mith wrote:The second puzzle simplifies with the same trick, though not to singles.
I was troubled when I tried, my first attempt was unsuccessful. Therefore, I tried to understand the blockage.
- Code: Select all
+-----------------------+-------------------+-----------------------+
| 1234 8 6 | 123 7 9 | 1234 1234 5 |
| 1234 1234 9 | 8 5 123 | 7 6 1234 |
| 7 5 123 | 123 4 6 | 9 8 123 |
+-----------------------+-------------------+-----------------------+
| 124 9 124 | 5 3 124 | 8 7 6 |
| 5 1234 7 | 6 8 124 | 1234 1234 9 |
| 8 6 1234 | 124 9 7 | 5 1234 1234 |
+-----------------------+-------------------+-----------------------+
| 9 234 8 | 7 1 34 | 6 5 234 |
| 6 134 5 | 9 2 8 | 134 134 7 |
| 1234 7 1234 | 34 6 5 | 1234 9 8 |
+-----------------------+-------------------+-----------------------+
The intermediate puzzle, without the knowledge of the arrangement of 1,2,3,4 in c5
- Code: Select all
+-----------------------+-----------------------+-----------------------+
| 1234 8 6 | 1234 7 9 | 1234 1234 5 |
| 1234 1234 9 | 8 5 1234 | 7 6 1234 |
| 7 5 1234 | 1234 1234 6 | 9 8 1234 |
+-----------------------+-----------------------+-----------------------+
| 1234 9 1234 | 5 1234 1234 | 8 7 6 |
| 5 1234 7 | 6 8 1234 | 1234 1234 9 |
| 8 6 1234 | 1234 9 7 | 5 1234 1234 |
+-----------------------+-----------------------+-----------------------+
| 9 1234 8 | 7 1234 1234 | 6 5 1234 |
| 6 1234 5 | 9 1234 8 | 1234 1234 7 |
| 1234 7 1234 | 1234 6 5 | 1234 9 8 |
+-----------------------+-----------------------+-----------------------+
This puzzle has 24 solutions (Andrew's Stuart site), and noticeable, 1, 2, 3, 4 are candidates in 36 cells alone, forming naked quads in all 27 sectors (no interaction with other digits, all placed in every sector).
- Code: Select all
+-----------------------+-------------------+-----------------------+
| abcd 8 6 | abcd 7 9 | abcd abcd 5 |
| abcd abcd 9 | 8 5 abcd | 7 6 abcd |
| 7 5 abcd | abcd abcd 6 | 9 8 abcd |
+-----------------------+-------------------+-----------------------+
| abcd 9 abcd | 5 abcd abcd | 8 7 6 |
| 5 abcd 7 | 6 8 abcd | abcd abcd 9 |
| 8 6 abcd | abcd 9 7 | 5 abcd abcd |
+-----------------------+-------------------+-----------------------+
| 9 abcd 8 | 7 abcd abcd | 6 5 abcd |
| 6 abcd 5 | 9 abcd 8 | abcd abcd 7 |
| abcd 7 abcd | abcd 6 5 | abcd 9 8 |
+-----------------------+-------------------+-----------------------+
This is exactly the same puzzle, written using a different notation.
yzfwsf wrote:Algebraic method, I think it is a general trial filling method, rather than some specific numbers.
Fully in line with yzfwsf
Now, following mith's idea, we can raise the questions:
- does the knowledge of the arrangement a, b, c, d in any of the 27 sectors yields a unique solution of the puzzle ?
- for sectors answsering "yes", are there some, "simplifying" the puzzle solution ? (using mith's wording)
- and last, but not least, how to spot these specifically ?
With 27 trials, the answer to those questions are:
- yes for all sectors, to the first,
- yes for 10 sectors (r2346, c346, b125), to the second.
The answer to this second question, is subjective. The quality "simplifying the puzzle solution" is just my own appreciation: I have selected the sectors yielding a solution with a few short chains (2-fishes, X-chains, and possibly wings).
Among the 10 sectors, three are notably simplifying the puzzle solution:
r6, c4, b5, with three X-chain solution, two X-chain solution, three X-chain solution resp.
- the answer to the third question is: I don't know...
Here is the solution for c4:
- Code: Select all
+-----------------------+-------------------+-----------------------+
| bcd*^ 8 6 | a 7 9 | bcd^ bcd^ 5 |
| abd-c* ab-c 9 | 8 5 cd^ | 7 6 abcd^ |
| 7 5 ac-d | b cd* 6 | 9 8 acd^ |
+-----------------------+-------------------+-----------------------+
| abcd* 9 abcd | 5 abd* abd | 8 7 6 |
| 5 ab 7 | 6 8 abd | abcd abcd 9 |
| 8 6 abd | c 9 7 | 5 abd bd |
+-----------------------+-------------------+-----------------------+
| 9 abcd 8 | 7 abc abc^ | 6 5 bcd^ |
| 6 abcd 5 | 9 abc 8 | abcd abcd 7 |
| abc 7 abc | d 6 5 | abc 9 8 |
+-----------------------+-------------------+-----------------------+
Skyscraper(*) (d)r12c1 = r4c1 - r4c5 = r3c5 => -d r3c3
X-chain(^) (c)r1c1 = r1c78 - r23c9 = r7c9 - r7c6 = r2c6 => -c r2c12; lclste
[NP(ab)r25c2 => -ab r78c2; NP(cd)r78c2 => -c r9c13; ste]
Solution:
- Code: Select all
+-----------------------+-------------------+-----------------------+
| c 8 6 | a 7 9 | b d 5 |
| d b 9 | 8 5 c | 7 6 a |
| 7 5 a | b d 6 | 9 8 c |
+-----------------------+-------------------+-----------------------+
| b 9 c | 5 a d | 8 7 6 |
| 5 a 7 | 6 8 b | d c 9 |
| 8 6 d | c 9 7 | 5 a b |
+-----------------------+-------------------+-----------------------+
| 9 c 8 | 7 b a | 6 5 d |
| 6 d 5 | 9 c 8 | a b 7 |
| a 7 b | d 6 5 | c 9 8 |
+-----------------------+-------------------+-----------------------+
Comparing to the givens in c5: d=4, a=3, b=1, c=2 (selecting 1 out of 24 ways to affect 1, 2, 3, 4 to a, b, c, d)
the simplification is there, but...
Note: the same study on the first puzzle gives similar results: ste for r23, c468, b2 [i.e. not every sector having NT(124)]
Any other suggestion than such tedious T&E ?