What's Next?

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What's Next?

Postby Yogi » Sun Jul 24, 2016 12:25 am

.8.64.9..6...9..4..94..7.6.7.9..62348.39.467146..73895.7846.3.954...9.86936...4..
Here we have a number of eliminations. There are a few locked pairs, but no skyscrapers.
The map says the only possible kites are in candidates 1 or 2. This firstly eliminated 1 from r2c4.
Then r8c5<>2 and r2c4<>2. By now r68c34 is an XWing in 2’s, eliminating 2 from everywhere else in these rows and columns anyway.
The next move I think I’ve got is r8c4<>1 as 1r8c4 produces a Deadly Pattern in the XWing.
But whether that’s right or not I’m still down to the dreaded bifurcation. Any ideas?
Yogi
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Re: What's Next?

Postby Leren » Sun Jul 24, 2016 1:11 am

Code: Select all
*--------------------------------------------------------------*
|c123   8    c57     | 6     4     15-2   | 9   ca12   c237    |
| 6     12   b57     | 358   9     1258   |a157   4     2378   |
| 123   9     4      | 1358  12358 7      |a15    6     238    |
|--------------------+--------------------+--------------------|
| 7     15    9      | 158   158   6      | 2     3     4      |
| 8     25    3      | 9     25    4      | 6     7     1      |
| 4     6     12     | 12    7     3      | 8     9     5      |
|--------------------+--------------------+--------------------|
| 12    7     8      | 4     6     125    | 3     125   9      |
| 5     4     12     | 1237  13    9      | 17    8     6      |
| 9     3     6      | 1578  1258  1258   | 4     125   27     |
*--------------------------------------------------------------*

About the simplest thing seen by my solver at your stuck point was an ALS chain :

ALS XY Wing: (2=7) r1c8, r23c7 - (7=5) r2c3 - (5=2) r1c1389 => - 2 r2c6

After that things get a bit easier :

Code: Select all
*--------------------------------------------------------------*
| 123   8     57     | 6     4     15     | 9     12    237    |
| 6     1-2   57     | 358   9    d1258   | 157   4     2378   |
| 123   9     4      | 1358 c12358 7      | 15    6     238    |
|--------------------+--------------------+--------------------|
| 7     15    9      | 158   158   6      | 2     3     4      |
| 8    a25    3      | 9    b25    4      | 6     7     1      |
| 4     6     12     | 12    7     3      | 8     9     5      |
|--------------------+--------------------+--------------------|
| 12    7     8      | 4     6     125    | 3     125   9      |
| 5     4     12     | 1237  13    9      | 17    8     6      |
| 9     3     6      | 1578  1258  1258   | 4     125   27     |
*--------------------------------------------------------------*

Empty Rectangle, or X chain => - 2 r2c2.

After that it's some basics, an Xwing in 1's c68 r19 => - 1 r9c45, and a Kite/Xchain in 2's => - 2 r9c9, which finishes the puzzle.

That's about the simplest solution I can see. Hodoku's solution was similar but included a 9 node discontinuous loop instead of the ALS chain. So you need at least one somewhat advanced move to help with the solution.

Leren
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Re: What's Next?

Postby JC Van Hay » Sun Jul 24, 2016 11:13 am

After Kite{2R5C3} -> -{2r8c4} -> Xwing{2R48} -> -{2r239c4}, it remains to check if there are other single digit eliminations for the digit 2 ! Which is the case ...
Code: Select all
+-------------+-------------------+----------------+
| 123  8   57 | 6     4      12'5 | 9    12   237  |
| 6    12' 57 | 358   9      1258 | 157  4    2378 |
| 123  9   4  | 1358  12358  7    | 15   6    2'38 |
+-------------+-------------------+----------------+
| 7    15  9  | 158   158    6    | 2    3    4    |
| 8    2"5 3  | 9     2'5    4    | 6    7    1    |
| 4    6   12'| 12"   7      3    | 8    9    5    |
+-------------+-------------------+----------------+
| 12'  7   8  | 4     6      125  | 3    125  9    |
| 5    4   12"| 12'37 13     9    | 17   8    6    |
| 9    3   6  | 578   1258   1258 | 4    12'5  27  |
+-------------+-------------------+----------------+
1. There is only one solution for the digit 2 associated with r5c5=r6c3=2' : r2c2=r7c1=r8c4=r1c6=r3c9=r9c8=2'
2. On the other hand, if r5c2=r6c4=r8c3=2", an exclusion could exist in the remaining unsolved boxes B12389 according to Keith's principle. Now, in B2389, Skyscraper{2"R27} -> -{2r1c8, 2r9c9} and that's all because of the now locked candidates in B39. Therefore, the common exclusions in both cases are 2r1c8 and 2r9c9 and stte.
Interpretation ;
Code: Select all
+-----------------+-----------------------+--------------------+
| 123    8     57 | 6     4        125    | 9    1-2    237    |
| 6      1(2)  57 | 358   9        158(2) | 157  4      378(2) |
| 13(2)  9     4  | 1358  1358(2)  7      | 15   6      38(2)  |
+-----------------+-----------------------+--------------------+
| 7      15    9  | 158   158      6      | 2    3      4      |
| 8      5(2)  3  | 9     5(2)     4      | 6    7      1      |
| 4      6     12 | 12    7        3      | 8    9      5      |
+-----------------+-----------------------+--------------------+
| 1(2)   7     8  | 4     6        15(2)  | 3    15(2)  9      |
| 5      4     12 | 1237  13       9      | 17   8      6      |
| 9      3     6  | 578   1258     1258   | 4    125    7-2    |
+-----------------+-----------------------+--------------------+
4-Fish{2R2357} -> ... -> [2r32c9==2r7c8] -> -{2r1c8, 2r9c9}

Where ... means, for example,
r5c2-r2c2=3-Fish[r2c9=r2c9-r3c5.r7c6=Skyscraper(r3c19,r7c18)]
||
r5c5-r3c5=3-Fish[r3c9=r3c1-r2c2.r7c1=Skyscraper(r2c69,r7c68)]
or
the set of candidates at the intersections of 2R2357 and 2C1256 has no solution

Concerning the digit 1, the same kind of analysis leads to
Code: Select all
+------------------+-----------------------+------------------+
| 123   8     57   | 6       4      25(1)  | 9    2(1)   237  |
| 6     2(1)  57   | 2358-1  9      258(1) | 157  4      2378 |
| 123   9     4    | 12358   12358  7      | 15   6      238  |
+------------------+-----------------------+------------------+
| 7     5(1)  9    | 158     158    6      | 2    3      4    |
| 8     25    3    | 9       25     4      | 6    7      1    |
| 4     6     2(1) | 2(1)    7      3      | 8    9      5    |
+------------------+-----------------------+------------------+
| 2(1)  7     8    | 4       6      25(1)  | 3    25(1)  9    |
| 5     4     2(1) | 1237    123    9      | 17   8      6    |
| 9     3     6    | 2578-1  1258   258(1) | 4    25(1)  27   |
+------------------+-----------------------+------------------+
Kite{1R6C2} -> -{1r2c4}
and
5-Fish{1R6C268B7} -> ... -> [1r9c68==1r6c4] -> -{1r9c4}
where ... means, for example,
||XWing(r1c68,r9c68)
||r2c6-r2c2=r4c2-r6c3=r6c4
||r7c68-r7c1=r8c3-r6c3=r6c4
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