SudoCue Users

Posted: **Wed Feb 22, 2006 4:42 pm**

Did anyone come up with a slick way to solve this puzzle? I can get to the solution, but I'm not very
happy with the approach I used. (A "4" in r1c2 cracks the puzzle wide open. But the best method I
could find to prove that r1c2 must be "4" involves ruling out the possible "2"s at r9c8 and r9c9, and
that's pretty messy.)

Just curious. dcb

Posted: **Wed Feb 22, 2006 6:43 pm**

Hi David,

The bottleneck in this nightmare is at this position:

Code: Select all

.------------------.------------------.------------------.
| 23    46    1    | 2789  2689  46789| 4689  3469  5    |
| 23    8     46   | 129   1269  5    | 469   1349  7    |
| 9     5     7    | 18    3     1468 | 2     146   1468 |
&#58;------------------+------------------+------------------&#58;
| 57    123   9    | 4     1568  1368 | 68    2567  268  |
| 6     124   248  | 1589  7     189  | 489   2459  3    |
| 57    34    48   | 3589  5689  2    | 1     45679 468  |
&#58;------------------+------------------+------------------&#58;
| 8     26    5    | 123   4     13   | 7     126   9    |
| 4     79    3    | 6     129   179  | 5     8     12   |
| 1     79    26   | 5789  589   789  | 3     246   246  |
'------------------'------------------'------------------'

SudoCue suggested a template elimination at this point, so I decided to find out what caused it.

There are 3 candidates for digit 6 in row 6. The first one is at R6C5. The other 2 would both force R3C6 to 6. This can be proven by looking at the pattern distribution for digit 6 only. R1C5, R2C5 and R4C6 are the shared peers for R6C5 and R3C6. Those can be eliminated.

This opens up the puzzle upto this point:

Code: Select all

.---------------.---------------.---------------.
| 23   46   1   | 7    289  46  | 89   39   5   |
| 23   8    46  | 129  129  5   | 469  1349 7   |
| 9    5    7   | 18   3    46  | 2    146  1468|
&#58;---------------+---------------+---------------&#58;
| 57   123  9   | 4    1568 138 | 68   257  268 |
| 6    12   248 | 1589 7    189 | 489  2459 3   |
| 57   34   48  | 3589 5689 2   | 1    4579 468 |
&#58;---------------+---------------+---------------&#58;
| 8    26   5   | 123  4    13  | 7    126  9   |
| 4    79   3   | 6    129  179 | 5    8    12  |
| 1    79   26  | 589  589  789 | 3    246  246 |
'---------------'---------------'---------------'

Examine the rectangle R89C26. When this rectangle would solve to candidates 7 and 9 only, there would be 2 solutions to this sudoku. However, column 6 has no candidates for digit 7 outside this rectangle. Digit 9 can therefore never be placed inside the rectangle. These two candidates can be eliminated. This situation is known as a type-4 unique rectangle.

After that, the puzzle falls to pieces.

Ruud.

Posted: **Wed Feb 22, 2006 9:34 pm**

I solved it the way David did -- and was also not satisfied. Plus it took
_hours_ ....

Laura, pleased to see a happier way

Posted: **Thu Feb 23, 2006 3:03 am**

Hi, Ruud! Thanks for the explanation.

After posting my message this morning I went back and looked at the SudoCue log for this puzzle. I spotted the pattern by
looking at columns, though.

Code: Select all

.------------------.------------------.------------------.
| 23    46    1    | 2789  2689  46789| 4689  3469  5    |
| 23    8     46   | 129   1269  5    | 469   1349  7    |
| 9     5     7    | 18    3     1468 | 2     146   1468 |
&#58;------------------+------------------+------------------&#58;
| 57    123   9    | 4     1568  1368 | 68    2567  268  |
| 6     124   248  | 1589  7     189  | 489   2459  3    |
| 57    34    48   | 3589  5689  2    | 1     45679 468  |
&#58;------------------+------------------+------------------&#58;
| 8     26    5    | 123   4     13   | 7     126   9    |
| 4     79    3    | 6     129   179  | 5     8     12   |
| 1     79    26   | 5789  589   789  | 3     246   246  |
'------------------'------------------'------------------'

I concentrated on column 7.

r1c7 = 6 ==> r2c3 = 6 ==> r3c6 = 6
r2c7 = 6 ==> r1c2 = 6 ==> r3c6 = 6
r4c7 = 6 ==> (either r1c6 or r3c6) = 6

In each of the three cases, then, r1c5 <>6, r2c5 <> 6, and r4c6 <> 6.

I'm curious -- can these "template" patterns usually be analyzed both row-wise and column-wise? Or is this example a
bit unusual? dcb

Posted: **Thu Feb 23, 2006 5:51 am**

David Bryant wrote:I spotted the pattern by looking at columns, though.

Good find! It took me a long time to find my method. There are a couple of misleading conjugate pairs that are of no use...

David Bryant wrote:I'm curious -- can these "template" patterns usually be analyzed both row-wise and column-wise? Or is this example a
bit unusual? dcb

I have not done a thorough analysis on the templating steps. The ones that I did check always had at least one alternative to the template elimination, so I felt safe to include them in my Nightmare series. No doubt there are different ways to tackle them. The solving methods that work on a single digit all overlap each other in various degrees.

I checked a few of these template patterns with the "Advanced Solving Techniques" team on the sudoku players forum. This group found several alternative solving techniques for each of my template situations. I will give them this one as well, see what they come up with.

As we speak, I am implementing a couple of these new solving techniques into SudoCue. You nightmares might even become a little more scarier in the near future

Ruud.

Posted: **Tue Oct 17, 2006 8:58 pm**

I realize this puzzle is long forgotten by this time, but I'm slowly working through the archives and consulting the forum for possible posts on any puzzles I've found interesting or more difficult than usual. Like some others, I found this one more time consuming than most Nightmares, so I'm happy to see that I was in good company. I did not find the template eliminations that Ruud explained or the approach that David and Laura used. For what it's worth, here's how I proceeded from the bottleneck position Ruud posted above (repeated below for convenience).

I observed the conjugate pair r67c4 for candidate 3 in column 4. It's clear that setting r6c4 to 3 has some immediate implications, in view of the bivalued peer cell r6c2. It's also clear that setting r7c4 to 1 or 2 has some immediate implications, in view of the bivalued peer cells r3c4 and r7c2, respectively. So further investigation looked promising.

Code: Select all

.------------------.------------------.------------------.
| 23    46    1    | 2789  2689  46789| 4689  3469  5    |
| 23    8     46   | 129   1269  5    | 469   1349  7    |
| 9     5     7    | 18    3     1468 | 2     146   1468 |
&#58;------------------+------------------+------------------&#58;
| 57    123   9    | 4     1568  1368 | 68    2567  268  |
| 6     124   248  | 1589  7     189  | 489   2459  3    |
| 57    34    48   | 3589  5689  2    | 1     45679 468  |
&#58;------------------+------------------+------------------&#58;
| 8     26    5    | 123   4     13   | 7     126   9    |
| 4     79    3    | 6     129   179  | 5     8     12   |
| 1     79    26   | 5789  589   789  | 3     246   246  |
'------------------'------------------'------------------'

Placing 2 in r7c4 would lead to a conflict since

r7c4=2 ==> r7c2=6 ==> r1c2=4 ==> r6c2=3 ==> r6c4 <> 3

leaving column 4 without a 3.

So 2 can be removed from r7c4, creating a naked pair of {1,3} in r7c46, which (among other eliminations) eliminates 1 from r8c6. Now r9c6 must equal 8 in order to avoid the non-unique rectangle which Ruud mentioned.

From the position above (independent of the previously mentioned eliminations) it can also be shown that candidate 1 can be eliminated from r7c4. Setting r7c4=1 would lead to a conflict of two "6" digits in box 6:

r7c4=1 ==> r6c4=3 ==> r6c2=4 ==> r6c3=8 ==> r6c9=6

Also, (making use of the conjugate pair in box 3 for candidate 8)

r7c4=1 ==> r3c4=8 ==> r3c9 <> 8 ==> r1c7=8 ==> r4c7=6.

So we conclude that r7c4=3, and it's straightforward from there, with nothing more advanced than naked pairs needed.

SudoCue Users

21 Feb, 2006

21 Feb, 2006

aha

Recognizing the "template" pattern

Another Approach