) each given or solved instance of the digit ( in one of the three Exocet cross lines) - represented by the vertical lines in the diagrams.

I am aware of the case (on p 18 of this thread, puzzle 25333) where the row count for a digit is allowed to exceed 2, if there is no instance of

a candidate or solution for the digit in a cross row (outside of the Exocet band). Is this a rare exception or does it need to be accounted for?

A few minor matters;

1) I'd collect all the statements on eliminations together to make the exposition clearer.

2) There is a change in terminology from JE etc to JP etc half way through - need to make consistent

3) The last paragraph would be clearer if it begins "When a base digit is known......"

Leren
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### Re: exocet pattern in hardest puzzles

daj95376,

For those last 5 puzzles I see all the exocets you list, but in 2715 and 2859 I see second Exocets

eg in 2859 I also see (to paraphrase your terminology): ### -2348- QExocet Base = r7c23 Target = r8c4==r9c9,r9c7

Do you see these ?

Leren
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### Re: exocet pattern in hardest puzzles

Leren wrote: but in 2715 and 2859 I see second Exocets
Leren

right for Jexocets
my solver sees more for exocets, but sometimes it's a quad in a box, so I have to check
champagne
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### Re: exocet pattern in hardest puzzles

daj95376 wrote:
ronk wrote:daj95376, would you please identify a handful of useful "JExocets" with 4 candidates in both base cells?
There are so many "similar" puzzles in champagne's collection, but maybe these will qualify as unique.

champagne wrote:
Leren wrote: but in 2715 and 2859 I see second Exocets
Leren
right for Jexocets
my solver sees more for exocets, but sometimes it's a quad in a box, so I have to check

Sorry, the possibility of triggering a side discussion never crossed my mind. I should have asked daj95376 for these examples via PM.
ronk
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### Re: exocet pattern in hardest puzzles

daj95376 wrote:
98.7..6....5.9........8..4.83..6.7.....3....6.....7...67...98....2.........6....1;2715

### -1245- QExocet Base = r56c5 Target = r1c6==r8c4,r7c4

98.7..6..5..9..7......84...6..5..9....3..........29...1......76.59...1.....1...9.;2859

### -2348- QExocet Base = r8c89 Target = r9c1==r7c6,r7c4

may-be some comments on these 2

as noticed Leren, these 2 puzzles have a double Jexocet quite classical.

Danny noticed r1c6==r8c4 for puzzle 2715 and r9c1==r7c6 for puzzle 2859

This is seen as other exocets by my solver working in "non Jexocet mode"
Here after the four exocets found by the solver for puzzle 2859 in that "non Jexocet mode".
Only the first and the last are seen in "Jexocet mode"

98.7..6..5..9..7......84...6..5..9....3..........29...1......76.59...1.....1...9. #2859
r7c2 r7c3 r8c4 r9c7 2348
r7c2 r7c3 r9c7 r9c9 2348
r8c8 r8c9 r7c4 r7c6 2348
r8c8 r8c9 r7c4 r9c1 2348
champagne
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### Re: exocet pattern in hardest puzzles

_____

My "additional" QExocets do not meet ronk's request for ... "4 candidates in both base cells".

[Edit: corrected reference to Exocet as a reference to QExocet.]
Last edited by daj95376 on Thu Jul 26, 2012 10:41 pm, edited 1 time in total.
daj95376
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### Re: exocet pattern in hardest puzzles

Leren thank you taking the time to provide your critique which has been very useful.

The aim was to produce an all-embracing definition of the various forms of JExocets for reference by players who, although familiar with the concepts, may have forgotten the details. Following this, I envisaged a separate post would provide a series of links to examples of each variant.

However I take your point that it is rather terse for newcomers. An opening post giving a preamble and an index may be a way to achieve both aims. Something along these lines:

This thread has been created to act as a summary of the effort of several contributors in analysing Junior Exocet patterns. Use the following links to fast track to various topics and examples of the different forms the pattern can take:
< Formal Definition >
< Newcomer walk-through > (yet to be written)
< JE Examples >
< JE+ Examples >
< Double JE Examples >
< Formal Proof >
etc

The other points you've made are all fair and I've edited the draft version accordingly as you will see.

You clearly found the description of condition 3 confusing, so I have almost completely reworded it to avoid over-using "lines". (In puzzle <2533> the houses (lines) covering all occurrences of each digit in these "S" cells are (1)c36, (2)c36, (3)r17, (5)c56, so each one therefore satisfies the condition.)
David P Bird
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### Re: exocet pattern in hardest puzzles

daj95376 wrote:If you take a closer look, these "additional" Exocet patterns do not meet ronk's request for ... "4 candidates in both base cells".

I wanted to put together "JExocet" exemplars with AAHSs in both of the "cross-lines", as David P Bird calls them. After experiencing no problem with the 3-digit variety, I was getting nowhere with the 4-digit case, i.e., Xsudo was no longer verifying exclusions.

Starting with the first of daj95376's examples (Thanks for those Danny) I was able to get somewhat further. The objective was to develop an exemplar with the minimum number of candidate voids, i.e., missing candidates, in r1c12, c3, c5 and c7.

___

[EDIT1: Note the base candidates of r1c12 are missing from r58c5 and r5c7 in the above. The [edit2: original exemplar and remaining text], in its entirety, is relegated to the land of the hidden.]

The older exemplar with probable flawed "JExocet" due to base candidates in r58c5 and r5c7: Show
I kept running into the "DLX A" limit of Xsudo, and so was unsuccessful. Even adding the clue 1r7c5 into the below could not be verified.

The "DLX A" limit also prevented me from proving that the extra SIS in c5 and c7 to construct the AAHSs could be repeated.

___

[edit2: r5c7 was typo r5c8, 2 plcs]
Last edited by ronk on Tue Jul 31, 2012 12:54 pm, edited 2 times in total.
ronk
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### Re: exocet pattern in hardest puzzles

daj95376 wrote:_____

If you take a closer look, these "additional" Exocet patterns do not meet ronk's request for ... "4 candidates in both base cells".

I confess I did not look at the prerequisite, but your remark surprises me.

The additional exocets share the base of your's. How can it be that a condition on the base cells is ok on one side and not on the other!!!

There is something I don't catch
champagne
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### Re: exocet pattern in hardest puzzles

David P Bird wrote:The aim was to produce an all-embracing definition of the various forms of JExocets for reference by players who, although familiar with the concepts, may have forgotten the details. Following this, I envisaged a separate post would provide a series of links to examples of each variant.

However I take your point that it is rather terse for newcomers. An opening post giving a preamble and an index may be a way to achieve both aims.

"An opening post?" The "JExocet" discussion had been in this thread since you introduced the term in mid-April. A separate thread 3 months ago might have been a good idea, but not now IMO. If we start another thread now, where is the "JExocet" discussion supposed to continue?
ronk
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### Re: exocet pattern in hardest puzzles

champagne wrote:I confess I did not look at the prerequisite, but your remark surprises me.

The additional exocets share the base of your's. How can it be that a condition on the base cells is ok on one side and not on the other!!!

There is something I don't catch

I corrected my above post to use QExocet instead of Exocet. My analyzer finds one "additional" QExocet per puzzle ... as opposed to the three additional Exocets that your solver finds. Here is my analyzer's output without ronk's restriction. You'll notice that the QExocets don't share the same base set.

Code: Select all
`98.7..6....5.9........8..4.83..6.7.....3....6.....7...67...98....2.........6....1;2715   c2b7  Locked Candidate 1              <> 9    r56c2 +--------------------------------------------------------------------------------+ |  9       8       134     |  7       12345   12345   |  6       1235    235     | |  12347   1246    5       |  124     9       12346   |  123     12378   2378    | |  1237    126     1367    |  125     8       12356   |  12359   4       23579   | |--------------------------+--------------------------+--------------------------| |  8       3       149     |  12459   6       1245    |  7       1259    2459    | |  12457   1245    1479    |  3       1245    12458   |  12459   12589   6       | |  1245    12456   1469    |  124589  1245    7       |  123459  123589  234589  | |--------------------------+--------------------------+--------------------------| |  6       7       134     |  1245    12345   9       |  8       235     2345    | |  1345    1459    2       |  1458    13457   13458   |  3459    6       34579   | |  345     459     8       |  6       23457   2345    |  23459   23579   1       | +--------------------------------------------------------------------------------+ # 189 eliminations remain ### -1245- QExocet   Base = r23c4   Target = r7c5==r5c6,r4c6 ### -1245- QExocet   Base = r56c5   Target = r1c6==r8c4,r7c4 *** double QExocet`

Code: Select all
`98.7..6..5..9..7......84...6..5..9....3..........29...1......76.59...1.....1...9.;2859 r1  b2  Locked Candidate 1              <> 5    r1c89 r9  b7  Locked Candidate 1              <> 6    r9c56 +--------------------------------------------------------------------------------+ |  9       8       124     |  7       135     1235    |  6       1234    1234    | |  5       12346   1246    |  9       136     1236    |  7       12348   12348   | |  237     12367   1267    |  236     8       4       |  235     1235    9       | |--------------------------+--------------------------+--------------------------| |  6       1247    12478   |  5       1347    1378    |  9       12348   123478  | |  2478    9       3       |  468     1467    1678    |  2458    124568  124578  | |  478     147     5       |  3468    2       9       |  348     13468   13478   | |--------------------------+--------------------------+--------------------------| |  1       234     248     |  2348    9       2358    |  23458   7       6       | |  23478   5       9       |  23468   3467    23678   |  1       2348    2348    | |  23478   23467   24678   |  1       3457    23578   |  23458   9       23458   | +--------------------------------------------------------------------------------+ # 174 eliminations remain ### -2348- QExocet   Base = r7c23   Target = r8c4==r9c9,r9c7 ### -2348- QExocet   Base = r8c89   Target = r9c1==r7c6,r7c4 *** double QExocet`

My QExocet is a subset of champagne's Exocet. David P. Bird's JExocet is a very large subset of my QExocet; i.e., almost identical.
Last edited by daj95376 on Thu Jul 26, 2012 10:44 pm, edited 1 time in total.
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