May 17, 2006 -- a "UR" and a "DIC"

[David Bryant]

I have noticed a frequent connection between "UR"s and double-implication chains rooted in cells that are conjugate to the exceptional cells at one or two corners of the rectangle. This puzzle (5/17/06) provides a particularly striking example.

After doing the usual stuff I arrived at this position.

Code: Select all

2689    5    29   2689  2489    3    27     1    247
  1    268    4     7    258  2568    9    36    23
  3    269    7     1    249   246    5    46     8
 469    3    15    259  2479   24    167    8   1567
 27    27     8    35     6     1     4     9    35
 469   469   15   3589  34789  48   1367    2   13567
  5    248    6    238  1238    7    123   34     9
 247   247    3    26    12     9     8     5   1246
 289    1    29     4   2358  2568   236    7    236

The "UR" lies in r58c12. Clearly one of r8c1 and r8c2 must contain the value "4", or else the solution to this "nightmare" would not be unique. So we can exclude "4" from r7c2, and from r8c9.

But look! We can immediately obtain a contradiction from the "conjugate cell" r8c9.

A. r8c9 = 4 ==> {2, 7} pair in r8c12 ==> r9c3 = 9
B. r8c9 = 4 ==> {2, 7} pair in r1c79 ==> r1c3 = 9

This contradiction leads directly to the same eliminations as the "UR" does. The symmetry and simplicity of this position are quite unusual. dcb