May 12, 2006: Do you UR?

[keith]

Today's Nightmare can be solved with basic methods and a single Unique Rectangle.

If you do not use the UR, the solution is much tougher.

David, is there a constellation lurking in this particular UR?

(I came late to solve May 11, and found I was early to May 12!)

Best wishes,

Keith

[David Bryant]

David, is there a constellation lurking in this particular UR?

Yes. There's always a "constellation" (or some similar configuration) buried outside every "UR" -- it has to be there, or else the solution would not be unique. The challenge is finding a pattern that's compact enough to be easily understood.

After a series of mostly routine moves (including some fancy coloring on the digit "6", and an XY-Wing from r5c6) I arrived at this position.

Code: Select all

  1    46    27    27     8     5     3     9    46
 39     5    29    234   36    46     7    18    18
 347   346    8     9    37     1    456   25    246
 239   239    4     5     1    789   28*    6    37
  8    239    1    47    679   46    24    37     5
  6     7     5    48     2     3     9    18    148
4579   489   79     6    359    2     1    357   378
2579   128    3    178   579   789   568    4   2678
 257   128    6   1378    4    78    58   2357    9

The "UR" is clearly visible at r2&6, c8&9. If we assume that the solution is unique we can set r6c9 = 4 and be merrily on our way. But what if we choose not to make that assumption?

In this case the chains proceeding from two of the "conjugate" cells (r1c9 & r6c4) are ridiculously long. I had my best luck (a "6-stars constellation") working from r5c7.

A. r5c7 = 4 ==> r5c4 = 7 ==> r5c8 = 3
B. r5c7 = 4 ==> r6c4 = 4
C. (r6c4 = 4 & r5c4 = 7) ==> r1c4 = 2 ==> r2c4 = 3

But with "3"s in columns 4 & 8 it becomes impossible to enter a "3" in row 9. Therefore r5c7 = 2, and the rest of the puzzle is straightforward.

(I came late to solve May 11, and found I was early to May 12!)

Our friend ravel presented a fascinating analysis of the May 11 nightmare in another discussion forum. Briefly, he found not only a "UR", but also a [non] Unique Hexagon on his way to the solution. dcb