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A Super Hard (26)

 
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keith



Joined: 19 Sep 2005
Posts: 3341
Location: near Detroit, Michigan, USA

PostPosted: Sat Dec 22, 2007 2:08 pm    Post subject: A Super Hard (26) Reply with quote

This is a pretty tough one. Be sure to bring the big bag of tools!

Code:
Puzzle: M5437354sh(26)
+-------+-------+-------+
| 7 . 3 | . . . | 4 . 5 |
| . 9 . | . . . | . 3 . |
| . 4 . | 6 . . | . 1 . |
+-------+-------+-------+
| . . 1 | 9 4 . | . . . |
| . . . | 1 . 8 | . . . |
| . . . | . 7 2 | 1 . . |
+-------+-------+-------+
| . 6 . | . . 7 | . 2 . |
| . 1 . | . . . | . 8 . |
| 2 . 4 | . . . | 9 . 7 |
+-------+-------+-------+


Keith
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Marty R.



Joined: 12 Feb 2006
Posts: 5770
Location: Rochester, NY, USA

PostPosted: Sat Dec 22, 2007 7:34 pm    Post subject: Reply with quote

Quote:
Be sure to bring the big bag of tools!

I did, including the one I just acquired the other day. I made seven moves using techniques that I'd known for some time and that are commonly discussed here. Then my newest tool, Medusa, eliminated a candidate, then, shortly thereafter, it exposed a contradiction which solved the puzzle. It kept me busy for at least an hour.
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keith



Joined: 19 Sep 2005
Posts: 3341
Location: near Detroit, Michigan, USA

PostPosted: Sat Dec 22, 2007 9:22 pm    Post subject: Reply with quote

After basic eliminations, there are a Type 1 and a Type 4 Unique Rectangle.
Then, two X-wings and a Type 6 UR bring you to here:
Code:
+-------------+-------------+-------------+
| 7   28  3   | 28  19  19  | 4   6   5   |
| 1   9   6   | 7   5   4   | 28  3   28  |
| 5   4   28  | 6   28  3   | 7   1   9   |
+-------------+-------------+-------------+
| 368 25  1   | 9   4   56  | 28  7   36  |
| 346 7   29  | 1   36  8   | 5   49  236 |
| 468 35  89  | 35  7   2   | 1   49  68  |
+-------------+-------------+-------------+
| 89  6   5   | 4   89  7   | 3   2   1   |
| 39  1   7   | 25  239 59  | 6   8   4   |
| 2   38  4   | 38  16  16  | 9   5   7   |
+-------------+-------------+-------------+


After that, Medusa coloring on <2>, <8>, and <3> solves R8C5 as <9>, and the puzzle is toast.

I have made a PowerPoint file that shows the Medusa coloring. If you view it in slide show mode, you can step through it to see how the pattern builds.

http://www.mediafire.com/?cnpu3txwotn

The puzzle is solved by the third coloring step, but it is interesting to complete the pattern. The candidate eliminations (X) do not solve the puzzle, though they do reduce it to a BUG+1. Only the contradictions that solve <9> in B8 lead directly to the puzzle solution.

Keith
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keith



Joined: 19 Sep 2005
Posts: 3341
Location: near Detroit, Michigan, USA

PostPosted: Sat Dec 22, 2007 9:26 pm    Post subject: Reply with quote

Marty R. wrote:
Quote:
Be sure to bring the big bag of tools!

I did, including the one I just acquired the other day. I made seven moves using techniques that I'd known for some time and that are commonly discussed here. Then my newest tool, Medusa, eliminated a candidate, then, shortly thereafter, it exposed a contradiction which solved the puzzle. It kept me busy for at least an hour.


Marty,

I think that solving this one in less than an hour is incredible, since I know you use no software and do it all by pencil and paper. I am impressed! Very Happy

Keith
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Marty R.



Joined: 12 Feb 2006
Posts: 5770
Location: Rochester, NY, USA

PostPosted: Sat Dec 22, 2007 10:36 pm    Post subject: Reply with quote

Quote:
After basic eliminations, there are a Type 1 and a Type 4 Unique Rectangle. Then, two X-wings and a Type 6 UR bring you to here:


I never saw a Type 1 or 6. I had one X-Wing, two simple colorings, two DPs broken up and a W-Wing with pincer coloring before starting Medusa.
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keith



Joined: 19 Sep 2005
Posts: 3341
Location: near Detroit, Michigan, USA

PostPosted: Sat Dec 22, 2007 11:32 pm    Post subject: Reply with quote

Marty,

I've looked at this a number of ways, and there are a number of ways to get there. But, I think my posted point is where most of us would have run out of steam.

Keith
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alanr555



Joined: 01 Aug 2005
Posts: 198
Location: Bideford Devon EX39

PostPosted: Sun Dec 23, 2007 3:16 am    Post subject: Reply with quote

keith wrote:

Code:
+-------------+-------------+-------------+
| 7   28  3   | 28  19  19  | 4   6   5   |
| 1   9   6   | 7   5   4   | 28  3   28  |
| 5   4   28  | 6   28  3   | 7   1   9   |
+-------------+-------------+-------------+
| 368 25  1   | 9   4   56  | 28  7   36  |
| 346 7   29  | 1   36  8   | 5   49  236 |
| 468 35  89  | 35  7   2   | 1   49  68  |
+-------------+-------------+-------------+
| 89  6   5   | 4   89  7   | 3   2   1   |
| 39  1   7   | 25  239 59  | 6   8   4   |
| 2   38  4   | 38  16  16  | 9   5   7   |
+-------------+-------------+-------------+


Only the contradictions that solve <9> in B8 lead directly to the puzzle solution.


This puzzle can be solved using Implication chains starting from
r8c6 being 5 or 9 leading to r1c4 must be 8.

The cells in each route are:
r8c6, r8c4, r1c4 (direct route).
r8c6, r7c5, r3c5, r1c4 (using diagonals)

Once r1c4 is set as 8, the remaining cells resolve easily.
The clue to spotting the chains is to notice that r1c4 has a link on 2
in a column to r8c4 and on a diagonal to r3c5 The 5 in r8c4 draws
attention to r8c6 and from there r7c5 completes the second chain.
Not the easiest to spot - but feasible with diligence! Once the
cells have been identified, it becomes apparent that the second
chain is "strong" only in one direction - fortunately the right one
to give a positive result, without needing to find a contradiction.
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Asellus



Joined: 05 Jun 2007
Posts: 865
Location: Sonoma County, CA, USA

PostPosted: Sun Dec 23, 2007 4:24 am    Post subject: Reply with quote

Alan,

Your chain is another way of expressing exactly the same Box 8 contradiction on <9>s that are revealed by Medusa coloring:
Code:
+-------------+---------------+-------------+
| 7   28  3   | 2r8g 19   19  | 4   6   5   |
| 1   9   6   | 7    5    4   | 28  3   28  |
| 5   4   28  | 6    28r  3   | 7   1   9   |
+-------------+---------------+-------------+
| 368 25  1   | 9    4    56  | 28  7   36  |
| 346 7   29  | 1    36   8   | 5   49  236 |
| 468 35  89  | 3r5g 7    2   | 1   49  68  |
+-------------+---------------+-------------+
| 89  6   5   | 4    8g9r 7   | 3   2   1   |
| 39  1   7   | 2g5r 239  5g9r| 6   8   4   |
| 2   38  4   | 38   16   16  | 9   5   7   |
+-------------+---------------+-------------+

Starting the coloring with the {59} bivalue at r8c6, we quickly arrive at two "red" <9>s in Box 8, which is not allowed. So, all of the "red" values are false and all of the "green" values (including the <8> in r1c4) are true.

It is two ways of revealing the same underlying structure. The advantage of Medusa is that it is a mechanical process that doesn't require any "if-then" reasoning. On the other hand, it perhaps requires a certain amount of (intuitive or better) insight into when it is likely to produce a useful result. A grid with lots of bivalues and strong links scattered about that involve lots of different digits is always a good candidate for Medusa.

That being said, it is possible to solve this puzzle without using Medusa. After the various X-Wings and URs (and, in my case, a Kite on <3>), a couple of XY Chains do the job.

[Edit to correct the Box number and a typo.]
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keith



Joined: 19 Sep 2005
Posts: 3341
Location: near Detroit, Michigan, USA

PostPosted: Sun Dec 23, 2007 9:41 am    Post subject: Reply with quote

alanr555 wrote:
keith wrote:

Code:
+-------------+-------------+-------------+
| 7   28  3   | 28  19  19  | 4   6   5   |
| 1   9   6   | 7   5   4   | 28  3   28@ |
| 5   4   28@ | 6   28  3   | 7   1   9   |
+-------------+-------------+-------------+
| 368 25  1   | 9   4   56  | 28  7   36  |
| 346 7   29# | 1   36  8   | 5   49  236#|
|46-8 35  89% | 35  7   2   | 1   49  68% |
+-------------+-------------+-------------+
| 89  6   5   | 4   89  7   | 3   2   1   |
| 39  1   7   | 25  239 59  | 6   8   4   |
| 2   38  4   | 38  16  16  | 9   5   7   |
+-------------+-------------+-------------+


Only the contradictions that solve <9> in B8 lead directly to the puzzle solution.


This puzzle can be solved using Implication chains ...


I made my statement, of course, in the context of the structure revealed by Medusa. But, if you look at the last PowerPoint slide, it is remarkable that almost all of the cells are consistently colored, and the only xx in the pattern involves <9> in B8 and / or R8C5.

I would guess that any other way to solve the puzzle would have to find that same xx in the pattern.

This is very easy to do on pencil and paper, though you might want to start on a fresh grid. Here is the one I used:



(Click on the thumbnail to see a larger version.)

I simply use a dot . or a circle o under each candidate to denote the colors. Other possibilities are v and ^, or + and x. You can see that I pretty quickly got to the contradiction oo in R8C5.

Where to start? Pick a candidate that has many strong links and two-candidate cells. In this case I noted that all the <2>'s can be connected by strong links, and that coloring on <8> would give two clusters, which <28> would connect. (In other words, color on <2>, then xx to coloring on <8> in the <28> cells.) This is not a big deal - starting with any of the colored cells in the last PowerPoint slide will eventually end up with the same complete pattern.

It is interesting to look at how that Medusa conclusions arise. For the contradiction in R8C5, it is easy to find a chain in C4 and C5 that says, essentially, if R8C5 is <2>, then R8C5 is <3>, and vice-versa.

I have marked above the elimination of <8> in R6C1. The cells @ are, at first glance a W-wing, connected by the strong link on <2> # in R5. However, all the links on <2> are strong, so the cells @ are a remote pair. One is <2>, the other is <8>. You can extend the remote pair by strong links on <8> in C2 and C9, to the pincer cells % in R6 that make the elimination. The chain is 89%=28@=29#=236#=28@=68%.

Keith
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ravel



Joined: 21 Apr 2006
Posts: 536

PostPosted: Sun Dec 23, 2007 4:10 pm    Post subject: Reply with quote

Code:
+--------------+-------------+-------------+
| 7    28  3   |@28  19  19  | 4   6   5   |
| 1    9   6   | 7   5   4   | 28  3   28  |
| 5    4   28  | 6  #28  3   | 7   1   9   |
+--------------+-------------+-------------+
| 368  25  1   | 9   4   56  | 28  7   36  |
| 346  7   29  | 1   36  8   | 5   49  236 |
| 468  35  89  | 35  7   2   | 1   49  68  |
+--------------+-------------+-------------+
| 89   6   5   | 4   89  7   | 3   2   1   |
| 39   1   7   | 25 #239 59  | 6   8   4   |
| 2    38  4   |@38  16  16  | 9   5   7   |
+--------------+-------------+-------------+

alanr555's chain can also be looked at as an xy-chain.
Either r8c4=2 or r8c4=5 => r8c6=9 => r7c5=8 => r3c5=2
=> r1c4<>2, r8c5<>2

There is another one eliminating 2 from r1c4:
Either r1c2=2 or r1c2=8 => r9c2=3 => r8c1=9 => r8c6=5 => r8c4=2

And (at least) another one solving the puzzle:
Either r5c5=3 or r5c5=6 => r4c6=5 => r8c6=9 => r7c5=8 => r9c4=3
=> r6c4<>3, r8c5<>3

I find them like xy-wings. If a pincer number does not match the other, i follow bivalue cells (and in harder grids strong links) .

Next i would try to find a "strong link chain" like the marked one with strong link for 2 in column 5:
Either r8c5=2 or r3c5=2 => r1c4=8 => r9c4=3
In both cases r8c5<>3.

All this (and much more) can be done systematically with Medusa coloring starting with any of the involved cells. But as long as i can, i do it in my head.

Happy Christmas
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keith



Joined: 19 Sep 2005
Posts: 3341
Location: near Detroit, Michigan, USA

PostPosted: Tue Dec 25, 2007 1:38 am    Post subject: Reply with quote

And so, ravel, do you propose that we will be able to solve Sudoku by bouncing balls on a grid?

http://www.youtube.com/watch?v=JSiffV1mBKo

A great video, and a Merry Christmas to all. Thank you!

Keith
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