## SSN -- SkyScraper Netherworld

For fans of Killer Sudoku, Samurai Sudoku and other variants

### SSN -- SkyScraper Netherworld

Here's a new possibility for Mathimagics to tackle. I'm not sure whether Kakuro or Sudoku would be a better choice here.

Each Skyscraper, rather than being 1 to 9 stories tall, is always actually 10 stories tall, but only 1 to 9 of the stories are above ground. The remaining 10-n stories are in basements and sub-basements below ground level.

The people above ground can see only the above-ground stories, and the people in the Netherworld can see only the below-ground stories. And, of course, a Skyscraper with P below-ground stories blocks the Netherperson's view of a Skyscraper behind it with only Q below-ground stories, if P>Q.

Each row and each column has four clues in it, two at each end of the row or column. Of the two clues at each end, one is the Realperson's view number and the other is the Netherperson's view number. Which is which can be distinguished by color, or by underscoring one and overscoring the other, or in some other manner.

Take that in your pipe and smoke it, Mathimagics!

Bill Smythe
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### Re: SSN -- SkyScraper Netherworld

Struth, mate!!

It's an intriguing problem (and a truly original idea) so I'll give it a shot ...

Watch this space!

Mathimagics
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### Re: SSN -- SkyScraper Netherworld

Hmm, I can now see a problem: if the below-ground values are complements of the upper, then the netherworld view is just the mirror image of the view above.

Or have i missed something?

Mathimagics
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### Re: SSN -- SkyScraper Netherworld

Yes, but the resulting Netherworld Skyscraper puzzle isn't really the complementary version of the one above ground.

For example, in straight Skyscraper it is entirely possible that a cell with a value of 1 might be invisible from all directions. In fact (at least in a Sudoku version, not with Kakuro), a 1 is visible from somewhere if and only if it is on one of the four edges.

But its complement, 9, is always visible from all directions.

So the symmetry between a configuration and its complement is not absolute.

Bill Smythe
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Joined: 11 February 2006

### Re: SSN -- SkyScraper Netherworld

Ok, I'll try harder!

Mathimagics
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### Re: SSN -- SkyScraper Netherworld

To put it another way, any Sudoku is isomorphic as a Sudoku to its Nether image, but the two are not isomorphic as Skyscrapers.

For example, in any Skyscraper the 9's are the easiest to solve, and the 1's are the hardest. If the upper and lower puzzles were isomorphic as Skyscrapers, then in the Nether version the 1's would be easiest and the 9's hardest. Yet, it's pretty clear that the 9's would still be the easiest and the 1's hardest.

So the Nether clues will provide additional information the solver wouldn't otherwise have.

Bill Smythe
Smythe Dakota

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Joined: 11 February 2006

### Re: SSN -- SkyScraper Netherworld

OK, I have a test rig set up - solver isn't ready yet but here is an example of presentation for a standard grid.

This has a unique solution for the above-ground clues, as it happens.
Attachments
Test6.jpg (64.38 KiB) Viewed 356 times

Mathimagics
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### Re: SSN -- SkyScraper Netherworld

I assume you are allowing digits 1-6, not 1-9, in this example.

Bill Smythe
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Joined: 11 February 2006

### Re: SSN -- SkyScraper Netherworld

Yes, sorry, I should have made that clear - I'm working with standard Skyscraper grids and rules initially.

And my solver is working! Here is a puzzle that requires some underground clues for uniqueness of solution! Perhaps you can work out which ones?

Test6_0024.jpg (54.87 KiB) Viewed 353 times

Solution
Hidden Text: Show
315246
152634
234561
426153
563412
641325

Mathimagics
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### Re: SSN -- SkyScraper Netherworld

I tested many Latin Squares for uniqueness of solution with and without the Skyscraper N clues.

Having both sets of clues makes a big difference to the pU (probability of uniqueness)
Code: Select all
`Size Tested  Uwith  Uwithout----------------------------  6   5023    4231     998  7   5013    3792     161  8   5008    3176      11`

My concern about properties of the complementary SS clues were obviously misplaced!

In my defence, it was very late at night, and the first 2 values I tested both gave the mirror image, so I leapt to conclusions ...

Note: the generator used for these tests was the Kakuro-grid generator, so these are by no means truly random grids. I will rerun these tests with unbiased RLS gen
Last edited by Mathimagics on Mon Apr 04, 2016 9:36 am, edited 2 times in total.

Mathimagics
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### Re: SSN -- SkyScraper Netherworld

Also, I think that the best setting for this puzzle is in a standard (LS/Sudoku) grid.

I could certainly place it in an extended (Kakuro-stye) grid but I see practical problems with cramming 8 clue values into one square which would greatly limit the size (and tidiness) of the grid, it seems to me.

Mathimagics
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### Re: SSN -- SkyScraper Netherworld

Using real RLS generation:

Code: Select all
`Size Tested   U(SSN)  pU    U(SS)  pU-------------------------------------     6   8821    7364    83%    615   7%  7   4157    2822    68%      6     8   2180    1007    46%      1     9   1884     394    21%      x  9S  1115     694    59%      x`

9S is Sudoku.

The combination of clues is a powerful fixer of uniqueness. One clue set only is a weak fixer.

Mathimagics
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### Re: SSN -- SkyScraper Netherworld

Sudoku example (should be easy)

SkyscraperN_9x9S.gif (27.76 KiB) Viewed 348 times

Solution:
Hidden Text: Show
261943578
743568192
895271436
352814967
416792853
987635214
124356789
539487621
678129345

Mathimagics
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### Re: SSN -- SkyScraper Netherworld

Mathimagics wrote: .... My concern about properties of the complementary SS clues were obviously misplaced! ....

You might say the Netherworld concept effectively levels the playing field between the 9's and the 1's.

Also, I think that the best setting for this puzzle is in a standard (LS/Sudoku) grid. .... I could certainly place it in an extended (Kakuro-stye) grid but I see practical problems with cramming 8 clue values into one square ....

Yes, that would be ugly -- which is why I posted earlier that Sudoku would be preferred over Kakuro as the basis for SSN puzzles, even though, in the "standard" world, I prefer standard Kakuro over standard Sudoku.

Meanwhile, even through all of this, I'm still having a little trouble getting my mind into Skyscraper mode at all. I'm just beginning to master elementary techniques. So please excuse me if I'm slow to respond to each new puzzle as you post it. I'll get there eventually (I think!).

Bill Smythe
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### Re: SSN -- SkyScraper Netherworld

Smythe Dakota wrote:Yes, that would be ugly -- which is why I posted earlier that Sudoku would be preferred over Kakuro as the basis for SSN puzzles, even though, in the "standard" world, I prefer standard Kakuro over standard Sudoku

Me too! Kakuro is my #1, Sudoku only when in Skyscraper or Futoshiki mode ...

Speaking of Kakuro - try KakuroPrime!

I'll leave SSN until I hear back from you regarding the solving experience. Meanwhile I have other irons in the fire ...

Cheers
Jim

Mathimagics
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