## Unique Rectangles: The Essentials

Advanced methods and approaches for solving Sudoku puzzles

### More on the last puzzle

In posting these puzzles, I am trying to find examples in which the UR is not only instructive, but also helpful in making the solution easier.

The previous puzzle is pretty tough! It is the Sudo Cue Nightmare of May 12. You can solve it yourself, or just believe my assertion that these patterns do happen in the real world.

The puzzle comes down to:
Code: Select all
`+-------------------+-------------------+-------------------+| 1     246   27    | 247   8     5     | 3     9     246   | | 23469 5     29    | 234   236   46    | 7     18    18    | | 23467 2346  8     | 9     2367  1     | 2456  25    246   | +-------------------+-------------------+-------------------+| 239   239   4     | 5     1     789   | 28    6     37    | | 8     2369  1     | 247   2679  46    | 24    37    5     | | 256   7     25    | 248   26    3     | 9     128   1248  | +-------------------+-------------------+-------------------+| 4579  489   579   | 6     3579  2     | 1     3578  378   | | 2579  1289  3     | 178   579   789   | 2568  4     2678  | | 257   128   6     | 1378  4     78    | 258   23578 9     | +-------------------+-------------------+-------------------+`

There is a Type 3 UR in on <18> in R26C89. The cells with "extra" candidates <24> are in R6 and B6.

In Box 6, there is a shared buddy, <24> in R5C7. This forms a reduced set with the two rectangle cells. So, no other cell in the shared buddies (Box 6) can be <2> or <4>. Remove <2> from R4C7, leaving only <8>.

In Row 6, there is a reduced set of four cells: The rectangle corners and R6C135 ; <24> and <256>, <25>, and <26>. So, we can remove <24> from R6C4, which must be <8>.

This illustrates an important point: If two corner cells of the UR are in the same block, they have two sets of common buddies: Those in the same line (row or column), and those in the same block. You can make eliminations separately in one set of buddies and / ot the other, but not together!

In this case, you may find it much simpler to make the Type 4 reductions: The only possibilities for <1> in R6 and B6 are in R6C89. Therefore, R6C89 cannot be <8>, forcing <8> in R4C7 and R6C4.

In general, Type 3 and Type 4 reductions are NOT equivalent.

This puzzle is not easy. If you choose to solve it without uniqueness arguments, it is qute difficult.

Keith
keith
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Posts: 215
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### Re: The rubber hits the road!

keith wrote:Let's try this one.
Code: Select all
`Puzzle: DSN051206+-------+-------+-------+| 1 . . | . 8 . | 3 . . | | . 5 . | . . . | 7 . . | | . . 8 | 9 . 1 | . . . | +-------+-------+-------+| . . 4 | 5 . . | . 6 . | | 8 . . | . . . | . . 5 | | . 7 . | . . 3 | 9 . . | +-------+-------+-------+| . . . | 6 . 2 | 1 . . | | . . 3 | . . . | . 4 . | | . . 6 | . 4 . | . . 9 | +-------+-------+-------+`

There is a Type 3 UR with two reductions based on two different sets of shared buddies. The UR is, by far, the simplest way to solve this puzzle.

Keith

I find the Type 1 UR in r26c89 easier to recognize and resolve. Naked/Hidden Singles complete the solution.

Code: Select all
`    b3  -  2456  Naked  Quadr1c8    =  9     Naked  Singler1c6    =  5     Hidden Single  c6    -  789   Naked  Triple    b6  -  1248  Naked  Quad    b6  -  1     Locked Candidate (1)r5c3    =  1     Hidden Singler4c5    =  1     Hidden Singler6      -  256   Naked  Triple  c8    -  18    Naked  Pair    b6  -  2     Locked Candidate (1)r5c6    ~  2     XY-Wingr6c5    =  2     Hidden Singler6c3    =  5     Naked  Singler6c1    =  6     Naked  Singler2      -  6     Locked Candidate (2)  c3    -  2     Locked Candidate (2)r3      -  2     Locked Candidate (2)r1      -  46    Naked  Pair    b2  -  4     Locked Candidate (1) *-----------------------------------------------------------* | 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     3579  2     | 1     357   378   | | 2579  1289  3     | 178   579   789   | 568   4     2678  | | 257   128   6     | 1378  4     78    | 58    2357  9     | *-----------------------------------------------------------*`
daj95376
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### Re: The rubber hits the road!

daj95376 wrote:I find the Type 1 UR in r26c89 easier to recognize and resolve.

You are correct. I would normally not look for XY-wings before Unique Rectangle, but I don't think that is the difference. If you look at the position I posted, you will see I have "missed" the triple "a" <256> <25> <26> in R6:
Code: Select all
`+-------------------+-------------------+-------------------+ | 239   239   4     | 5     1     789   | 28    6     37    | | 8     2369  1     | 247#  2679# 46b   | 24b   37    5     | | 256a  7     25a   | 248*  26ab  3     | 9     128*# 1248*#| +-------------------+-------------------+-------------------+ `

This eliminates <2> in the other cells in R6, "*", and reveals the Type 1 UR you have pointed out.

Or, in my position there is an XY-wing "b" on <2> centered on the <46> in R5C6. It takes out <2> from R5C45 and R6C89, "#".

Keith
keith
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Posts: 215
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### The power of UR's

Keith wrote :
In posting these puzzles, I am trying to find examples in which the UR is not only instructive, but also helpful in making the solution easier.

As an illustration of the power of UR's associated to strong-links I propose the following two puzzles.
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`Puzzle #1+-------+-------+-------+| . . . | . . 6 | 8 . 5 | | . . . | . . 1 | . 6 7 | | . . 4 | . 5 . | . . . | +-------+-------+-------+| . . . | . . 7 | . . 3 | | . 6 . | . 1 . | . 4 . | | 5 . . | 2 . . | . . . | +-------+-------+-------+| . . . | . 9 . | 2 . . | | 9 1 . | 5 . . | . . . | | 8 . 7 | 3 . . | . . . | +-------+-------+-------+`

After the usual basic steps (excluding x-wing, XY-wing,...) you reach the following situation :
Code: Select all
`+-------------------+-------------------+-------------------+| 1237  2379  1239  | 4     23    6     | 8     39    5     | | 23    2359  2359  | 89    238   1     | 4     6     7     | | 6     8     4     | 7     5     39    | 139   1239  129   | +-------------------+-------------------+-------------------+| 124   249   1289  | 6     48    7     | 159   12589 3     | | 237   6     2389  | 89    1     5     | 79    4     289   | | 5     479   189   | 2     348   39    | 1679  189   1689  | +-------------------+-------------------+-------------------+| 34    345   356   | 1     9     48    | 2     7     468   | | 9     1     26    | 5     7     248   | 36    38    468   | | 8     24    7     | 3     6     24    | 159   159   19    | +-------------------+-------------------+-------------------+`

Then you will need forcing chains, coloring, ... to progress.

But look at R78-C69.

You can apply here 2 variants of UR+3/2SL types (from Mike Barker / Uniqueness Type 6 - UR meets X-Wing) :
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`   ab     abX       UR+3C/2SL            |                  a|        - "b" can be removed from "abZ"        a   |          abY-----abZ       and   ab-----abX       UR+3N/2SL        a   |                  b|        - "a" can be removed from "abY"            |        - Y+Z can be used as a quantum cell as for UR type 3 (or 2 if Y=Z)   abY     abZ`

Here, you have 3 stacked URs:
Code: Select all
`  48------486       - UR+3C/2SL : removes "4" in the lower-right corner    |   8   |        - upper-right UR+3N/2SL : removes "8" in the lower-left corner,   |8     4|          and the quantum cell 26 + R8C3 removes "6" from R8C7   |   4   |        - lower-left UR+3N/2SL : removes "8" in the upper-right corner, 482------486         and as type 2 removes "6" from R6C9.`

... and the puzzle collapses !

The effectiveness of UR's on the second puzzle is even more impressive :
Code: Select all
`Puzzle #2 +-------+-------+-------+| . . . | . . 9 | . . 1 || . 6 . | . 4 . | . . . || 5 2 . | . . . | . . 8 |+-------+-------+-------+| 3 . . | . 8 . | 9 . . || . 8 . | . 9 . | . 6 . || . . 9 | . 6 . | . . 7 |+-------+-------+-------+| 7 . . | . . . | . 4 5 || . . . | . 5 . | . 1 . || 4 . . | 3 . . | . . . |+-------+-------+-------+`

After the preliminary steps (excluding x-wing, XY-wing,...) the situation is :
Code: Select all
`+-------------------+-------------------+-------------------+| 8     47    347   | 25    37    9     | 6     25    1     | | 9     6     1     | 258   4     258   | 57    37    23    | | 5     2     37    | 6     137   17    | 4     9     8     | +-------------------+-------------------+-------------------+| 3     457   6     | 1     8     457   | 9     25    24    | | 12    8     47    | 2457  9     23457 | 15    6     34    | | 12    45    9     | 245   6     2345  | 158   38    7     | +-------------------+-------------------+-------------------+| 7     1     8     | 9     2     6     | 3     4     5     | | 6     3     2     | 478   5     478   | 78    1     9     | | 4     9     5     | 3     17    178   | 2     78    6     | +-------------------+-------------------+-------------------+`

Look at R13-C35. You can see here an UR+2D/1SL type :
Code: Select all
`  ab     abY        UR+2D/1SL  |                   |a                - "a" can be removed from "abY"  |  abX     ab`

Is this case we have :
Code: Select all
` 374------37        - UR+2D/1SL can be applied twice, removing "3" in the     |   3   |          lower-right upper-left corners, and R3C3=R1C5=3    |3     3|   |   3   |  37------371`

You can immediately follow with a type 1 UR in R39-C56, solving R9C6=8,
... and again the puzzle collapses.

Without the UR+2D/1SL, Susser needs a lot of forcing chains ....

Mage
Mage

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Location: France

There's something I just don't quite get about unique rectangles:

If one has the following situation -

Code: Select all
`0 0   0  0 0  0  0 0 00 0   0  0 0  0  0 0 00 12  0  0 12 0  0 0 00 0   0  0 0  0  0 0 00 124 0  0 12 0  0 0 0`

the UR tenet tells us that the '4' has to be in r4c2. But doesn't that presuppose that we've narrowed down the choices as far as we possibly can?

For example, what if we have the above, and there's some kind of complicated fish thingamajig that eliminates the 1 in r4c2. Now if we get rid of that 1 in r4c2, we eliminate the possibility of a non-unique solution, do we not? So we can have a 2 in r4c2, which means a 1 in r2c2, 2 in r2c5, and a 1 in r4c5. We don't require the 4 to be in r4c2.

But what if we failed to notice the fish thingamajig, and simply eliminated the 1 and 2 from the candidate possibilites for r4c2?

I'm sure there's a flaw in my reasoning somewhere. Please be patient when you folk point it out...
boweasel

Posts: 2
Joined: 04 June 2006

boweasel wrote:If one has the following situation -
Code: Select all
`0 0   0  0 0  0  0 0 0 0 0   0  0 0  0  0 0 0 0 12  0  0 12 0  0 0 0 0 0   0  0 0  0  0 0 0 0 124 0  0 12 0  0 0 0 `

...and there's some kind of complicated fish thingamajig that eliminates the 1 in r4c2.

First of all, in the situation you show you cannot make any reduction, complicated fish thingamajig or not. The four cells of the UR must be defined by two rows, two columns AND two boxes. Your pattern is divided over four boxes.

Suppose the pattern is only in two boxes and there is a complicated fish thingamajig, then you can still make the uniqueness reduction. Actually you can then make it "safely", without even knowing if there is an unique solution or not. A more detailed explanation can be found here.

RW
RW
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### Another example

Here is another example. It is the Michael Mepham Diabolical from June 4:
Code: Select all
`Puzzle: MM060406 Diabolical+-------+-------+-------+| . . . | 1 . 8 | . . . | | . 5 . | . . . | . . . | | 6 . 2 | . . . | 1 . 9 | +-------+-------+-------+| . 3 . | 7 . 1 | . 5 2 | | 4 . . | . . . | . . . | | 2 8 . | 4 . 6 | . 7 . | +-------+-------+-------+| 8 . . | . . . | 5 . 1 | | . . . | . . . | . 3 . | | . . . | 9 . 2 | . . . | +-------+-------+-------+`

http://www.sudoku.org.uk/DailySudoku.asp?day=04/06/2006

After using only naked and hidden singles, you get here:
Code: Select all
`+----------------+----------------+----------------+| 3    9    7    | 1    24   8    | 6    24   5    | | 1    5    8    | 26   2469a 49  | 3    24   7    | | 6    4    2    | 5    37   37   | 1    8    9    | +----------------+----------------+----------------+| 9    3    6    | 7    8    1    | 4    5    2    | | 4    7    5    | 23   239  39   | 8    1    6    | | 2    8    1    | 4    5    6    | 9    7    3    | +----------------+----------------+----------------+| 8    2    34   | 36   3467b 347c| 5    9    1    | | 7    6    9    | 8    1    5    | 2    3    4    | | 5    1    34   | 9    34   2    | 7    6    8    | +----------------+----------------+----------------+`

There is a Type 1 UR on <24> in R12C58. It eliminates <24> in the cell labeled "a".

There is a Type 1 UR on <34> in R79C35. It eliminates <34> in the cell labeled "b".

There is a Type 4 UR on <37> in R37C56. It eliminates <3> from "b" and "c".

Note how the last two UR's overlap, and that making the eliminations from only one UR will destroy the other UR.

If you make the eliminations for all three UR's, the puzzle has only one cell, R5C5, with three possibilities, <239>. This is another uniqueness pattern, BUG or BUG+1. R5C5 must be <3>, and that solves the puzzle.

If you only spot the one of the two overlapping UR's, you have to find an XY-wing to complete the solution. (And, the X-wings are different, depending on which UR you used.)

If you do not use the Unique Rectangles at all, this puzzle is much more difficult to solve.

The lesson: Scan the puzzle for all possible eliminations before making any!

Keith
keith
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Posts: 215
Joined: 03 April 2006

### Re: Another example

keith wrote:The lesson: Scan the puzzle for all possible eliminations before making any!

Code: Select all
`+----------------+----------------+----------------+| 3    9    7    | 1    24   8    | 6    24   5    | | 1    5    8    | 26   69   49   | 3    24   7    | | 6    4    2    | 5    37   37   | 1    8    9    | +----------------+----------------+----------------+| 9    3    6    | 7    8    1    | 4    5    2    | | 4    7    5    | 23   239  39   | 8    1    6    | | 2    8    1    | 4    5    6    | 9    7    3    | +----------------+----------------+----------------+| 8    2    34   | 36   67   347  | 5    9    1    | | 7    6    9    | 8    1    5    | 2    3    4    | | 5    1    34   | 9    34   2    | 7    6    8    | +----------------+----------------+----------------+`

I found it like you (knowing that i had to look for UR's), but i am sure, that RW and others would spot the deadly pattern also after the eliminations. The 37 pair is "suspicious", also the strong link for 7 in rc7c56 holding a 7 in the "diagonal cell".
ravel

Posts: 998
Joined: 21 February 2006

### Re: Another example

ravel wrote:
Code: Select all
`+----------------+----------------+----------------+| 3    9    7    | 1    24   8    | 6    24   5    | | 1    5    8    | 26   69   49   | 3    24   7    | | 6    4    2    | 5    37   37   | 1    8    9    | +----------------+----------------+----------------+| 9    3    6    | 7    8    1    | 4    5    2    | | 4    7    5    | 23   29+3 39   | 8    1    6    | | 2    8    1    | 4    5    6    | 9    7    3    | +----------------+----------------+----------------+| 8    2    34   | 36   67   47+3 | 5    9    1    | | 7    6    9    | 8    1    5    | 2    3    4    | | 5    1    34   | 9    34   2    | 7    6    8    | +----------------+----------------+----------------+`
RW and others would spot the deadly pattern also after the eliminations. The 37 pair is "suspicious" ...

Since [edit: candidate] 3 exists elsewhere in r7, c5, and b8 ... I believe one can pretend it still exists in r7c5. But for those with the uniqueness BUG , IMO it's just as easy to apply BUG+2 for r5c6<>3 [edit: and r9c5<>3].
Last edited by ronk on Tue Jun 06, 2006 9:01 pm, edited 1 time in total.
ronk
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### Missing candidates?

ronk,

I agree about the BUG+2. Now, who is going to write: "De-BUGging Sudokus: The Essentials"?

Ravel,

I am in two minds. If you are sure that all naked and hidden singles are identified, I see no reason not to "add back" candidates to cells. But, Mike Barker has shown that you can destroy useful patterns: If applied early, UR's will solve otherwise unsolvable puzzles. If applied as a last resort, they will not.

Keith

(I may get a lot of responses for not including all the legal fine print and explanations in these comments!)
keith
2017 Supporter

Posts: 215
Joined: 03 April 2006

### A Dual Type 3 from Ruud

Ruud posted this one in another forum:

Starting from here:

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

you get to here:
Code: Select all
`+-----------------+----------------+-----------------+| 4568 1459  148  | 7    3    2    | 1569  459  149  | | 456  2     3    | 49   15   16   | 8     7    149  | | 7    1459  14   | 49   158  168  | 1569  2    3    | +-----------------+----------------+-----------------+| 1    46a   46b  | 8    2    7    | 39g   39h  5    | | 9    8     7    | 3    6    5    | 4     1    2    | | 2    3     5    | 1    4    9    | 7     8    6    | +-----------------+----------------+-----------------+| 3    1456c 1468d| 2    9    18   | 15    45e  7    | | 48f  7     9    | 5    18   3    | 2     6    148n | | 58k  15    2    | 6    7    4    | 1359i 359j 189m | +-----------------+----------------+-----------------+`

abcd is a potential UR on <46>. The "extra" candidates on cd are <158>. In R8 they form a reduced set with <18> and <15>. Therfore, e is not <5>, it must be <4>.

In B7, there is also a reduced set with <58> and <15>. So, f is not <8>, it must be <4>. These are Type 3 reductions.

There are equivalent Type 4 reductions: In R7, and in B4, the only canditates for <6> are the corners of the rectangle, cd. So, c or d cannot be <4>; e and f must be <4>.

There is more! Another UR on <39> is in ghij. The "extra" candidates in ij are <15>. In R9 they form a reduced set with <15>. So, k is not <5>, and m is not <1>.

In B9 there is also a reduced set with <15>. So, e is not <5> (again), n is not <1>, and m is not <1> (again).

The rest of the puzzle is naked singles.

If you do not use UR's, this puzzle is quite difficult.

Keith
keith
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Posts: 215
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### Uniqueness by initial value

I noticed this in my morning newspaper. It is the David Bodycombe puzzle of June 19, 2006:
Code: Select all
`Puzzle: DB061906  *----+-------+-------+-------+| 9 . . | 7 . . | 2 4 . | | 4 . . | 1 . . | . . 8 | | . 5 . | . 8 9 | . 6 3 | +-------+-------+-------+| . 1 4 | 6 . . | . . . | | . . . | . . . | . . . | | . . . | . . 2 | 1 7 . | +-------+-------+-------+| 1 9 . | 8 5 . | . 2 . | | 3 . . | . . 7 | . . 6 | | . 7 2 | . . 4 | . . 5 | +-------+-------+-------+`

This puzzle is very easy, and you do not need Unique Rectangles to solve it. However, it has the solution (in R58C56) of:
Code: Select all
`7  1----1  7`

The solution is unique, because R8C6 = <7> is an initial value. This illustrates the assertion, in the Introduction of my Guide:

This is not a unique solution*
...
(*Unless one of the values in the above diagram is given as an initial clue. Then, the solution is unique.)

Here is the solution:
Code: Select all
`+-------+-------+-------+| 9 6 8 | 7 3 5 | 2 4 1 | | 4 3 7 | 1 2 6 | 9 5 8 | | 2 5 1 | 4 8 9 | 7 6 3 | +-------+-------+-------+| 7 1 4 | 6 9 8 | 5 3 2 | | 5 2 9 | 3 7 1 | 6 8 4 | | 6 8 3 | 5 4 2 | 1 7 9 | +-------+-------+-------+| 1 9 6 | 8 5 3 | 4 2 7 | | 3 4 5 | 2 1 7 | 8 9 6 | | 8 7 2 | 9 6 4 | 3 1 5 | +-------+-------+-------+`

If you remove R8C6 as an initial value, this puzzle has multiple solutions. If you substitute R8C6 = <1> as an initial value, I believe it then has two solutions.

Keith
keith
2017 Supporter

Posts: 215
Joined: 03 April 2006

### Re: Uniqueness by initial value

For the given puzzle, the clue 7[86] is not so special in this regard -- there are several other clues that likewise have the property of being the only clue in some unavoidable rectangle. I've probably missed some, but each clue in brackets is like that ...
Code: Select all
`+-------+-------+-------+| 9 . . |[7]. . | 2 4 . || 4 . . |[1]. . | . . 8 || .[5]. | . 8 9 | . 6 3 |+-------+-------+-------+| . 1 4 |[6]. . | . . . || . . . | . . . | . . . || . . . | . . 2 | 1 7 . |+-------+-------+-------+| 1 9 . | 8 5 . | . 2 . ||[3]. . | . .[7]| . . 6 || . 7 2 | . . 4 | . . 5 |+-------+-------+-------+`

This situation is typical, rather than unusual.
r.e.s.

Posts: 337
Joined: 31 August 2005

### Re: The Guide

keith wrote:5.3 Short Forcing Chains

If you encounter a possible Unique Rectangle it is worthwhile to consider the candidates of the cells in the set of shared buddies. For example, suppose we have
Code: Select all
`+--------------+|  -   -    -  || 12   -   123 ||  -   -    -  |+--------------+| 12   -   124 ||  -   -    -  ||  -   -   13  |+--------------+`

The extra cell contains one of the UR's extra candidates and one of the deadly candidates. It turns out the lower right corner of the UR cannot be <1>!

The easiest way to see this is to note that
Code: Select all
`+--------------+|  -   -    -  || 12   -   123 ||  -   -    -  |+--------------+| 12   -    1  ||  -   -    -  ||  -   -   13  |+--------------+`

forces the deadly pattern. This is a chain, in which a certain candidate value in one cell results in a contradiction. So, that candidate can be eliminated.

Another way to look at this: If the <13> cell is <1>, the <124> cell is not <1>. If the <13> cell is <3>, there is a Type 1 UR, and the <124> cell must be <4>. In either case, the <124> cell is not <1>. (This is an excellent technique: Tabulate the possible solutions.)

(Version 3.07: Last edit 053006 5:00 pm EDT)

keith, A great presentation. However, I'm confused by the deadly pattern paragraph in the above quote. I don't see it. Fortunately, your Another way to look at this makes perfect sense.
daj95376
2014 Supporter

Posts: 2624
Joined: 15 May 2006

### Re: The Guide

daj95376"][quote="keith wrote:5.3 Short Forcing Chains

If you encounter a possible Unique Rectangle it is worthwhile to consider the candidates of the cells in the set of shared buddies. For example, suppose we have
Code: Select all
`+--------------+|  -   -    -  || 12   -   123 ||  -   -    -  |+--------------+| 12   -   124 ||  -   -    -  ||  -   -   13  |+--------------+`

The extra cell contains one of the UR's extra candidates and one of the deadly candidates. It turns out the lower right corner of the UR cannot be <1>!

The easiest way to see this is to note that
Code: Select all
`+--------------+|  -   -    -  || 12   -   123 ||  -   -    -  |+--------------+| 12   -    1  ||  -   -    -  ||  -   -   13  |+--------------+`

forces the deadly pattern. This is a chain, in which a certain candidate value in one cell results in a contradiction. So, that candidate can be eliminated.

keith, A great presentation. However, I'm confused by the deadly pattern paragraph in the above quote. I don't see it. Fortunately, your Another way to look at this makes perfect sense.

I asserted that the lower right cell is not <1>. If we assume it is <1>, we have
Code: Select all
`+--------------+|  -   -    -  || 12   -   123 ||  -   -    -  |+--------------+| 12   -    1  ||  -   -    -  ||  -   -   13  |+--------------+`

which is easily reduced to
Code: Select all
`+--------------+|  -   -    -  ||  1   -    2  ||  -   -    -  |+--------------+|      -    1  ||  -   -    -  ||  -   -    3  |+--------------+`

which is the deadly solution. So, the lower right cell is not <1>.

Keith
keith
2017 Supporter

Posts: 215
Joined: 03 April 2006

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