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[ArkieTech]

Code: Select all

 *-----------*
 |.54|..8|7..|
 |1..|...|...|
 |.3.|5.7|6..|
 |---+---+---|
 |7..|...|.85|
 |...|672|...|
 |91.|...|..7|
 |---+---+---|
 |..5|3.1|.7.|
 |...|...|..2|
 |..9|7..|51.|
 *-----------*


Play/Print this puzzle online

[Cenoman]

Initial (messy) solution

Hidden Text: Show

Code: Select all

 +-------------------+------------------+---------------------+
 |  6     5     4    |  19   139   8    |  7      2     139   |
 |  1     9     7    |  2    346   36   |  348    5     348   |
 |  28    3     28   |  5    149   7    |  6      49    149   |
 +-------------------+------------------+---------------------+
 |  7    a24    6    |  1-4  13    9    |  234    8     5     |
 |  5    b48#   38   |  6    7     2    |  1      349   349   |
 |  9     1     23   | d48  c358#  35   |  234    6     7     |
 +-------------------+------------------+---------------------+
 |  248   268*  5    |  3    289*  1    |  489    7     468   |
 |  348   7     1    |  89   56    56   |  3489   34    2     |
 |  238   268*  9    |  7    28*   4    |  5      1     368   |
 +-------------------+------------------+---------------------+

UR(28)r79c25 using externals
(4)r4c2 - (4-8)r5c2==(8)r6c5 - (8=4)r6c4 => -4 r4c4: ste
A chain for 2r4c2 is missing (as spotted by SpAce)

Code: Select all

 +-------------------+------------------+---------------------+
 |  6     5     4    |  19   139   8    |  7      2     139   |
 |  1     9     7    |  2    346   36   |  348    5     348   |
 |  28    3     28   |  5    149   7    |  6      49    149   |
 +-------------------+------------------+---------------------+
 |  7   Ba24#   6    |  1-4  13    9    |  234    8     5     |
 |  5    A48#   38   |  6    7     2    |  1      349   349   |
 |  9     1    b23   |zd48  y358#  35   | c234    6     7     |
 +-------------------+------------------+---------------------+
 |  248   268*  5    |  3    289*  1    |  489    7     468   |
 |  348   7     1    |  89   56    56   |  3489   34    2     |
 |  238   268*  9    |  7    28*   4    |  5      1     368   |
 +-------------------+------------------+---------------------+

UR(28)r79c25 using externals
(2)r4c2 - r6c3 = (2-4)r6c7 = (4)r6c4
(8-4)r5c2 = (4)r4c2
(8)r6c5 - (8=4)r6c4
=> -4 r4c4: ste

[SpAce]

Code: Select all

.--------------.-------------.-------------------------.
| 6    5    4  | 19  139  8  |   7        2       139  |
| 1    9    7  | 2   346  36 |   348      5       348  |
| 28   3    28 | 5   149  7  |   6        49      149  |
:--------------+-------------+-------------------------:
| 7   b24   6  | 14  13   9  | a(2)4-3    8       5    |
| 5   b48   38 | 6   7    2  |   1      c(39)4  c(39)4 |
| 9    1    23 | 48  358  35 |   234      6       7    |
:--------------+-------------+-------------------------:
| 248  268  5  | 3   289  1  |   489      7       468  |
| 348  7    1  | 89  56   56 |   3489     34      2    |
| 238  268  9  | 7   28   4  |   5        1       368  |
'--------------'-------------'-------------------------'

(2)r4c7 = (24)r45c2 - (4=93) => -3 r4c7; stte

[SpAce]

UR(28)r79c25 using externals
(4)r4c2 = (4-8)r5c2==(8)r6c5 - (8=4)r6c4 => -4 r4c4: ste

Isn't that missing the third external (#2)r4c2 or am I missing something? I can see a path for it but with that the UR isn't needed at all:

(4=2)r4c2 - r4c7 = (24)r6c74 => -4 r4c4

Added: Anyway, this seems to be an example of the situation we talked about. Externals can solve the puzzle easily (even if simpler ways exist), but at least I can't do much with the internals (which again have a locked digit). Do you agree?

[Ngisa]

Code: Select all

+---------------------+--------------------+----------------------+
| 6       5        4  | 19      139     8  | 7        2       139 |
| 1       9        7  | 2       346     36 | 348      5       348 |
| 28      3        28 | 5       149     7  | 6        49      149 |
+---------------------+--------------------+----------------------+
| 7      c24       6  |b14      13      9  |d234      8       5   |
| 5       48       38 | 6       7       2  | 1        349     349 |
| 9       1        23 |a48      358     35 |e23-4     6       7   |
+---------------------+--------------------+----------------------+
| 248     2468     5  | 3       289     1  | 489      7       468 |
| 348     7        1  |89      56       56 | 3489     34      2   |
| 238     268      9  | 7       28      4  | 5        1       368 |
+---------------------+--------------------+----------------------+


(4)r6c4 = r4c4 - (4=2)r4c2 - r4c7 = (2)r6c7 => - 4r6c7; stte

Clement

[SteveG48]

Code: Select all

 *-----------------------------------------------------------*
 | 6     5     4     | 19    139   8     | 7     2     139   |
 | 1     9     7     | 2     346   36    | 348   5     348   |
 | 28    3     28    | 5     149   7     | 6     49    149   |
 *-------------------+-------------------+-------------------|
 | 7    b24    6     | 1-4   13    9     | 234   8     5     |
 | 5     48    38    | 6     7     2     | 1     349   349   |
 | 9     1    a23    |a48   a358  a35    | 234   6     7     |
 *-------------------+-------------------+-------------------|
 | 248   268   5     | 3     289   1     | 489   7     468   |
 | 348   7     1     | 89    56    56    | 3489  34    2     |
 | 238   268   9     | 7     28    4     | 5     1     368   |
 *-----------------------------------------------------------*


(4=2358)r6c3456 - (2=4)r4c2 => -4 r4c4 ; stte

[Cenoman]

Isn't that missing the third external (#2)r4c2 or am I missing something?

Right. Edited post

Anyway, this seems to be an example of the situation we talked about. Externals can solve the puzzle easily (even if simpler ways exist), but at least I can't do much with the internals (which again have a locked digit). Do you agree?

Agreed. It is true for the UR in my post [(28)r79c25] but also for UR(68)r79c29 and DP(356)r268c56
UR(68)r79c29: no weak link with internals 4r7c9 nor 3r9c9, no weak link either with external 8r2c9, but chains can be built with externals (box 9) 8r7c7 and 8r8c7.
DP(356)r268c56: no weak link with internal 4r2c5, but chains can be built with externals (row 2) 3r2c7 and 3r2c9
(Considered only guardians of troublesome pairs)

EDIT: obvious derived weak links exist between 4r2c5 and possible targets (cf. SpAce's and eleven's comments below). Yet another missed opportunity for me to shut up !

[SpAce]

Agreed. It is true for the UR in my post [(28)r79c25] but also for UR(68)r79c29 and DP(356)r268c56

Excellent! I guess we now have a definite answer to that lingering question. Although it seemed very probable, it's nice to know for sure.

UR(68)r79c29: no weak link with internals 4r7c9 nor 3r9c9, no weak link either with external 8r2c9, but chains can be built with externals (box 9) 8r7c7 and 8r8c7.

Wow! Thanks! This explains why you're the master of this stuff. I would have never even considered testing the 8r78c7 after seeing the more obvious guardians, including 8r2c9, fail. I guess this also answers the logical follow-up question -- seems that all three kinds of external possibilities need be considered separately if one wants to be thorough. That yet increases the possible combinations, though I guess in most cases at least one of the three (like in this case the row externals) can be dropped pretty easily.

DP(356)r268c56: no weak link with internal 4r2c5, but chains can be built with externals (row 2) 3r2c7 and 3r2c9

Another nice observation indeed! Seems to me that we could alternately use the box-based variant 3r1c5 in this case? Or 3r14c5 for a (column-based) all-externals solution? (Even the row-externals 3r6c37 work, so lots of working externals options here.) [Added: actually, isn't there a chain with 4r2c5, too: (4-6)r2c5 = (63)r26c6 - (3=24)b4p92 ? ]

Can we also predict that mixed guardians can produce results not reasonably achievable with pure internal or external combinations? Seems like a pretty safe bet, but an example of that would be nice to see too.

Edit: added "reasonably".

[eleven]

Generally you can't say, that results could be "achievable" with external/mixed, but not with internal, or other external candidates. It can be easier, and maybe simpler written as AIC, but you can't have the ones without the others.

In the 2 samples you easily get from the internals to the externals:

UR68: externals 8r5c2, 8r78c7
2r79c2-(2=48)r45c2
34r79c9,r8c8 - (3|4=89)r78c7

UR356: externals 3r14c5
4r2c5 - (4=3)r134c5
8r6c5 - (8=9)r79c5 - (9=3)r134c5

[SpAce]

Generally you can't say, that results could be "achievable" with external/mixed, but not with internal, or other external candidates. It can be easier, and maybe simpler written as AIC, but you can't have the ones without the others.

Of course. I should have said "reasonably achievable" (which I now edited). I guess with enough branching everything is achievable with anything (except falsehoods with true candidates, and solutions with false candidates). In this case demonstrating the equivalency didn't even require such measures, though. Thanks for pointing it out!

34r79c9,r8c8 - (3|4=89)r78c7

Thanks, I missed that one!

So, perhaps the question is still somewhat open, but I guess the answer also depends on the definition of "reasonable"?