The New Sudoku Players' Forum

[ArkieTech]

Code: Select all

 *-----------*
 |...|4..|3..|
 |9.4|...|.5.|
 |.7.|358|.2.|
 |---+---+---|
 |..1|...|..6|
 |4..|...|..2|
 |857|.1.|...|
 |---+---+---|
 |...|..2|...|
 |...|.95|1..|
 |...|6..|..9|
 *-----------*


Play/Print this puzzle online

[SpAce]

Code: Select all

.---------------.-----------------.------------.
| 25    28   58 | 4     6    9    | 3   7    1 |
| 9     3    4  |d1(7)  2    1-7  | 6   5    8 |
| 1     7    6  | 3     5    8    | 9   2    4 |
:---------------+-----------------+------------:
| 23    29   1  | 59    347  347  | 57  8    6 |
| 4     6   a39 | 59    8   a3(7) | 57  1    2 |
| 8     5    7  | 2     1    6    | 4   9    3 |
:---------------+-----------------+------------:
| 367  c19  b39 |c17    347  2    | 8   346  5 |
| 36    4    2  | 8     9    5    | 1   36   7 |
| 357   18   58 | 6     347  1347 | 2   34   9 |
'---------------'-----------------'------------'

(7=39)r5c63 - (9)r7c3 = (91)r7c24 - (1=7)r2c4 => -7 r2c6; stte

[Ngisa]

Code: Select all

+--------------------+---------------------+------------------+
| 25      28     58  | 4      6       9    | 3      7       1 |
| 9       3      4   |a17     2      b17   | 6      5       8 |
| 1       7      6   | 3      5       8    | 9      2       4 |
+--------------------+---------------------+------------------+
| 23      29     1   | 59     347     347  | 57     8       6 |
| 4       6     d39  | 59     8      c37   | 57     1       2 |
| 8       5      7   | 2      1       6    | 4      9       3 |
+--------------------+---------------------+------------------+
| 367    f19    e39  |g1-7    347     2    | 8      346     5 |
| 36      4      2   | 8      9       5    | 1      36      7 |
| 357     18     58  | 6      347     1347 | 2      34      9 |
+--------------------+---------------------+------------------+


(7)r2c4 = r2c6 - (7=3)r5c6 - r5c3 = (3-9)r7c3 = (9-1)r7c2 = (1)r7c4 => - 7r7c4; stte

Clement

[SpAce]

(7)r2c4 = r2c6 - (7=3)r5c6 - r5c3 = (3-9)r7c3 = (9-1)r7c2 = (1)r7c4 => - 7r7c4; stte

You could also eliminate -1 r2c4 with that chain (AIC Type 2). Since our chains are essentially the same, I could do that too if I added a node:

(1=7)r2c6 - (7=39)r5c63 - (9)r7c3 = (91)r7c24 - (1=7)r2c4 => -1 r2c4, -7 r2c6; stte

(Of course the extra elimination is unnecessary in this case, but it's good to be aware of that possibility.)

[eleven]

Code: Select all

+-------------------+-------------------+-------------------+
| 25    28    58    | 4     6     9     | 3     7     1     |
| 9     3     4     | 17    2     17    | 6     5     8     |
| 1     7     6     | 3     5     8     | 9     2     4     |
+-------------------+-------------------+-------------------+
| 23    29    1     | 59    347   347   | 57    8     6     |
| 4     6     39    | 59    8     37    | 57    1     2     |
| 8     5     7     | 2     1     6     | 4     9     3     |
+-------------------+-------------------+-------------------+
|#367   19    39    | 17   #347   2     | 8    #346   5     |
|#36    4     2     | 8     9     5     | 1    #36    7     |
|#37+5  18    58    | 6    #347   1347  | 2    #34    9     |
+-------------------+-------------------+-------------------+

MUG 3467 r79c158, r8c18 => r9c1=5, stte

[SpAce]

MUG 3467 r79c158, r8c18 => r9c1=5, stte

Hi eleven! Could you please explain how that works? I can see three intertwined URs (34, 36, 37) but I still can't see how the combined MUG is supposed to work. Shouldn't it produce multiple solutions if the sole guardian 5r9c1 is removed? I can't see that. Instead I can easily see that both 3r9c1 and 7r9c1 quickly produce contradictions, i.e. no solutions. Is it then really a MUG? Or am I just missing something (very possible)?

[eleven]

MUG 3467 r79c158, r8c18 => r9c1=5, stte

Hi eleven! Could you please explain how that works? I can see three intertwined URs (34, 36, 37) but I still can't see how the combined MUG is supposed to work.

But that's it. Combine them and you have this MUG. You could add another 36 UR with 6 both in r9c1 and r9c8, and that would be a MUG too.

Shouldn't it produce multiple solutions if the sole guardian 5r9c1 is removed? I

Yes, there are 8 solutions for this MUG, 2 for each UR.

Instead I can easily see that both 3r9c1 and 7r9c1 quickly produce contradictions, i.e. no solutions.

Don't know, what you mean. There is no contradiction in the MUG. In the (unique) puzzle all digits produce contradictions, which are not part of the solution.


PS: Maybe this helps for the understanding:
If you have one of the 8 MUG solutions in the solution grid, then (at least) one of the 3 UR's is part of it. Therefore another solution would exist to the given puzzle (with the 2 digits flipped).
So there must be another digit in the MUG cells to get a unique solution.

[Cenoman]

Code: Select all

 +------------------+--------------------+------------------+
 |  25    28   58   |  4    6     9      |  3    7     1    |
 |  9     3    4    |  17   2     17     |  6    5     8    |
 |  1     7    6    |  3    5     8      |  9    2     4    |
 +------------------+--------------------+------------------+
 |  23    29   1    |  59   47+3  34+7   |  57   8     6    |
 |  4     6    39   |  59   8     37     |  57   1     2    |
 |  8     5    7    |  2    1     6      |  4    9     3    |
 +------------------+--------------------+------------------+
 |  67+3  19   39   |  17   34+7  2      |  8    46+3  5    |
 |  36    4    2    |  8    9     5      |  1    36    7    |
 |  57+3  18   58   |  6    37+4  4+37-1 |  2    34    9    |
 +------------------+--------------------+------------------+

BUG+9
(3)r4c5 - r45c6 = (3)r9c6
(7)r4c6 - (7=1)r4c6
(3)r7c18 - r7c3 = r5c3 - r5c6 = (34)r49c6
(7)r7c5 - (7=1)r7c4
(3-7)r9c1 = r7c1 - (7=1)r7c4
(4)r9c5 - (4=3)r9c8 - r9c56 = r7c5 - r7c3 = r5c3 - r5c6 = (34)r49c6
(37)r9c6
=> -1 r9c6; ste

[SpAce]

Thanks, eleven! I'm still confused but I'm sure it's not your fault. This MUG just seems to be a bit out of my depth. I seem to be exceptionally blind today anyway. I originally wanted to use the large BUG (just for the kick of it) but somehow thought there was no way to remove all guardians to form a valid BUG so I gave up that idea. As Cenoman showed I was of course wrong, and I don't know why I failed to see that the first time. That was momentary blindness, however, as I had no problem when I tried again after seeing BUG mentioned in Cenoman's solution. On the other hand, the MUG logic still seems elusive to me.

Yes, there are 8 solutions for this MUG, 2 for each UR.

How can 3x2 be 8? Is there a fourth UR?

Don't know, what you mean. There is no contradiction in the MUG.

My mistake. The contradictions appear external to the MUG, of course.

If you have one of the 8 MUG solutions in the solution grid, then (at least) one of the 3 UR's is part of it. Therefore another solution would exist to the given puzzle (with the 2 digits flipped).

But that's exactly what I fail to see. If I remove 5r9c1 and try the remaining two digits (3|7), I would get these two MUG solutions (ignoring any external cells):

Code: Select all

.---------------.-----------------.--------------.
| 25     28  58 | 4   6      9    | 3   7      1 |
| 9      3   4  | 17  2      17   | 6   5      8 |
| 1      7   6  | 3   5      8    | 9   2      4 |
:---------------+-----------------+--------------:
| 23     29  1  | 59  347    347  | 57  8      6 |
| 4      6   39 | 59  8      37   | 57  1      2 |
| 8      5   7  | 2   1      6    | 4   9      3 |
:---------------+-----------------+--------------:
| 36(7)  19  39 | 17  3(4)7  2    | 8   34(6)  5 |
| 3(6)   4   2  | 8   9      5    | 1  (3)6    7 |
|(3)57   18  58 | 6   34(7)  1347 | 2   3(4)   9 |
'---------------'-----------------'--------------'

.---------------.-----------------.--------------.
| 25     28  58 | 4   6      9    | 3   7      1 |
| 9      3   4  | 17  2      17   | 6   5      8 |
| 1      7   6  | 3   5      8    | 9   2      4 |
:---------------+-----------------+--------------:
| 23     29  1  | 59  347    347  | 57  8      6 |
| 4      6   39 | 59  8      37   | 57  1      2 |
| 8      5   7  | 2   1      6    | 4   9      3 |
:---------------+-----------------+--------------:
| 3(6)7  19  39 | 17  34(7)  2    | 8   3(4)6  5 |
|(3)6    4   2  | 8   9      5    | 1   3(6)   7 |
| 35(7)  18  58 | 6   3(4)7  1347 | 2  (3)4    9 |
'---------------'-----------------'--------------'

I can't see any flippable digits contained within those two solutions. Should I? What am I missing?

[eleven]

Sorry for the confusion, my fault. I knew that it is a MUG, and thought, i could simplify things, but that was wrong.
(It is not the case, that the last 2 solutions contain 2 UR's, as i had thought)
So i have to look at all 8 cases, listed below (in lexicographical order), your examples are 6 and 8.
All others have a UR in them, and these 2 can be flipped in a solution without changing the rest of the grid.
In fact there are only 3 different "footprints", defined by the pair in c5, the others can be switched to them: cases 134, 257, 68.

Code: Select all

1.
:------------+------------+-----------:
| 3    .  .  |  .  4   .  | .  6    . |
| 6    .  .  |  .  .   .  | .  3    . |
| 7    .  .  |  .  3   .  | .  4    . |
'------------'------------'-----------'
2.
:------------+------------+-----------:
| 3    .  .  |  .  7   .  | .  6    . |
| 6    .  .  |  .  .   .  | .  3    . |
| 7    .  .  |  .  3   .  | .  4    . |
'------------'------------'-----------'
3.
:------------+------------+-----------:
| 6    .  .  |  .  3   .  | .  4    . |
| 3    .  .  |  .  .   .  | .  6    . |
| 7    .  .  |  .  4   .  | .  3    . |
'------------'------------'-----------'
4.
:------------+------------+-----------:
| 6    .  .  |  .  4   .  | .  3    . |
| 3    .  .  |  .  .   .  | .  6    . |
| 7    .  .  |  .  3   .  | .  4    . |
'------------'------------'-----------'
5.
:------------+------------+-----------:
| 6    .  .  |  .  7   .  | .  3    . |
| 3    .  .  |  .  .   .  | .  6    . |
| 7    .  .  |  .  3   .  | .  4    . |
'------------'------------'-----------'
6.
:------------+------------+-----------:
| 6    .  .  |  .  7   .  | .  4    . |
| 3    .  .  |  .  .   .  | .  6    . |
| 7    .  .  |  .  4   .  | .  3    . |
'------------'------------'-----------'
7.
:------------+------------+-----------:
| 7    .  .  |  .  3   .  | .  6    . |
| 6    .  .  |  .  .   .  | .  3    . |
| 3    .  .  |  .  7   .  | .  4    . |
'------------'------------'-----------'
8.
:------------+------------+-----------:
| 7    .  .  |  .  4   .  | .  6    . |
| 6    .  .  |  .  .   .  | .  3    . |
| 3    .  .  |  .  7   .  | .  4    . |
'------------'------------'-----------'

[SpAce]

Thanks a lot, eleven! I appreciate it. I think I get it now. I'd still probably have a hard time seeing or verifying that MUG myself in a live grid. A very cool solution indeed, and I'm glad you brought it up!