## ultimate fish and nishio

Advanced methods and approaches for solving Sudoku puzzles
daj95376 wrote:Q1: If you revisit your Exemplars for fish, you'll see that there are several examples where you don't perform the Locked Candidates eliminations because it restricts the pattern excessively when fin cells are included.

That shouldn't be true for the non-degenerative patterns (the ones shown without fin cells). And that shouldn't be true for degenerative patterns unless all the fin cells are false. If it is, I must have made a mistake.

Q2: I'm not sure, but I thought that some of my other posts using tilted pairs was followed by posts where someone used grouped coloring to perform the same eliminations. That's why I said that I believe that grouped coloring might also work with the general pattern I produced. My belief could easily be wrong!!!

Although I didn't study them in great detail, I saw no issue with the coloring deductions. They all appeared to be direct deductions (using udosuk's term).

Unlike fish patterns, the tilted pairs pattern can perform eliminations in cells that are not directly seen by the X cells. Maybe that's why I've been able to get some of the NoFish puzzles to work with it.

I still don't know if the tilted pairs pattern is just some variant of Broken Wings. If someone who understands Broken Wings has time to check, I'd appreciate knowing. TIA!!!

The "tilted pairs" pattern is a composite of broken wing patterns. Every tilted pairs pattern involving four [edit: boxes, rows and columns] has eight broken wings in the form of invalid turbot fish. DanG's name of "school of turbots" is thus very appropriate. For the case of NoFish20 (below), all eight invalid turbots yield the very same exclusion -- r8c3<>5. In each illustration, at least one of the "guardian cells" (@) must be true in order to prevent the invalid turbot.

Code: Select all
`NoFish20 ....97.5.1.......9....2.1.4..34.8....8...2...9...7..2.....4.7......6..48.29..36..After SSTS:.---------------.---------------.---------------. | 38   34   248 | 1    9    7   | 23   5    6   | | 1    357  2567| 368  35   4   | 23   78   9   | | 3567 9    567 | 368  2    56  | 1    78   4   | :---------------+---------------+---------------: | 2    6    3   | 4    1    8   | 5    9    7   | | 57   8    157 | 9    35   2   | 4    6    13  | | 9    45   145 | 36   7    56  | 8    2    13  | :---------------+---------------+---------------: | 68   1    68  | 5    4    9   | 7    3    2   | | 357  357  57  | 2    6    1   | 9    4    8   | | 4    2    9   | 7    8    3   | 6    1    5   | '---------------'---------------'---------------'  /  .  . |  .  .  / |  .  .  .        /  /  . |  .  .  / |  .  .  . /  5  5 |  .  5  / |  .  .  .        / *5 @5 |  .  5  / |  .  .  .*5  / @5 |  /  / *5 |  /  /  /       *5  / @5 |  /  / *5 |  /  /  /---------+----------+----------      ---------+----------+---------- /  /  . |  .  .  / |  .  .  .        .  /  . |  .  .  / |  .  .  .*5  / @5 |  .  5  / |  .  .  .        5  /  5 |  .  5  / |  .  .  . / *5 @5 |  /  / *5 |  /  /  /        / *5 @5 |  /  / *5 |  /  /  /---------+----------+----------      ---------+----------+---------- /  .  . |  .  .  / |  .  .  .        .  /  . |  .  .  / |  .  .  .@5  5 -5 |  .  .  / |  .  .  .        5 @5 -5 |  .  .  / |  .  .  . /  .  . |  .  .  / |  .  .  .        .  /  . |  .  .  / |  .  .  . invalid turbot r36c16b4              invalid turbot r36c26b1 .  /  . |  .  /  . |  .  .  .        /  /  . |  .  /  . |  .  .  . / *5 @5 |  / *5  / |  /  /  /        / *5 @5 |  / *5  / |  /  /  / 5  /  5 |  .  /  5 |  .  .  .       *5  / @5 |  .  /  5 |  .  .  .---------+----------+----------      ---------+----------+---------- /  /  . |  .  /  . |  .  .  .        /  .  . |  .  /  . |  .  .  .*5  / @5 |  / *5  / |  /  /  /       *5  / @5 |  / *5  / |  /  /  / / *5 @5 |  .  /  5 |  .  .  .        /  5  5 |  .  /  5 |  .  .  .---------+----------+----------      ---------+----------+---------- .  /  . |  .  /  . |  .  .  .        /  .  . |  .  /  . |  .  .  . 5 @5 -5 |  .  /  . |  .  .  .       @5  5 -5 |  .  /  . |  .  .  . .  /  . |  .  /  . |  .  .  .        /  .  . |  .  /  . |  .  .  . invalid turbot r25c25b4              invalid turbot r25c15b1 /  .  . |  /  /  / |  .  .  .        /  .  . |  .  .  / |  .  .  .  /  5  5 |  / *5  / |  .  .  .        /  5  5 |  .  5  / |  .  .  . *5  / @5 |  /  / *5 |  /  /  /       *5  / @5 |  /  / *5 |  /  /  / ---------+----------+----------      ---------+----------+---------- /  .  . |  .  /  . |  .  .  .        /  .  . |  /  /  / |  .  .  . *5  / @5 |  / *5  / |  /  /  /       *5  / @5 |  / *5  / |  /  /  /  /  5  5 |  .  /  5 |  .  .  .        /  5  5 |  /  / *5 |  .  .  . ---------+----------+----------      ---------+----------+---------- /  .  . |  .  /  . |  .  .  .        /  .  . |  .  .  / |  .  .  . @5  5 -5 |  .  /  . |  .  .  .       @5  5 -5 |  .  .  / |  .  .  .  /  .  . |  .  /  . |  .  .  .        /  .  . |  .  .  / |  .  .  .  invalid turbot r35c15b2              invalid turbot r35c16b5 .  /  . |  /  /  / |  .  .  .        .  /  . |  .  /  . |  .  .  .  / *5 @5 |  / *5  / |  /  /  /        / *5 @5 |  / *5  / |  /  /  /  5  /  5 |  /  / *5 |  .  .  .        5  /  5 |  .  /  5 |  .  .  . ---------+----------+----------      ---------+----------+---------- .  /  . |  .  .  / |  .  .  .        .  /  . |  /  /  / |  .  .  .  5  /  5 |  .  5  / |  .  .  .        5  /  5 |  / *5  / |  .  .  .  / *5 @5 |  /  / *5 |  /  /  /        / *5 @5 |  /  / *5 |  /  /  / ---------+----------+----------      ---------+----------+---------- .  /  . |  .  .  / |  .  .  .        .  /  . |  .  /  . |  .  .  .  5 @5 -5 |  .  .  / |  .  .  .        5 @5 -5 |  .  /  . |  .  .  .  .  /  . |  .  .  / |  .  .  .        .  /  . |  .  /  . |  .  .  .  invalid turbot r26c26b2              invalid turbot r26c25b5`

[edit: corrected r35c16b5 figure]
Last edited by ronk on Tue Apr 17, 2007 5:59 am, edited 2 times in total.
ronk
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RW wrote:
ronk wrote:

Same as DanG's reasoning for r9c3 but with a hidden single instead of the locked candidates (if r9c8=9 clears r9;c8 => r8c3=9 => clears c3).

I think we have to be more discipline: the technique here is what Jeff called Error Net or SIN (Single Implication Network)
You cannot choose your exclusion as you like, it may start with "if cell A is true" but then it must be followed by some sort of implication net for finding other cells ( B then C etc.) which must be true in order to strip the extra candidates till only impossible pattern remains.
In a way is like saying solution A,B,C.. is not possible because leads to invalid pattern, therefore "A" must be false.
Note: the reverse may not be true, i.e. B=true does NOT imply A must be is true!

Next:
RW wrote:
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`T9  .  . | T9 T9 T9 |  .  .  . . T9  . |  .  .  . | T9  .  . .  .  . | T9 T9 T9 |  .  . T9---------+----------+--------- . T9  . |  .  .  . |  .  . T9T9  .  . |  .  .  . | T9  .  . .  .  . |  .  .  . |  .  .  .---------+----------+--------- .  .  . |  .  .  . |  .  .  . .  .  . |  .  .  . |  .  .  . .  .  . |  .  .  . |  .  .  .`

The vertical pairs in box 2 form a 3 way X-wing which lead to multiple solutions! It should be only one vertical pair there or what I called a dead end (flipping the tower to one side).
DanG

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DanG wrote:I think we have to be more discipline...
You cannot choose your exclusion as you like...
Why ? What is correct , i can use. If its a pattern to spot, the better.
ravel

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ravel wrote:
DanG wrote:I think we have to be more discipline...
You cannot choose your exclusion as you like...
Why ? What is correct , i can use. If its a pattern to spot, the better.

I agree. DanG, in what way was my elimination in r9c8 less disciplined than your eliminatinon in r9c3?
DanG wrote:
RW wrote:
Code: Select all
`T9  .  . | T9 T9 T9 |  .  .  .  . T9  . |  .  .  . | T9  .  .  .  .  . | T9 T9 T9 |  .  . T9 ---------+----------+---------  . T9  . |  .  .  . |  .  . T9 T9  .  . |  .  .  . | T9  .  .  .  .  . |  .  .  . |  .  .  . ---------+----------+---------  .  .  . |  .  .  . |  .  .  .  .  .  . |  .  .  . |  .  .  .  .  .  . |  .  .  . |  .  .  . `

The vertical pairs in box 2 form a 3 way X-wing which lead to multiple solutions! It should be only one vertical pair there or what I called a dead end (flipping the tower to one side).

Multiple solutions? Maybe you could start by showing me one solution to that single digit candidate pattern.

RW
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ravel wrote:
DanG wrote:I think we have to be more discipline...
You cannot choose your exclusion as you like...

Why ? What is correct , i can use. If its a pattern to spot, the better.

Code: Select all
`NoFish4 ..72....62...4..758.....1..1.4.6.8...2.8....3.....9....3...4..14..6..2.....7..... After SS eliminations: T9  .  . |  . T9  . |  .  .  .  . T9  9 |  .  .  . | T9  .  .  .  .  9 | T9  .  . |  .  9 T9 ---------+----------+---------  . T9  . |  .  .  . |  .  9 T9 T9  .  9 |  .  .  . | T9  .  .  .  .  . |  .  .  . |  .  .  . ---------+----------+---------  .  .  . | T9  .  . |  . T9  .  .  .  9 |  . T9  . |  . T9  .  9  9 -9 |  .  9  . |  9  9  .  if r9c3=9 clears r9;c3 => r78c8= locked => clears c8  then b1234689 = School of Burbots (size 5x11)  therefore r9c3<>9 `

You cannot assume both r9c3=true and r9c8=true because they belong to same unit (house) r9.
Actually I think the only way one may do that is when the two boxes do NOT see each other, otherwise it should be some implication net there.
On the other hand, starting with r9c8=true cannibalizes into the School of Turbots (SoT) which defeats the purpose. The goal here is to use SoT as an exploit for the elimination.

RW wrote:
Code: Select all
` T9  .  . | T9 T9 T9 |  .  .  .  . T9  . |  .  .  . | T9  .  .  .  .  . | T9 T9 T9 |  .  . T9 ---------+----------+---------  . T9  . |  .  .  . |  .  . T9 T9  .  . |  .  .  . | T9  .  .  .  .  . |  .  .  . |  .  .  . ---------+----------+---------  .  .  . |  .  .  . |  .  .  .  .  .  . |  .  .  . |  .  .  .  .  .  . |  .  .  . |  .  .  . `

Sorry, my mistake, each vertical pair in box 2 makes a locked pair (not X-wing), therefore the other two pairs should be eliminated. Now tell me which pair is a keeper?

I will prepare a new post here in a day or two with new revelations about SoT and the exclusion method.
DanG

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DanG wrote:You cannot assume both r9c3=true and r9c8=true because they belong to same unit (house) r9.

Ehm... that is one weird argument. You are not assuming both to be true at the same time.
DanG wrote:On the other hand, starting with r9c8=true cannibalizes into the School of Turbots (SoT) which defeats the purpose. The goal here is to use SoT as an exploit for the elimination.

I repeat, box 8 and 9 are not part of this school of turbots. From the elimination point of view it does not matter at all what is in those two boxes. Assuming r9c8=9 does not cannibalize anything, it doesn't touch the SoT, but it leads to a contradiction exactly in the same way as your elimination (only a bit simpler, as your one needs locked candidates).

DanG wrote:Sorry, my mistake, each vertical pair in box 2 makes a locked pair (not X-wing), therefore the other two pairs should be eliminated. Now tell me which pair is a keeper?

Eliminate pairs? I'm not sure if you understand here... The pattern in itself is a form of SoT , it cannot exist in any puzzle. Here's another one:
Code: Select all
`T9  .  . | T9 T9 T9 |  .  .  .  . T9  . |  .  .  . | T9  .  .  .  .  . | T9 T9 T9 |  .  . T9 ---------+----------+---------  . T9  . | T9 T9 T9 |  .  .  . T9  .  . |  .  .  . | T9  .  .  .  .  . | T9 T9 T9 |  .  . T9 ---------+----------+---------  .  .  . |  .  .  . |  .  .  .  .  .  . |  .  .  . |  .  .  .  .  .  . |  .  .  . |  .  .  . `

It works because of the strong links in r2 and r5. Maybe it should rather be drawn like this:
Code: Select all
` 9  /  / |  .  .  . |  /  /  /  /  9  / |  /  /  / |  9  /  /  /  /  / |  .  .  . |  /  /  9 ---------+----------+---------  /  9  / |  .  .  . |  /  /  /  9  /  / |  /  /  / |  9  /  /  /  /  / |  .  .  . |  /  /  9 ---------+----------+---------  .  .  . |  .  .  . |  .  .  .  .  .  . |  .  .  . |  .  .  .  .  .  . |  .  .  . |  .  .  . `

RW
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DanG wrote:
Code: Select all
`NoFish4 ..72....62...4..758.....1..1.4.6.8...2.8....3.....9....3...4..14..6..2.....7..... After SS eliminations: T9  .  . |  . T9  . |  .  .  .  . T9  9 |  .  .  . | T9  .  .  .  .  9 | T9  .  . |  .  9 T9 ---------+----------+---------  . T9  . |  .  .  . |  .  9 T9 T9  .  9 |  .  .  . | T9  .  .  .  .  . |  .  .  . |  .  .  . ---------+----------+---------  .  .  . | T9  .  . |  . T9  .  .  .  9 |  . T9  . |  . T9  .  9  9 -9 |  .  9  . |  9  9  .  if r9c3=9 clears r9;c3 => r78c8= locked => clears c8  then b1234689 = School of Burbots (size 5x11)  therefore r9c3<>9 `

RW and I have our own tests, but by what test do you know that that "school of turbots" is an illegal pattern.
ronk
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### Re: School of Turbots

[Withdrawn: Not really relevant.]
Last edited by daj95376 on Tue Apr 17, 2007 6:33 pm, edited 2 times in total.
daj95376
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ronk wrote:but by what test do you know that that "school of turbots" is an illegal pattern.

It's a loop of an uneven amount of strong links.

RW
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After a second look, none of the examples in this thread are any different from Rod Hagglunds broken wing with loops of five strong links and guardian cells.
Code: Select all
`NoFish2  3 -3  3 |  .  . #3 |  .  .  .  .  .  . |  .  .  . |  .  .  .  3  3  3 |  3  3  . |  .  .  . ---------+----------+--------- *3 #3  . |  . *3  . | #3  .  .  .  .  3 |  .  . *3 |  3  .  .  .  3  . |  .  . .  | +3  .  . ---------+----------+---------  .  .  . |  .  .  . |  .  .  .  .  .  3 |  3  .  . |  .  .  . *3 #3  . |  .  . *3 |  .  .  . NoFish3  .  .  . |  . #9 #9 | -9  .  9  .  . *9 |  . *9  . |  9  .  9  9  .  . |  .  . *9 |  .  .  9 ---------+----------+---------  .  9  . |  .  9  . |  9  .  .  .  . *9 |  .  . *9 | #9  .  .  .  9 #9 |  .  9 #9 |  .  . +9 ---------+----------+---------  .  .  . |  .  .  . |  .  .  .  .  .  . |  .  .  . |  .  .  .  9  9  . |  .  .  . |  .  .  . NoFish4  9  .  . |  .  9  . |  .  .  .  . *9 #9 |  .  .  . | *9  .  .  .  .  9 |  9  .  . |  . #9 *9 ---------+----------+---------  . *9  . |  .  .  . |  . #9 *9  9  .  9 |  .  .  . |  9  .  .  .  .  . |  .  .  . |  .  .  . ---------+----------+---------  .  .  . |  9  .  . |  . +9  .  .  . +9 |  .  9  . |  . +9  .  9 #9 -9 |  .  9  . |  9 -9  . NoFish5  . #9  . | #9 *9  . | *9  .  .  .  9  . |  .  .  9 |  .  . *9  .  9 +9 |  . #9  . |  .  . #9 ---------+----------+---------  .  .  . |  .  .  . |  .  .  .  .  .  . | #9 *9  . |  .  . *9  .  .  . |  .  .  9 |  .  9  . ---------+----------+---------  .  .  . |  9 #9  . |  9  9  .  .  .  9 | -9 #9  . |  9  .  .  .  9  . |  9 #9  9 |  .  .  . `

RW
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Where is the Broken Wing and Guardian cells in this pattern, please? TIA!!!

Code: Select all
`^-----------------------------------^|  9  /  9  |  9  9  9  |  /  9  /  ||  /  9  9  |  /  /  /  |  9  9  /  ||  /  /  9  |  9  9  9  |  /  9  9  ||-----------+-----------+-----------||  /  9  9  |  9  9  9  |  /  9  9  ||  9  /  9  |  9  9  9  |  9  9  /  ||  /  /  9  |  9  9  9  |  /  9  /  ||-----------+-----------+-----------||  /  /  9  |  9  9  9  |  /  9  /  ||  /  /  9  |  9  9  9  |  /  9  /  ||  9  9 -9  | -9 -9 -9  |  9 -9  9  |^-----------------------------------^`
daj95376
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daj95376 wrote:Where is the Broken Wing and Guardian cells in this pattern, please? TIA!!!

Good one, which I believe requires a kraken broken wing. Have you invented that one already
Code: Select all
`^-----------------------------------^ |  9  /  9  |  9  9  9  |  / @9  /  | |  / *9 @9  |  /  /  /  | *9 @9  /  | |  /  /  9  |  9  9  9  |  / @9 *9  | |-----------+-----------+-----------| |  / *9 @9  | @9 @9 @9  |  / @9 *9  | |  9  /  9  |  9  9  9  |  9  9  /  | |  /  /  9  |  9  9  9  |  /  9  /  | |-----------+-----------+-----------| |  /  /  9  |  9  9  9  |  /  9  /  | |  /  /  9  |  9  9  9  |  /  9  /  | |  9 @9 -9  | -9 -9 -9  |  9 -9 @9  | ^-----------------------------------^  * -- Turbot fish @ -- Guardians`
ronk
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ronk wrote:
daj95376 wrote:Where is the Broken Wing and Guardian cells in this pattern, please? TIA!!!

Good one, which I believe requires a kraken broken wing. Have you invented that one already
Code: Select all
`^-----------------------------------^ |  9  /  9  |  9  9  9  |  / @9  /  | |  / *9 @9  |  /  /  /  | *9 @9  /  | |  /  /  9  |  9  9  9  |  / @9 *9  | |-----------+-----------+-----------| |  / *9 @9  | @9 @9 @9  |  / @9 *9  | |  9  /  9  |  9  9  9  |  9  9  /  | |  /  /  9  |  9  9  9  |  /  9  /  | |-----------+-----------+-----------| |  /  /  9  |  9  9  9  |  /  9  /  | |  /  /  9  |  9  9  9  |  /  9  /  | |  9 @9 -9  | -9 -9 -9  |  9 -9 @9  | ^-----------------------------------^  * -- Turbot fish @ -- Guardians`

So, the Broken Wing is the chain

Code: Select all
`[r2c2]=[r2c7]-[r3c9]=[r4c9]-[r4c2] => [c2]=invalid`

being blocked by at least one Guardian cell. All that remains is to show that (any of) [r9c34568]=9 results in all of the Guardian cells being eliminated. I've already done that.

Thanks Ron!!!

I did not invent this pattern from scratch. I took NoFish4 and worked my tail off searching for the most general pattern I could find that encompassed it. The above was my results. When I sat down and reviewed the grid, it took Locked Candidates (1) & (2) plus Colors for me to reach an invalid pattern.

Me invent kraken broken wing? LOL!!! No, I'm shooting for overlapping broken wings!
daj95376
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RW wrote:I repeat, box 8 and 9 are not part of this school of turbots. From the elimination point of view it does not matter at all what is in those two boxes. Assuming r9c8=9 does not cannibalize anything, it doesn't touch the SoT, but it leads to a contradiction exactly in the same way as your elimination (only a bit simpler, as your one needs locked candidates).

Well I agree, however NoFish4 forces me to consider Box 8,9.
Code: Select all
`NoFish4 ..72....62...4..758.....1..1.4.6.8...2.8....3.....9....3...4..14..6..2.....7..... After SS eliminations: T9  .  . |  . T9  . |  .  .  .  . T9  9 |  .  .  . | T9  .  .  .  .  9 | T9  .  . |  .  9 T9 ---------+----------+---------  . T9  . |  .  .  . |  .  9 T9 T9  .  9 |  .  .  . | T9  .  .  .  .  . |  .  .  . |  .  .  . ---------+----------+---------  .  .  . | T9  .  . |  . T9  .  .  .  9 |  . T9  . |  . T9  .  9  9 -9 |  .  9  . |  9  9  . `

Cells in box2 see other cells in their respective columns, Box8 (otherwise they are singles). Same story between box8 and box9. Actually, Box 2,8,9 could be consider as a dead end which does not change the impossible pattern, just the size of the Turbots.

RW wrote:After a second look, none of the examples in this thread are any different from Rod Hagglunds broken wing with loops of five strong links and guardian cells.

I agree with that too because:
1. Both require a SoT pattern recognition.
2. Both lead to the same Candidate Elimination Cell (CEC). Actually it can be proofed that both methods are equivalent.
The difference is how one finds CEC.
Broken Wing require finding Guardians and then fishing for CEC among a bunch of cells.
With SoT method one may spot CEC right of the bat.

ronk wrote:RW and I have our own tests, but by what test do you know that that "school of turbots" is an illegal pattern.

My test is described here.
DanG

Posts: 20
Joined: 28 March 2007

DanG wrote:Broken Wing require finding Guardians and then fishing for CEC among a bunch of cells.
With SoT method one may spot CEC right of the bat.

No. With SoT you do consider guardian cells as well.
Code: Select all
`T9  .  . |  . T9  . |  .  .  .  . T9 #9 |  .  .  . | T9  .  .  .  . #9 | T9  .  . |  . #9 T9 ---------+----------+---------  . T9  . |  .  .  . |  . #9 T9 T9  . #9 |  .  .  . | T9  .  .  .  .  . |  .  .  . |  .  .  . ---------+----------+---------  .  .  . | T9  .  . |  . T9  .  .  .  9 |  . T9  . |  . T9  .  9  9 -9 |  . #9  . | #9 #9  . `

The cells marked with '#' are guardians that prevent the SoT. Then you have looked for a placement that eliminates all of the guardians. Even if you choose not to call these cells 'guardians', it doesn't change the fact that they are such.

After some thinking I'm sure the SoT will never provide any more eliminations than the broken wing. And as your example in NoFish 4 shows, SoT leads to fewer eliminations than the broken wing pattern that only needs a small part of the SoT cells (your SoT needs 14 pattern cells and 8 guardian cells and leads to one elimination, broken wing needs 5 pattern cells and 4 guardian cells to give two eliminations...).

RW
RW
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