## BUG+2 or Z-color wing?

Advanced methods and approaches for solving Sudoku puzzles

### BUG+2 or Z-color wing?

When I first made my post on the Z-Turbot Fish, I didn't have any examples of the Z-color wing. Since then I have found about a dozen puzzle-cracking examples. One of these was an alternate solution to a BUG+2 puzzle which surprized me. In this example, there is a fin x-wing in rows 6 and 9 which would eliminate 2 from r5c5 and create a BUG+2, but the alternate solution does not use this. There is an xy-wing with a pivot at r6c2 and 8 Z-conjugates at r4c3 and r6c6. By itself it does not produce any eliminations but when it is combined with the 8 conjugate pair r8c34, it becomes a Z-color wing. Consider conjugate pair r48c3. If r4c3 is not 8, its Z-conjugate r6c6 is 8. If r8c3 is not 8, its conjugate r8c4 must be 8. Consequently either r6c6 or r8c4 must be 8 and 8 can be eliminated from their peer r6c4. This cracks the puzzle and the fin x-wing and BUG+2 are redundant. Of course, if you use the fin x-wing and the BUG+2 technique, the Z-color wing is redundant. I don't want to argue the relative merits of the two solutions, since this is a personal choice. The important point here is that in this case, you do have a choice. This does not imply that this will work on all BUG+2 puzzles

Z-color wing/BUG+2 Example
Code: Select all
` |-----------------+-----------------+-----------------| |   1    7    9   |   3    5    2   |   4    8    6   | |   2    8    6   |   7    9    4   |   5    3    1   | |   3    4    5   |   6   18   18   |   7    2    9   | |-----------------+-----------------+-----------------| |  58    6   28   |   4    7    3   |   9    1   25   | |  49    3    7   |  29   126  15   |   8   46   25   | |  49   25    1   |  289  268  58   |   3   46    7   | |-----------------+-----------------+-----------------| |   6    1    3   |   5    4    9   |   2    7    8   | |   7    9   28   |  28    3    6   |   1    5    4   | |  58   25    4   |   1   28    7   |   6    9    3   | |-----------------+-----------------+-----------------| `

This is Sudoku9981 Expert Book 27 puzzle #8
Last edited by Bud on Mon Nov 03, 2008 9:53 am, edited 2 times in total.
Bud

Posts: 56
Joined: 24 August 2008

Bud There is a problem with that grid you have duplicate cells in various houses. eg 1's in r1, 7's in box 3.
Glyn

Posts: 357
Joined: 26 April 2007

Bud, there are still two 5s in r3 (r3c9 should be 9 not five).
hobiwan
2012 Supporter

Posts: 321
Joined: 16 January 2008
Location: Klagenfurt

Hi Bud,

In this case, your Z-color wing is exactly the same as xy-transport. But, what I'm really interested is the your BUG+2 after finned x-wing step. Typically, if there is a step that will reduce you to BUG+2, there is a BUG+n for some n > 2 available. Consider the following:

Code: Select all
`  *--------------------------------------------------* | 1    7    9    | 3    5    2    | 4    8    6    | | 2    8    6    | 7    9    4    | 5    3    1    | | 3    4    5    | 6    18   18   | 7    2    9    | |----------------+----------------+----------------| | 58   6    28   | 4    7    3    | 9    1    25   | | 49   3    7    | 29   16+2 15   | 8    46   25   | | 49   25   1    | 89+2 26+8 58   | 3    46   7    | |----------------+----------------+----------------| | 6    1    3    | 5    4    9    | 2    7    8    | | 7    9    28   | 28   3    6    | 1    5    4    | | 58   25   4    | 1    28   7    | 6    9    3    | *--------------------------------------------------*`

Here we have a BUG+3 grid which immediately (utilizing r9c5) yields r6c5<>2. This doesn't crack the puzzle completely. However, a more careful examination yields r5c9<>2, solving the puzzle. Okay, cool, whatever. But here is what is really interesting to me. Let's say you didn't catch r5c9<>2. There was a choice made when I set up my BUG+3 grid. I (think) I could have also wrote it as:
Code: Select all
` *--------------------------------------------------* | 1    7    9    | 3    5    2    | 4    8    6    | | 2    8    6    | 7    9    4    | 5    3    1    | | 3    4    5    | 6    18   18   | 7    2    9    | |----------------+----------------+----------------| | 58   6    28   | 4    7    3    | 9    1    25   | | 49   3    7    | 9+2  126  15   | 8    46   25   | | 49   25   1    | 289  6+28 58   | 3    46   7    | |----------------+----------------+----------------| | 6    1    3    | 5    4    9    | 2    7    8    | | 7    9    28   | 28   3    6    | 1    5    4    | | 58   25   4    | 1    28   7    | 6    9    3    | *--------------------------------------------------*`

from which we immediately see (again using r9c5) that r5c5<>2. This, coupled with the r6c5<>2 exclusion solves the puzzle.

So, I'm a little worried about whether my second grid is a valid BUG+3 setup (r5c4 is where I'm worried). In any case, another alternative solution is the semi-remote naked pair r4c1-r6c6<8>, eliminating 5 from r6c2.
re'born

Posts: 551
Joined: 31 May 2007

### BUG+2 or Z-color wing?

Hi Re'born,

Thanks for your reply. I think we share a common interest, I recognized when I made this post that it would be great to find a BUG+3 example and if I find one I will post it. I have a few BUG+1 examples including one which can be solved either with the Z-color wing or the Z-2-string kite, which could be useful for someone who does not want to assume uniqueness. I don't think a potential BUG+2 should be considered a BUG+3.

Hi Glynn and Hobiwan,
Thanks for pointing out my grid errors.
Bud

Posts: 56
Joined: 24 August 2008

### BUG+2 or Z-color wing?

Hello Again Re'born,
When you first mentioned the name "Transport" I had never heard of it before since my knowledge of solving techniques comes mostly from solving techniques sites. Today I finally found your post on transport at Sudoku.org.uk and I realized that the technique is exactly the same as my Z-Turbot Fish. So there is really nothing new on my original post on these. I liked your example on the xyz-wing. Thanks for your help.

Bud
Bud

Posts: 56
Joined: 24 August 2008