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[ArkieTech]

Code: Select all

 *-----------*
 |...|.37|.1.|
 |..5|8..|36.|
 |...|...|.5.|
 |---+---+---|
 |4..|1.9|...|
 |1..|.4.|..2|
 |...|7.8|..5|
 |---+---+---|
 |.9.|...|...|
 |.3.|..5|7..|
 |.2.|39.|..1|
 *-----------*


Play/Print this puzzle online

[Leren]

Code: Select all

*-------------------------------------*
| 29 48  6   | 5  3 7  | 249  1   489 |
| 29 47  5   | 8  1 24 | 3    6   479 |
| 78 1   3   | 9  6 24 | 24   5   78  |
|------------+---------+--------------|
| 4  78  2   | 1  5 9  | 68   37  36  |
| 1  5  a78  | 6  4 3  | 89  b79  2   |
| 3  6   9   | 7  2 8  | 1    4   5   |
|------------+---------+--------------|
| 58 9   48  | 24 7 1  | 456  23  36  |
| 6  3   1   | 24 8 5  | 7   c29 d49  |
| 57 2  f4-7 | 3  9 6  |e45   8   1   |
*-------------------------------------*

(7) r5c3 = (7-9) r5c8 = r8c8 - (9=4) r8c9 - r9c7 = (4) r9c4 => - 7 r9c3; stte

Leren

Code: Select all

*----------------------------------------*
| 29  4-8 6   | 5   3 7  | 249  1   489  |
| 29  7-4 5   | 8   1 24 | 3    6   49-7 |
| 8-7 1   3   | 9   6 24 | 24   5   7-8  |
|-------------+----------+---------------|
| 4   8-7 2   | 1   5 9  | 6-8  7-3 3-6  |
| 1   5   7-8 | 6   4 3  | 8-9  9-7 2    |
| 3   6   9   | 7   2 8  | 1    4   5    |
|-------------+----------+---------------|
| 5-8 9-4 8-4 | 2   7 1  | 4-56 3-2 6-3  |
| 6   3   1   | 4-2 8 5  | 7    2-9 9-4  |
| 7-5 2   4-7 | 3   9 6  | 5-4  8   1    |
*----------------------------------------*

PS Just for fun I thought I'd try Medusa Coloring. My solver says : "Medusa Coloring Case 2: 5 & 6 in Cell r7c7 have the same parity (contradiction). Eliminate all members of their parity group value."

Leading to the 25 eliminations as shown; stte. Does anybody understand any of that ? I don't

Leren

[Cenoman]

Code: Select all

 +-----------------+-----------------+-------------------+
 |  29   48   6    |  5    3    7    |  29+4  1    48+9  |
 |  29   47   5    |  8    1    24   |  3     6    79+4  |
 |  78   1    3    |  9    6    24   |  24    5    78    |
 +-----------------+-----------------+-------------------+
 |  4    78   2    |  1    5    9    |  68    37   36    |
 |  1    5    78   |  6    4    3    |  89    79   2     |
 |  3    6    9    |  7    2    8    |  1     4    5     |
 +-----------------+-----------------+-------------------+
 |  58   9    48   |  24   7    1    |  56+4  23   36    |
 |  6    3    1    |  24   8    5    |  7     29   49    |
 |  57   2    47   |  3    9    6    |  45    8    1     |
 +-----------------+-----------------+-------------------+

(5)r7c7 = (5-8)r7c1 = r7c3 - r5c3 = r5c7 - (8=6)r4c7 => -6 r7c7; ste
@Leren
Written that way: 5r7c7 = 5r7c1 - 8r7c1 = 8r7c3 - 8r5c3 = 8r5c7 - 8r4c7 = 6r4c7 - 6r7c7
my chain proves that 6r7c7 has the same parity as 5r7c7 (the rationale is: all the weak links are provided by bivalues or bilocations)

[SpAce]

Two steps:

Code: Select all

.-------------.-----------.------------------------.
| b29  48  6  | 5   3  7  | ab29[+(4)]  1  b48+9   |
|  29  47  5  | 8   1  24 |   3         6  a79[+4] |
|  78  1   3  | 9   6  24 |   2-4       5   78     |
:-------------+-----------+------------------------:
|  4   78  2  | 1   5  9  |   68        37  36     |
|  1   5   78 | 6   4  3  |   89        79  2      |
|  3   6   9  | 7   2  8  |   1         4   5      |
:-------------+-----------+------------------------:
|  58  9   48 | 24  7  1  |  a56[+4]    23  36     |
|  6   3   1  | 24  8  5  |   7         29  49     |
|  57  2   47 | 3   9  6  |   45        8   1      |
'-------------'-----------'------------------------'

Step 1. BUG+4: (4)r17c7,r2c9 == (9,2,4)r1c917 => -4 r3c7 (5 placements)

Code: Select all

.------------.----------.--------------------.
| 2   48  6  | 5   3  7 | 9-4     1   89(+4) |
| 9   47  5  | 8   1  2 | 3       6   47     |
| 78  1   3  | 9   6  4 | 2       5   78     |
:------------+----------+--------------------:
| 4   78  2  | 1   5  9 | 68      37  36     |
| 1   5   78 | 6   4  3 | 89      79  2      |
| 3   6   9  | 7   2  8 | 1       4   5      |
:------------+----------+--------------------:
| 58  9   48 | 24  7  1 | 56(+4)  23  36     |
| 6   3   1  | 24  8  5 | 7       29  9-4    |
| 57  2   47 | 3   9  6 | 45      8   1      |
'------------'----------'--------------------'

Step 2. BUG+2: (4)r1c9,r7c7 => -4 r1c7,r8c9; stte

PS. It's interesting what happens in r1c9: the extra candidate is different in the two BUGs.

Added. One-step BUG:

Code: Select all

.--------------.-----------.----------------------.
| 29  b48   6  | 5   3  7  | b29+-4   1   b48+9   |
| 29   47   5  | 8   1  24 |  3       6   a79[+4] |
| 78   1    3  | 9   6  24 |  2-4     5    78     |
:--------------+-----------+----------------------:
| 4   c78   2  | 1   5  9  |  68      37   36     |
| 1    5   d78 | 6   4  3  |  89      79   2      |
| 3    6    9  | 7   2  8  |  1       4    5      |
:--------------+-----------+----------------------:
| 58   9   d48 | 24  7  1  | a56[+4]  23   36     |
| 6    3    1  | 24  8  5  |  7       29   9-4    |
| 57   2   e47 | 3   9  6  | f(4)5    8    1      |
'--------------'-----------'----------------------'

BUG+4:

(4)r2c9,r7c7 == (48)r1c72|(98)r1c92 - r4c2 = (84)r57c3 - r9c3 = (4)r9c7 => -4 r13c7,r8c9; stte

[SpAce]

PS Just for fun I thought I'd try Medusa Coloring. My solver says : "Medusa Coloring Case 2: 5 & 6 in Cell r7c7 have the same parity (contradiction)."

Not only in cell r7c7 but also (at least) in 4r7 and 4b9, though seeing any one of them is enough.

Eliminate all members of their parity group value.
Leading to the 25 eliminations as shown; stte. Does anybody understand any of that ? I don't

Are you serious? How did you ever code it if you don't understand how it works? Anyway, it's very simple. I tried to explain it before but I'm not sure if you read it.

Code: Select all

.-----------------------.-----------------.----------------------------.
| 2-9    4"-8'  6       |  5        3  7  | +9"-24     1      +8"-49   |
| 29     7"-4'  5       |  8        1  24 |  3         6       4-7'9   |
| 8"-7'  1      3       |  9        6  24 |  24        5       7"-8'   |
:-----------------------+-----------------+----------------------------:
| 4      8"-7'  2       |  1        5  9  |  6"-8'     7"-3'   3"-6'   |
| 1      5      7"-8'   |  6        4  3  |  8"-9'     9"-7'   2       |
| 3      6      9       |  7        2  8  |  1         4       5       |
:-----------------------+-----------------+----------------------------:
| 5"-8'  9     *8"-(4') | *2"-(4')  7  1  | *4-[5'6']  3"-2'   6"-3'   |
| 6      3      1       |  4"-2'    8  5  |  7         2"-9'  *9"-{4'} |
| 7"-5'  2      4"-7'   |  3        9  6  | *5"-{4'}   8       1       |
'-----------------------'-----------------'----------------------------'

3D-Medusa Wrap ('-contradictions in 4r7, 4b9, r7c7)
=> 25 eliminations + 2 direct placements (9r1c7, 8r1c9) (with 6 more eliminations); stte (only 4 singles left)
or => 24 direct placements (with 28 eliminations) + 3 more eliminations (-7 r2c9, -56 r7c7)
(either way, a total of 31 immediate eliminations out of 35 possible)

The coloring maps two opposing parities (' and "), of which one must be true and the other false. If two digits of the same parity end up in the same cell (5'6' in r7c7), or if two of the same digit of the same parity end up in the same house (4' in r7c34 and also in b9p67), it's obviously a contradiction for that parity. Thus that parity must be false (eliminating all candidates of that parity) and the other true (all of its candidates can be placed). Any of those contradictions can be easily mapped into a chain. The coloring just displays the results of multiple chains at the same time. It could also have trap (pincer) eliminations where the same non-colored candidate sees both parities and thus must be false. Those are even easier to map into chains.

Coloring is the easiest way to find eliminations in manual solving, and the very first thing I learned when I started solving non-basic puzzles. Of course 3D Medusa, being restricted to conjugate pairings, is very limited in its solving potential, but it can be easily extended with GEM -- and then it solves pretty much everything that can be solved with chains or nets.

One more point. The only cells that aren't directly touched by the Medusa are (29)r12c1 and (24)r23c6,r3c7 -- the exact same group that is solved by my first BUG+4 (which is why it isn't an stte solution alone). In fact, that group has its own isolated Medusa, but it doesn't solve anything (without non-conjugate extensions):

Code: Select all

.--------------.-------------.----------------.
| 2'9"  48  6  | 5   3  7    | 2"49  1   489  |
| 2"9'  47  5  | 8   1  2'4" | 3     6   479" |
| 78    1   3  | 9   6  2"4' | 2'4"  5   78   |
:--------------+-------------+----------------:
| 4     78  2  | 1   5  9    | 68    37  36   |
| 1     5   78 | 6   4  3    | 89    79  2    |
| 3     6   9  | 7   2  8    | 1     4   5    |
:--------------+-------------+----------------:
| 58    9   48 | 24  7  1    | 456   23  36   |
| 6     3   1  | 24  8  5    | 7     29  49   |
| 57    2   47 | 3   9  6    | 45    8   1    |
'--------------'-------------'----------------'

With GEM extensions it solves the Medusa cluster itself (via contradictions) but nothing else:

Code: Select all

.---------------------.----------------.-------------------------.
| 2†-9‡  4"8.    6    | 5     3  7     | 4.9.-2‡  1      4.8"9.  |
| 9†-2‡  4.7"    5    | 8     1  2†-4‡ | 3        6      4.7.-9‡ |
| 7.8"   1       3    | 9     6  4†-2‡ | 2†-4‡    5      7"8.    |
:---------------------+----------------+-------------------------:
| 4      7.(8")  2    | 1     5  9     | 6.(8")  [3"7"]  3.6"    |
| 1      5       7"8. | 6     4  3     | 8.9"    {7.9.}  2       |
| 3      6       9    | 7     2  8     | 1        4      5       |
:---------------------+----------------+-------------------------:
| 5"8.   9       4.8" | 2.4"  7  1     | 4.5.6"   2"3.   3"6.    |
| 6      3       1    | 2"4.  8  5     | 7        2.9"   4"9.    |
| 5.7"   2       4"7. | 3     9  6     | 4.5"     8      1       |
'---------------------'----------------'-------------------------'

GEM: Contradictions on ‡ : (8")r4c27, (3"7")r4c8, (7.9.)r5c8 => -2 r1c7,r2c1,r3c6, -4 r2c6,r3c7, -9 r1c1,r2c9 (5 placements)

(The actual contradictions may vary depending on the coloring order. On the other hand, pure Medusa always produces the same contradictions.)