## BUG3/Mutant Z-Wing Example

Advanced methods and approaches for solving Sudoku puzzles

### BUG3/Mutant Z-Wing Example

I think this is an interesting example first because it contains a Mutant Z-Wing (which I think is a rara avis} and second because the Mutant Z-Wing reduces a BUG+3 puzzle to a BUG+1 puzzle. The puzzle is Sudoku9981 Expert Book 42 #7. I used a 7 color wing to reach this point in the puzzle. The xy-wing of interest has a pivot at r5c5 and a 5 z-conjugate at r3c5 and r5c8. The Mutant z-wing occurs when this z-conjugate is combined with the 5 conjugate pair in row1. This forces the Z-conjugate to be a true conjugate and also forces the other candidates (78) in the xy-wing to actually be in the xy-wing. Consequently, 5 can be eliminated from r2c4 and 8 can be removed from r9c5. The BUG+1 technique then reveals that 7 must be in r2c6.

BUG3/Mutant Z-Wing Example
Code: Select all
` |-----------------+-----------------+-----------------| |   8    4    6   |  57    1    9   |   2   57    3   | |   9    1   57   | -578   2    3   |  457  48    6   | |   2    3   57   |   4   58    6   |   9    1   78   | |-----------------+-----------------+-----------------| |   6   58    2   |   3    4   78   |  57    9    1   | |   3   58    4   |   9   78    1   |   6   57    2   | |   1    7    9   |   2    6    5   |   8    3    4   | |-----------------+-----------------+-----------------| |   5    6    3   |   1    9    2   |  47   48   78   | |   7    9    8   |   6    3    4   |   1    2    5   | |   4    2    1   |  58   57-8 78   |   3    6    9   | |-----------------+-----------------+-----------------| `
Bud

Posts: 56
Joined: 24 August 2008

Hi Bud,

Would you post the original puzzle? At the point you presented, there is an xy-wing with pivot r3c5 that eliminates 7 from r1c8, solving the puzzle.
re'born

Posts: 551
Joined: 31 May 2007

### BUG3/Mutant Z-Wing Example

Hi Re'born,
Now that I see it, I can't believe I missed it. Thanks for pointing it out to me. So this is another bad example. Here is the original puzzle.

000 009 000
010 000 000
230 000 010
000 300 000
004 000 000
009 005 804
060 100 000
008 004 005
000 000 009

I have another Z-Wing Example which solves a puzzle, and so far I haven't found a simpler pattern technique that can be used. If you're interested in checking it out, I will post it here.
Bud

Posts: 56
Joined: 24 August 2008

### Re: BUG3/Mutant Z-Wing Example

Bud wrote:Hi Re'born,
Now that I see it, I can't believe I missed it. Thanks for pointing it out to me. So this is another bad example. Here is the original puzzle.

000 009 000
010 000 000
230 000 010
000 300 000
004 000 000
009 005 804
060 100 000
008 004 005
000 000 009

Bud wrote:I have another Z-Wing Example which solves a puzzle, and so far I haven't found a simpler pattern technique that can be used. If you're interested in checking it out, I will post it here.

Please do. Incidentally, before you apply coloring on 7, there is a transported xyz-wing that makes the same elimination as my xy-wing (it's the same pivot and prong set, just the added digit in the pivot to complicate matters), and solves the puzzle at once.
Code: Select all
`  *--------------------------------------------------* | 8    4    6    | 57*  1    9    | 2    5-7  3    | | 9    1    57   | 785  2    3    | 457  48   6    | | 2    3    57   | 4    578* 6    | 9    1    78*  | |----------------+----------------+----------------| | 6    58   2    | 3    4    78   | 57   9    1    | | 3    58   4    | 9    78a  1    | 6    57A  2    | | 1    7    9    | 2    6    5    | 8    3    4    | |----------------+----------------+----------------| | 5    6    3    | 1    9    2    | 47   48   78   | | 7    9    8    | 6    3    4    | 1    2    5    | | 4    2    1    | 578  578  78   | 3    6    9    | *--------------------------------------------------*`
re'born

Posts: 551
Joined: 31 May 2007

### BUG3/Mutant Z-Wing Example

OK, Here it is, Sudoku9981 Expert Book 42 puzzle #6. I used a 9 xy-wing and a 9 w-wing in order to get to this point in the puzzle. The xy-wing of interest has a pivot at r3c7 and a 6 z-conjugate at r3c3 and r6c7. The z-wing is formed when this is combined with the 6 conjugate pair in r9. This eliminates 6 from r1c3 and 1 from r3c469. Thanks for checking it out for me.

Z-Wing Example
Code: Select all
` |-----------------+-----------------+-----------------| |   5    7  12-6  |   4    8   16   |   9   26    3   | |  26    9    4   |   3    5   17   |   8   26   17   | |   8    3   16   | -19    2 -1679  |  15    4  -157  | |-----------------+-----------------+-----------------| |   3    2    9   |   5    6    4   |   7    1    8   | |   1    6    5   |   8    7    3   |   4    9    2   | |   4    8    7   |  12    9   12   |  56    3   56   | |-----------------+-----------------+-----------------| |  29   15    8   |   6    4   259  |   3    7   19   | |  69    4    3   |   7    1    8   |   2    5   69   | |   7   15   26   |  29    3   259  |  16    8    4   | |-----------------+-----------------+-----------------| `

The original puzzle is.

570 000 003
090 050 800
030 020 040
300 560 000
165 800 490
000 000 000
008 640 070
000 010 250
700 000 000
Last edited by Bud on Sat Nov 08, 2008 12:17 pm, edited 1 time in total.
Bud

Posts: 56
Joined: 24 August 2008

Bud Thanks for the example. As a practice of Nice Loops for me I would represent this as the continuous loop
Code: Select all
`-r3c3-1-r3c7=1=r9c7=6=r9c3-6-r3c3- => r3c469<>1,r1c3<>6`

You missed one elimination of digit 1.
Glyn

Posts: 357
Joined: 26 April 2007

### Re: BUG3/Mutant Z-Wing Example

Bud wrote:
The original puzzle is.

570 000 003
090 050 800
030 020 040
300 560 000
165 800 490
000 000 000
008 640 070
000 010 250
700 000 000

Alternatively, after your xy-wing (but before the w-wing), there is the following xy-chain:
(1=7)r2c6 - (7=1)r2c9 - (1=9)r7c9 - (9=2)r7c1 - (2=6)r2c1 - (6=1)r3c3, => r3c46<>1, solving the puzzle. (Note: This chain also excludes 1 from r3c9, but this is seemingly unimportant).
re'born

Posts: 551
Joined: 31 May 2007

### BUG3/Mutant Z-Wing Example

Hi Reborn,
Thanks for checking out this for me. I don't think that an alternate xy-chain solution means that this is a bad example, but of course this is just my opinion. Normally my progression is to exhaust my pattern technique toolbox before I even begin looking for chains. I do think that the W-wing and Z-wing are easier to find than an xy-chain. Of course many of the pattern techniques are chains.

I have another example that you might be interested in. It is a Z-2-string-kite that solves a puzzle. One of the strings is a z-conjugate from an x-y wing and the other is a z-conjugate from a W-wing.
Bud

Posts: 56
Joined: 24 August 2008

Bud, would you please clarify the origin of the z-wing term

ronk
2012 Supporter

Posts: 4764
Joined: 02 November 2005
Location: Southeastern USA

### BUG3/Mutant Z-Wing Example

Hi Ron,
Let's start with the term Z-conjugate. In my first post on the Z-Wing and Z-Turbot Fish I was using a link in patterns which is neither a strong link or a week link and I needed a name for it. Specifically, if two cells A and B or are z-conjugates, then Z is A inclusive or B. For a conjugate pair this relationship is A exclusive-or B. Since Z is the digit of interest in the XY-wing, and since these can replace one or both of the conjugate pairs in a turbot fish pattern, I picked this name. If there is an estblished name for this type of link or if someone has a better name, I will use it.
As for the Z-wing, again I needed a name for a pattern in which one of the conjugate pairs in the x-wing pattern is replaced by a z-conjugate. If there is an established specific name for this or if anyone has a better name for this, I will use it. You pointed out to me that this is a continuous loop, but that name is not specific. I think that you know more about the background on the actual pattern than I do. I don't think that the pattern is new, but it needs a name.
Bud

Posts: 56
Joined: 24 August 2008

### Re: BUG3/Mutant Z-Wing Example

Bud wrote:Hi Reborn,
Thanks for checking out this for me. I don't think that an alternate xy-chain solution means that this is a bad example, but of course this is just my opinion. Normally my progression is to exhaust my pattern technique toolbox before I even begin looking for chains. I do think that the W-wing and Z-wing are easier to find than an xy-chain. Of course many of the pattern techniques are chains.

Just so that you don't get the wrong idea about my intentions, I only offer alternative solutions because I'm obsessive compulsive, not because I am trying to discredit your examples. I appreciate, very much, your point of view (especially because transport is a stategy I've tried very hard to popularize) and I wouldn't begin to tell anyone which techniques they should use or in which order. My priority on every puzzle is to find a one-step solution. Other find this a silly exercise. I say, whatever keeps you playing.

Bud wrote:I have another example that you might be interested in. It is a Z-2-string-kite that solves a puzzle. One of the strings is a z-conjugate from an x-y wing and the other is a z-conjugate from a W-wing.

As usual, I would love to see it. In fact, next time, don't even ask; just post it.
re'born

Posts: 551
Joined: 31 May 2007

### BUG3/Mutant Z-Wing Example

I wanted you to check it out. The fact that you found an alternate xy-chain solution, but didn't find a simpler pattern technique solution makes it a good example. If you were not able to find anything but a tedious or contraversial alternate solution it would mke it an excellent example. I'm happy with good. Actually we are allies on the transport principle. I have only been studying advanced solving techniques for less than 3 years, and most of what I learned came from the solving sites such as sudopedia. There is a lot of good information at this forum, but the fact that it is not organized makes it hard for someone who is just beginning to learn new techniques. I started using the transport principle about 6 months ago on the classic turbot fish patterns, but I didn't know there was a name for it. I called these patterns extended 2-string kite, extended ER, etc because that's what they are. I've lost track of the number of puzzles i've solved using the extended 2-string kite. I've checked the terminology section of Sudopedia and the name Transport doesn't even appear. I have already written up the extended 2-string kite and the extended ER at Sudopedia with diagrams. I am planning on editing this to say these techniques are based on the Transport principle and add actual puzzle examples. The transport principle has such broad application, that there has to be a specific name for the varius patterns that can result from it. When the dust settles I am going to write a page in Sudopedia on the Z-Wing and Z-Turbot Fish. The name xy-transport isn't descriptive enough although the individual patterns belong to that group.
I'm still looking for that example. It could take awhile.
Bud

Posts: 56
Joined: 24 August 2008

### Re: BUG3/Mutant Z-Wing Example

Bud wrote:The transport principle has such broad application, that there has to be a specific name for the varius patterns that can result from it.

I have to admit it has never occurred to me to write down all possible patterns arising from using transport (it seems like a long and thankless task). I've always viewed it more as a strategy: Use an existing pattern to obtain a derived strong inference amongst a set of candidates. Choose a candidate and start moving away from it in steps of 2, a weak, then a strong inference. This gives a new strong inference between the set of candidates. Look for a deductions. Rinse, repeat.

Having said that, there is certainly a dearth of well-diagrammed and explained examples in the sudoku universe. Any contribution you make will most likely be appreciated (even if no one ever says 'thank you').
re'born

Posts: 551
Joined: 31 May 2007