A Siamesed Pair of Unique Rectangles

Everything about Sudoku that doesn't fit in one of the other sections

A Siamesed Pair of Unique Rectangles

Postby keith » Sat Jul 08, 2006 8:51 pm

Ruud's SudoCue Nightmare for July 8 is interesting:

000001607908070010020000000036005000004060900000200460000000050090030108802700000

http://www.sudocue.net/daily.php

Applying the usual, plus an XY-wing and a Type 2 UR, you get here:

Code: Select all
+----------------------+----------------------+----------------------+
| 34     45     35     | 89     289    1      | 6      289    7      |
| 9      6      8      | 345    7      24     | 25     1      234    |
| 7      2      1      | 345689 4589   4689   | 58     3489   349    |
+----------------------+----------------------+----------------------+
| 12     3      6      | 49     49     5      | 78     78     12     |
| 125    78     4      | 18     6      37     | 9      23     1235   |
| 15     78     9      | 2      18     37     | 4      6      135    |
+----------------------+----------------------+----------------------+
| 346    14     37     | 14689  12489  24689  | 27     5      2469   |
| 46     9      57     | 456    3      246    | 1      247    8      |
| 8      15     2      | 7      15     469    | 3      49     469    |
+----------------------+----------------------+----------------------+

Take a look at R456C19, and consider the following:

The pattern
Code: Select all
12  | 12
125 | 125
15  | 15

Has two solutions:
Code: Select all
1 | 2
2 | 5
5 | 1

and

2 | 1
5 | 2
1 | 5

Note how three values in the columns may be interchanged to get the two solutions.

So, in the position posted above, one of R56C9 must be <3>; R5C8 and R23C9 cannot be <3>.

These eliminations solve the puzzle. (Actually, eliminating <3> from only one of R5C8 or R2C9 solves the puzzle!)

Otherwise, it seems, the path to the solution is still quite difficult. Sudoku Susser needs four chains (with 6, 11, 4, and 5 links) and an XY-wing.

Keith
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Postby ravel » Sun Jul 09, 2006 10:43 am

Nice, see also there for the generalized pattern.
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Posts: 998
Joined: 21 February 2006

BUGged again!

Postby keith » Sun Jul 09, 2006 12:10 pm

Thank you Ravel. Ruud pointed out to me that this is a BUG-lite, and I believe it was the intent of his puzzle. Personally, I have never before found one of these in a real puzzle.

Coincidentally, there is another nice (not "Nice") pattern in one from the Australian site yesterday:

Code: Select all
Puzzle: AU070806
+-------+-------+-------+
| . . 8 | . . . | . 2 . |
| . 9 7 | 3 . . | . . . |
| . . . | 5 8 . | 1 . . |
+-------+-------+-------+
| 6 4 . | . . . | 2 . . |
| . . . | . 7 . | . . . |
| . . 2 | . . . | . 5 4 |
+-------+-------+-------+
| . . 5 | . 1 6 | . . . |
| . . . | . . 7 | 8 9 . |
| . 8 . | . . . | 5 . . |
+-------+-------+-------+

The basic techniques get you to here:
Code: Select all
+----------------+----------------+----------------+
| 345  1    8    | 7    46   49   | 3469 2    569  |
| 245  9    7    | 3    246  1    | 46   68   568  |
| 234  236  46   | 5    8    249  | 1    37   79   |
+----------------+----------------+----------------+
| 6    4    39   | 1    39   5    | 2    78   78   |
| 8    5    39   | 24   7    24   | 369  136  169  |
| 1    7    2    | 6    39   8    | 39   5    4    |
+----------------+----------------+----------------+
| 9    23   5    | 8    1    6    | 7    4    23   |
| 234  236  146  | 24   5    7    | 8    9    1236 |
| 7    8    146  | 9    24   3    | 5    16   126  |
+----------------+----------------+----------------+

Note the swordfishy thing on <39> in R456C357. So, R5C7 must be <6>.

This does not crack the puzzle, but it's a neat pattern!

Keith
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