dailysudoku.com

[keith]

Revision: May 27, 2006

An improved version of this guide is now posted at:

http://www.sudoku.com/forums/viewtopic.php?p=29105#29105

Keith

=========================================

Unique Rectangles: An Introduction
Terminology: In the following,
cell = square
possibility = candidate

Most people consider a Sudoku puzzle to be "valid" only if it has a single solution.

If one is willing to take advantage of this uniqueness, there are solution methods that can be applied.

Consider the following fragment of a puzzle solution:

[Marty R.]

Hey there, "Mr. Rectangle", very nice primer. 8)

That thread you linked to looks very intimidating at first glance. I wonder about some of these arcane techniques; are they necessary for one to become a pretty good solver or are they of interest mostly to the Sudoku "wonks" ( I'm not using that word in a negative sense)?

[keith]

Marty,

Thanks for the kind words. I wrote this because of a comment you made!

Shameless self-promotion: I am pleased that Mike Barker has listed this primer as his first resource on Unique Rectangles. It is in the link I gave in the original post.

You wrote:

[Robert Ferre]

I'm missing some point of logic here, Keith. Above you said this:


+-------------+-------------+-------------+
| 249 7 8 | 369 5 39 | 46 1 29 |
| 5 239 239 | 68 4 1 | 68 27 279 |
| 1 49 6 | 89 7 2 | 48 3 5 |
+-------------+-------------+-------------+
| 8 5 239 | 4 239 6 | 1 27 237 |
| 23 6 4 | 5 1 7 | 9 8 23 |
| 7 239 1 | 29 239 8 | 5 6 4 |
+-------------+-------------+-------------+
| 349 349 7 | 1 8 5 | 2 49 6 |
| 249 8 5 | 239 6 39 | 7 49 1 |
| 6 1 29 | 7 29 4 | 3 5 8 |
+-------------+-------------+-------------+

There is a unique rectangle on <27> in R24 C89. R2C9 and R4C9 cannot be <2>. In C8, the only possibilities for <7> are the corners of the rectangle. Therefore, neither of these corners can be <2>.

***********

Now, if C8 contains the only two possibilities for 7, and they are on the corners of the rectangle, wouldn't only one of them become a 7? Why wouldn't the other corner then become a 2? You are saying that 2 can be eliminated from both corners. I don't get it. Same for R24 C9 which you have also eliminated. So you have eliminated 2 from all four corners. I'm missing something here.

Robert

[keith]

Robert,

Thank you! You have found a typo. I said:

[Guest]

[George Woods]

I am too slow to understand this part of your argument You say::
+--------------+
| - - x |
| 46 - 246 |
| - - x |
+--------------+
| 46 - 469 |
| - - x |
| - - 269 |
+--------------+
| - - x |
| - - x |
| - - x |
+--------------+
[/code]
In this case, we can eliminate 2 and 9 as possibilities in any of the cells marked "x".


If I put a 2 into one of your "forbidden" cells I come up with the solution



0 0 0
6 0 4
0 0 0

4 0 9
0 0 0
0 0 6

0 0 2
0 0 0
0 0 0

and similarly for a 9 a different but apparently valid solution!

So where have I got it wrong?

I now realise the answer lies in the posting above

[keith]

"Guest" and George,

Thank you. I have edited it to make the corrections.

Keith

[George Woods]

Keith - Don't you find that knowledge of advanced techniques can sometimes be a "nuisance"

Today's puzzle (tuesday) has a unique rectangle that allows the placing of a 2 in box8 - so when the puzzle was solved I asked myself- Did it rely on spotting the unique rectangle? -- answer NO-- so if someone asks "how" was it done - it must be better to give the solution that uses "simple" logic!

[keith]

George said:

[Marty R.]

[keith]

Marty,

I have always thought this is a very weak explanation. I think I have figured out how to explain it better. I was planning to rewrite and expand this Introduction, and post a revised article this weekend.

Let me try this:

If you write the UR as

[Marty R.]

As always, thank you Keith. I have been fascinated with rectangles since I first learned about them. I have no quantitative evidence to base this on, but rectangles seem to be one of the most powerful techniques in the "toolbox."

One of the reasons I ask such detailed questions is that the way my brain is wired is such that I am unable to deduce all this stuff. Any success I may have at solving has come from learning "rules" and applying them when I can recognize a situation.

I look forward to the next rectangles primer.

[David Bryant]

[keith]

Marty,

As always, it is a pleasure. I think I have answered your question, with more to come in my revised Introduction to UR's.

I think this is very powerful, especially when combined with other techniques. Some people talk about "Almost Unique Rectangles" (AUR's) in which you do not have a cookbook pattern. Rather, you have a strong clue to look for a short elimination chain. (Keith's interpretation!)

Best wishes,

Keith

[guest]

I'd like to know if anybody had problems with sudoku from May 18. I can not find reasoning for given hints.
sudoku lover

[keith]

The last question does not really belong here.

May 18: Look for a triple <356> in column 5, and in block 5, and in block 8.

Keith

[keith]

A much improved guide to unique rectangles is posted here:

http://www.sudoku.com/forums/viewtopic.php?p=29105#29105

Keith

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