## Hidden Skyscraper Sudoku

For fans of Killer Sudoku, Samurai Sudoku and other variants
If you don't want numerical clues you can consider adding 1 or more "greater than" signs between adjacent cells to show their relationship. Or, like in "consecutive sudoku" add a thick bar between some cells which are consecutive (in official consecutive sudoku puzzles there is the "bar=consecutive & no-bar=non-consecutive" rule but in your case I think you can drop the latter half of it, i.e. "no-bar can mean consecutive or non-consecutive").

I don't know... adding another RULE to the game seems even less elegant than using an existing rule to add a clue. And if I add a new rule, then I want it to either be in line with more classical sudoku (say a diagonal constraint) or involve skyscraper constraints. For example, I've been tossing around the idea of shading squares which accurately identify how many total skyscrapers can be seen in every direction. This has the potential to add quite a bit of information (and it's negative rule will add even more), so you wouldn't need nearly as many arrows (at the moment, I think that 25 squares with arrows is really pushing the lower limits, but 32+ squares is a bit too simple)

Speaking of which, I also think you might try to drop the "no-arrow" rule so that you can have more leeway in creating the puzzles. Just a random thought...

Yeah, I see your point, but I actually like the "no-arrow" rule. I like using it even when it may not be strictly required. And it does resolve some uniqueness issues in some puzzles. It's a bit hard to plan for it to show up, though, when constructing the puzzles.

I think I've resolved the problem in the puzzle I mentioned. I'm pretty sure it requires the no-arrow rule at least at the end, and I'd place the difficulty on par with the second puzzle. It's definitely more difficult than number 3. I'm running through one more solve (I had to make one minor change after the 2nd solve through) before I post it.
stigant

Posts: 14
Joined: 10 January 2007

ok, here it is. This will probably be my last HSS for a while.
[/img]
stigant

Posts: 14
Joined: 10 January 2007

Regarding the latest (#4) puzzle:

Code: Select all
__.__ __.__ __.__ __.__ __.__ __.v_ __.__ __.__ __.__
__.v> __.__ <^.__ __.__ __.v_ __.__ __.__ __.v_ __.__
__.__ __._> __.v> <_.__ <_._> __.__ __.__ <^._> __.__
__.__ _^.v_ __._> __.__ __.__ __.__ __.__ __.__ __.__
__.__ __.__ __.__ __.__ __.__ _^.v> __.v_ __.__ <_.v_
__.__ <_.v_ __.__ __.__ __.__ _^._> __.v> <_.__ __.__
_^.__ __.__ __.__ __.__ __.v> __.__ _^.__ __.__ _^.v_
_^.__ __.__ _^._> __.__ __.__ __.__ __.v> __.__ _^.__
__.__ __.__ __.__ _^._> <_._> _^.__ __.__ __.__ __.__

After all analysis via "existence of arrows", I reached this state:

Code: Select all
+----------------+----------------+----------------+
| 7    3    9    | 46   8    1    | 56   2    456  |
| 2    8    1    | 46   5    9    | 67   3    467  |
| 6    4    5    | 2    3    7    | 8    1    9    |
+----------------+----------------+----------------+
| 9    2    3    | 1    4    8    | 567  67   567  |
| 8    7    4    | 5    6    2    | 3    9    1    |
| 5    1    6    | 9    7    3    | 2    4    8    |
+----------------+----------------+----------------+
| 3    9    7    | 8    1    6    | 4    5    2    |
| 4    6    2    | 7    9    5    | 1    8    3    |
| 1    5    8    | 3    2    4    | 9    67   67   |
+----------------+----------------+----------------+

r1c6=1 without right arrow: r1c7 can't be 6, must be 5
r3c4=2 without up arrow: r12c4 can't be [64], must be [46]

Highlight to view the solution I wrote:739481526
281659734
645237819
923148675
874562391
516973248
397816452
462795183
158324967

Thanks for the nice puzzle!
udosuk

Posts: 2698
Joined: 17 July 2005

Ok, here's one more, but this really is the last one. I was working on this one and #4 at the same time, and I didn't think it was going to come out, but it did. I used a randomly generated grid again (like puzzle #1) so I can't claim much creativity. There are less arrows (22 boxes) than any of the previous puzzles, and I required a bit more indirect reasoning (not too deep though... still enough to do in my head), and a lot of no-arrow applications. Over all, I would rate this one as the most difficult to solve.

stigant

Posts: 14
Joined: 10 January 2007

Thanks! #5 is definitely the toughest among the lot. Here is a brief run down:

After some intensive analysis on the existence of the arrows, I came to this state:

Code: Select all
+----------------------+----------------------+----------------------+
| 9      5678   45     | 3      2      1      | 45678  45678  5678   |
| 2      3567   1      | 4567   8      4567   | 45679  5679   35679  |
| 35678  35678  45678  | 4567   9      4567   | 2      1      35678  |
+----------------------+----------------------+----------------------+
| 34     35678  5678   | 459    67     2      | 56789  56789  1      |
| 4567   1     *5678   | 459    3      45     | 567    2     *789    |
| 567    2      9      | 8      1      67     | 3      56     4      |
+----------------------+----------------------+----------------------+
| 567    567   -87     | 1      4      3      | 56789  56789  2      |
| 1      9      3      | 2      567    8      | 4567   4567   567    |
|*5678   4      2      | 67     56     9      | 1      3     *5678   |
+----------------------+----------------------+----------------------+

turbot fish ("skyscraper" - not the constraint but the technique)
8 @ r5 locked @ r5c39
8 @ r9 locked @ r9c19
one of r59c9 must NOT be 8
=> one of r5c3 & r9c1 must be 8
=> r7c3, seeing r5c3+r9c1, can't be 8

Then after further analysis it comes to this state:

Code: Select all
+-------------------+-------------------+-------------------+
| 9     567   45    | 3     2     1     | 4578  46    5678  |
| 2     3     1     | 567   8     467   | 457   67    9     |
| 567   5678  4568  | 567   9     467   | 2     1     3     |
+-------------------+-------------------+-------------------+
| 3     5678  568   | 4     67    2     | 689   89    1     |
| 4     1     68    | 9     3     5     | 67    2     78    |
| 67    2     9     | 8     1     67    | 3     5     4     |
+-------------------+-------------------+-------------------+
| 56    56    7     | 1     4     3     | 89    89    2     |
| 1     9     3     | 2     567   8     | 457   467   567   |
| 8     4     2     | 67    56    9     | 1     3     567   |
+-------------------+-------------------+-------------------+

Here I could have used another turbot fish and an x-chain plus an xy-wing to make more eliminations.
However, I decide to make use of the "non-existence of arrows" to finish it off:

r3c7=1 without down arrow: r4c7 can't be 9, must be 8
r7c9=2 without down arrow: r8c9 can't be 5

And all singles from here.

Highlight to view the solution I wrote:975321468
231684579
684597213
356472981
418935627
729816354
567143892
193258746
842769135

udosuk

Posts: 2698
Joined: 17 July 2005

Hmmmm, I didn't use any complicated standard Sudoku theorems (ex:skyscraper technique). However, from the first state that you showed, r9c9 can't be 8 (or in fact 7 either)because r7c9 is a 2 (no-arrow rule). This directly implies that r9c1 is 8 and r7c2 is 7.

Also, since r1c1 = 9 and r1c6 = 1 and has no >, r1c6 != 8, and similarly, using r1c4=3 with no >, r1c7 != 8 leaving a pointing pair of 8's in the top right 3x3 (r13c9) which eliminate the 8's in r59c9.

Of course, it's a matter of taste whether you'd prefer to use skyscraper technique or no-arrow rule, I suppose.
stigant

Posts: 14
Joined: 10 January 2007

stigant wrote:Of course, it's a matter of taste whether you'd prefer to use skyscraper technique or no-arrow rule, I suppose.

Yes you're spot on there.

I tried to do as much as I could before applying the no-arrow rule. In that case I figured that without the no-arrow rule the puzzle would have 7 solutions (though I'm not certain ). And using the turbot fish/skyscraper I was able to reduce the puzzle to 7 solutions. That's why I picked that approach.

Of course using the no-arrow rule early we don't need any advanced techniques.
udosuk

Posts: 2698
Joined: 17 July 2005

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