## SSX (SkyScraperXtra, V2, New Improved....)

For fans of Killer Sudoku, Samurai Sudoku and other variants

### SSX (SkyScraperXtra, V2, New Improved....)

We can eXtend the classic SkyScraper puzzle type from being restricted to Latin Square and/or Sudoku format, by employing Kakuro-style grids. The rules for SSX are simple:

• clues, where given, are SkyScraper clues (see here for a useful guide to classic SkyScraper)
• the digits in each run of length N need to form a set of consecutive values. For example, for run length 3, the valid sets are {1,2,3}, {2,3,4} ... {6,7,8}, {7,8,9}

The variable-domain feature solves a problem I had with a previous version (SSK), whereby solvable grids of larger size became increasingly rare, and (conversely) the solutions easier to find.

With SSX, solvable grids are now easier to find, well-formed puzzles on those grids likewise relatively easy to find (in fact I have far more trouble generating regular Kakuro), but they can still be truly challenging.

For a gentle introduction, please see example see below (reply 3). I give here a moderately sized, but non-trivial example:

KT1111_T008.jpg (107.4 KiB) Viewed 906 times

Solution:
Hidden Text: Show
342x32145
453x46753
567921834
6748539xx
2314x4352
xx6795483
598367241
68957x564
47568x675
Last edited by Mathimagics on Fri Apr 01, 2016 12:23 pm, edited 1 time in total.

Mathimagics
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Location: Canberra

### SSX 25x15 Challenger

... and move straight on to a real challenger on a full-page grid.

Beware this one unless you want an extreme challenge!

It's really a demonstration of the existence of, and ability of my system to find, puzzles of varying difficulty on any size grid.

A puzzle on the same grid, but much simpler, is given below:

KP2515_0001.gif (103.02 KiB) Viewed 904 times

Solution:
Hidden Text: Show
x546x657xxx65
31578462x1324
23x897354621x
4235x34657x34
x45x65x462753
xx897x43x5876
4321x65x23145
564327x1342xx
x46578231x564
x9786x32x1423
32x2354x48675
213456x152346
58674x4213x12
4576x65x6543x
678x54623978x
xx2435x124356
52134x23x6897
6547x21x675xx
745263x45x21x
43x15432x4635
x896451372x23
6785x12683754
56xxx645x564x
Last edited by Mathimagics on Sat Apr 02, 2016 5:29 am, edited 2 times in total.

Mathimagics
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Location: Canberra

### Re: SSX (SkyScraperXtra, V2, New Improved....)

Smythe Dakota wrote:My comments so far (re: SSK):

• The simple example is a good way to get started, as it illustrates some important concepts.
• Your second example is perfect, in its degree of difficulty and interest.
• Your "challenging" example: No fair !! All words should be labeled with their clues. Otherwise it's just plain too hard. I'm not one of those people who likes to spend 10 hours on a single puzzle, or who somehow uses a computer to help solve it.

Bill Smythe

First of all, my apologies ... I am in fact one of those nutters who like to spend hours on a single puzzle!

I draw the line at using a computer to help me though!

I take great pains (and many long nights) writing the software to generate these puzzles, and my primary objective is to spend those few spare hours left on solving puzzles created by my own system.

I will provide some simple SSX examples shortly. I do note that the relaxing of domain restrictions in SSX tends to make things harder (no simple givens for SS clues of 1, for example). Hopefully the provision of bi-directional clues compensates somewhat!

I'll be interested to know if the first example above is doable, as I have not tested it myself. I will provide a better introductory set of examples in any case.

Smythe Dakota wrote:In general I love the concept. Skyscraper Kakuro is even better than Skyscraper Sudoku (just as Kakuro is more fun than Sudoku). Thank you!

I have some more variations on this theme in the pipeline, you'll be pleased to know

Mathimagics
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### SSX - A Gentle Example

This should have been given first up, but better late than never.

This example has DD (degree of difficulty) = 1, which means that the computer claims that all cell values are implied, so no trial and error should be required. Hopefully Bill can confirm this!

KT0909_T003_DD1.jpg (62.36 KiB) Viewed 899 times

Solution
Hidden Text: Show
2314x56
3245x89
xx56478
132x567
24351xx
32x3214
45x4325
Last edited by Mathimagics on Fri Apr 01, 2016 12:36 pm, edited 1 time in total.

Mathimagics
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### SSX - Moderate Exercise

This example on the same grid has DD = 41, which means it's harder than the previous one ... moderately so?

KT0909_T003_DD41.jpg (61.36 KiB) Viewed 899 times

Solution:
Hidden Text: Show
1432x23
2341x32
xx23514
465x645
35647xx
23x6978
54x5867

Mathimagics
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### SSX 25x15 Easy

As promised, here is a puzzle on the same 25x15 grid as the challenger above, but which is much simpler, requiring (as far as I can tell), no trial and error (ie: no DFS).

KP2515_T001_0003.gif (107.7 KiB) Viewed 890 times

Solution:
Hidden Text: Show
x546x657xxx65
31578462x1324
23x897354621x
4235x34657x34
x45x65x462753
xx897x43x5876
4321x65x23145
564327x1342xx
x46578231x564
x9786x32x1423
32x2354x48675
213456x152346
58674x4213x12
4576x65x6543x
678x54623978x
xx2435x124356
52134x23x6897
6547x21x675xx
745263x45x21x
43x15432x4635
x896451372x23
6785x12683754
56xxx645x564x

Mathimagics
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### SSX with Futoshiki clues

Here is an example which is made easier by the provision of a couple of key cell-pair relationship clues.

The result is an easy puzzle (no trial and error).

KT1111_T012_0012a.gif (35.97 KiB) Viewed 886 times

Solution
Hidden Text: Show
3456x32xx
12643578x
21x324516
43251x345
x6125743x
675x48657
783465x65
x54376928
xx65x9876

Mathimagics
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### Re: SSX - A Gentle Example

Mathimagics wrote: .... This example has DD (degree of difficulty) = 1, which means that the computer claims that all cell values are implied, so no trial and error should be required. Hopefully Bill can confirm this! ....

So far I have gotten nowhere with this DD1 puzzle, but that probably proves only that I am a neophyte at this skyscraper stuff.

I have, however, been able to make a rather peculiar observation.

If there were no 9's anywhere in the puzzle, then as soon as you have found "the" solution, you can create another valid solution just by adding 1 to the values in each cell.

Likewise, if there are no 1's, you can create an additional valid solution by subtracting 1 from each value.

Therefore, by uniqueness, there must be at least one 1 and at least one 9 somewhere in the puzzle.

Proofs by uniqueness are generally considered, by the purist in all of us, to be a scandalous dirty trick. One feels that one of the duties of a (human) solver is to establish uniqueness, not to assume it in order to reach a solution.

The possibility of being able to use a uniqueness argument comes up all the time in Kakuro and Sudoku. Suppose, for example, that you have already established that r2c5,r3c5,r6c5 all must be 1, 2, or 4 (and that all of these are in the same word), and that r2c6,r3c6 must both be either 1 or 2. Then you can conclude that r6c5 cannot be a 4, because then all four cells in the rectangle r2c5,r3c5,r2c6,r3c6 would have to be either 1 or 2, which means that, given one solution, you could create another by replacing the two 1's with 2's and vice versa. (Please note that everything in this entire paragraph applies to both Kakuro and Sudoku.)

Anyway, the existence of the uniqueness assumption temps me to try to solve this latest SSX puzzle by first looking for someplace a 1 can be, and someplace a 9 can be. If there is only one of each, then I'm on my way. But that would be a dirty trick!

Bill Smythe
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### Re: SSX - A Gentle Example

Smythe Dakota wrote:If there were no 9's anywhere in the puzzle, then as soon as you have found "the" solution, you can create another valid solution just by adding 1 to the values in each cell.

Likewise, if there are no 1's, you can create an additional valid solution by subtracting 1 from each value.

Therefore, by uniqueness, there must be at least one 1 and at least one 9 somewhere in the puzzle.

Good thinking!

Smythe Dakota wrote:Proofs by uniqueness are generally considered, by the purist in all of us, to be a scandalous dirty trick. One feels that one of the duties of a (human) solver is to establish uniqueness, not to assume it in order to reach a solution.

Not by me! I don't buy this argument at all, I'm afraid.

Puzzle-solving is all about logical deduction, inference and implication. So why on earth should I ignore the perfectly legitimate implications of the US property?

If I am doing a particularly hard Kakuro, and in some last area I have to resort to trial-and-error, then I stop as soon as I have a valid solution, I don't waste time going back and proving to myself that all the other combinations must be wrong!
I'd rather start a new puzzle ...

If I want proof-of-uniqueness, I ask my computer!

Mathimagics
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### Re: SSX - A Gentle Example

Earlier, I wrote: .... Therefore, by uniqueness, there must be at least one 1 and at least one 9 somewhere in the puzzle. ....

I just noticed this same argument can be applied to all the digits, not just 1 and 9.

For example, if there are no 5's, then any valid solution can generate another, just by adding 1 to all the values below 5 (and a third, just by subtracting 1 from all the values above 5).

This technique will never mess up the "every word must be a sequence" rule -- in fact, it would tend to correct any violations of this rule that might somehow have existed.

Corollary: No SSX puzzle with fewer than 9 cells can have a unique solution.

Bill Smythe
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Joined: 11 February 2006

### Re: SSX (SkyScraperXtra, V2, New Improved....)

Smythe Dakota wrote:So far I have gotten nowhere with this DD1 puzzle, but that probably proves only that I am a neophyte at this skyscraper stuff

Come on, Bill, try harder!

Mathimagics
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