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A very tough minimal sudoku

 
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David Bryant



Joined: 29 Jul 2005
Posts: 559
Location: Denver, Colorado

PostPosted: Thu Mar 09, 2006 4:18 am    Post subject: A very tough minimal sudoku Reply with quote

Here's a very difficult 17-clue sudoku puzzle. Someone_somewhere sent me this puzzle in a private message. I'm not yet sure how to reason my way through to the solution.
Code:
6.2.5....
.....4.3.
.........
43...8...
.1....2..
......7..
5..27....
.......81
...6.....

Oh -- s_s called this a "zen" sudoku, apparently implying that it can be solved if one thinks about it (becomes it?) hard enough. dcb
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someone_somewhere



Joined: 07 Aug 2005
Posts: 275
Location: Munich

PostPosted: Sun Mar 12, 2006 2:12 pm    Post subject: Reply with quote

Yes, I called it Zen.
No single response in this forum is confirming it's name.
I know how to get to the solution, but it takes a long time, like in the Zen training. No brute force is needed.

If someone makes/made some progress on it, share it with us.

see u,
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keith



Joined: 19 Sep 2005
Posts: 3355
Location: near Detroit, Michigan, USA

PostPosted: Sun Mar 12, 2006 10:09 pm    Post subject: Position 1 Reply with quote

This is about the toughest puzzle I have ever tried.

I found six pinned squares, and a pair in column 9. Then after another seven moves (intersections to remove possibilities, not to solve squares), I get to this:

Code:



+----------------------+----------------------+----------------------+
| 6      4789   2      | 13789  5      379    | 1489   179    49     |
| 1789   5789   15789  | 1789   169    4      | 1689   3      2      |
| 13789  4789   134789 | 1789   169    2      | 14689  15679  4569   |
+----------------------+----------------------+----------------------+
| 4      3      5679   | 1579   2      8      | 169    1569   569    |
| 789    1      56789  | 34579  349    35679  | 2      4569   38     |
| 289    25689  5689   | 13459  1349   3569   | 7      14569  38     |
+----------------------+----------------------+----------------------+
| 5      4689   134689 | 2      7      139    | 3469   69     469    |
| 2379   2679   3679   | 3459   349    359    | 3569   8      1      |
| 139    49     1349   | 6      8      1359   | 3459   2      7      |
+----------------------+----------------------+----------------------+



I have played with this for a while, and I have no clue! The best I can suggest is to examine the squares that could be <4>.

Let's call this "Position 1". What next?

Keith
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David Bryant



Joined: 29 Jul 2005
Posts: 559
Location: Denver, Colorado

PostPosted: Sun Mar 12, 2006 11:12 pm    Post subject: There's a "coloring" move on the "7"s Reply with quote

Hi, Keith!

I think we can eliminate one more possibility from the grid using "standard" methods. After that I think we'll have to start using "double-implication chains."
Code:
+----------------------+----------------------+----------------------+
| 6      4789   2      | 13789  5      379B   | 1489   179A   49     |
| 1789   5789   15789  | 1789   169    4      | 1689   3      2      |
| 13789  4789   134789 | 1789   169    2      | 14689  15679a 4569   |
+----------------------+----------------------+----------------------+
| 4      3      5679c  | 1579C  2      8      | 169    1569   569    |
| 789    1      56789  | 34579  349    35679b | 2      4569   38     |
| 289    25689  5689   | 13459  1349   3569   | 7      14569  38     |
+----------------------+----------------------+----------------------+
| 5      4689   134689 | 2      7      139    | 3469   69     469    |
| 2379   2679   3679   | 3459   349    359    | 3569   8      1      |
| 139    49     1349   | 6      8      1359   | 3459   2      7      |
+----------------------+----------------------+----------------------+

There are three binary chains in the "7"s, marked Aa, Bb, and Cc above.

1. r1c8 = 7 ==> r1c6 <> 7 ==> r5c6 = 7 ==> r4c4 <> 7 ==> r4c3 = 7 ==> r3c3 <> 7.
2. r1c8 <> 7 ==> r3c8 = 7 ==> r3c3 <> 7.

So we can eliminate the possible "7" at r3c3.

I'm still working on my next move. dcb
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David Bryant



Joined: 29 Jul 2005
Posts: 559
Location: Denver, Colorado

PostPosted: Mon Mar 13, 2006 12:30 am    Post subject: R1C9 = 9 Reply with quote

I'm starting from the position previously posted, with the "7" removed from r3c3. We can demonstrate that r1c9 <> 4 as follows.

r1c9 = 4 ==> {6, 9} pair in r7c8&9 ==> {3, 4, 5, 9} quad in r8c456&7.

Making these eliminations the matrix looks like this.

Code:
  6    789     2    13789   5    379    189    179    4
1789  5789   15789  1789   169    4    1689     3     2
13789 4789   13489  1789   169    2    1689   15679  569
  4     3    5679   1579    2     8     169   1569   569
 789    1    56789  34579  349  35679    2    4569   38
 289  25689  5689   13459 1349  3569     7    14569  38
  5    48    1348     2     7    13     34     69    69
 27    267    67    3459   349   359    35      8     1
 139   49    1349     6     8   1359    345     2     7


But now we can spot another quad, {4, 7, 8, 9} in column 2. Making this elimination in the column also reveals that the "7" in column 2 must lie in the top left 3x3 box, so we can simplify the matrix like this.

Code:
  6    789     2    13789   5    379    189    179    4
 189    5     189   1789   169    4    1689     3     2
1389  4789   13489  1789   169    2    1689   15679  569
  4     3    5679   1579    2     8     169   1569   569
 789    1    56789  34579  349  35679    2    4569   38
 289   26    5689   13459 1349  3569     7    14569  38
  5    48    1348     2     7    13     34     69    69
 27    26     67    3459   349   359    35      8     1
 139   49    1349     6     8   1359    345     2     7


And now we have a series of forced moves:

1. r2c4 = 7 (unique horizontal)
2. r5c6 = 7 (unique vertical)
3. r6c6 = 6 (unique vertical)
4. r6c2 = 2 (sole candidate)
5. r4c3 = 7 (unique horizontal -- because of the "7" at r2c4)
6. r8c3 = 6 (sole candidate)
7. r8c2 = 2 (sole candidate)

But we can't have two "2"s in column 2, so r1c9 <> 4, and r1c9 = 9. This leaves the matrix looking like this.

Code:
  6    478     2    1378    5    37     148    17     9
1789  5789   15789  1789   169    4    1689     3     2
13789 4789   13489  1789   169    2    14689  15679  456
  4     3    5679   1579    2     8     169   1569   56
 789    1    56789  34579  349  35679    2    4569   38
 289  25689  5689   13459 1349  3569     7    14569  38
  5   4689  134689    2     7    139   3469    69    46
2379  2679   3679   3459   349   359   3569     8     1
 139   49    1349     6     8   1359   3459     2     7


From here we can see right away that r7c9 = 4, because if we put in the "6" it will be just like the case with the {6, 9} pair that we analyzed above. So here's where I'm going to leave it for now:

Code:
  6    478     2    1378    5    37     148    17     9
1789  5789   15789  1789   169    4     168     3     2
13789 4789   13489  1789   169    2    1468   1567   56
  4     3    5679   1579    2     8     169   1569   56
 789    1    56789  34579  349  35679    2    4569   38
 289  25689  5689   13459 1349  3569     7    14569  38
  5    689   13689    2     7    139    369    69     4
2379  2679   3679   3459   349   359   3569     8     1
 139   49    1349     6     8   1359    359     2     7
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someone_somewhere



Joined: 07 Aug 2005
Posts: 275
Location: Munich

PostPosted: Mon Mar 13, 2006 8:59 am    Post subject: Reply with quote

Thank you, guys.
I used also double implication chains, but ... I had to travel to a couple of other galaxies, until I got to the solution.
It is ZEN.
Hope that you are finding a way, your way ...

Wake me up, when you have something for someone_somewhere

see u,
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