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cage shape
http://www.calcudoku.org/forum/viewtopic.php?f=16&t=187
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Author:  sneaklyfox  [ Sat Apr 28, 2012 6:52 am ]
Post subject:  Re: cage shape

pnm wrote:
Here's another one, not easy (!)
[scared]


I dunno... doesn't seem hard. Maybe on par with a 5x5 medium?

Author:  clm  [ Sun Apr 29, 2012 1:05 pm ]
Post subject:  Re: cage shape

picklepep wrote:
Have you ever thought about using a mandelbrot equation to generate cage shapes?


I assume you are thinking of developing a Mandelbrot set and after that getting the "boundary" of the set to create the cage shape (by similarity)?.

The idea is very interesting though I think it would require a very powerful computer (and graphics) and a lot of time (due to the huge number of iterations) to generate each cage; when the module of the "seed", the complex "c" in the equation, is < 1 or even << 1 then we usually arrive to very small (in module) vectors which require calculations that quickly go beyond the limits of the normal software and computers (we generally need supercomputers), and the same happens to the graphics in order to represent the points (the "tips of the arrows" of the vectors).

I think that there are easy ways to obtain random figures anyway.

I do not know what algorithm is Patrick actually using but considering that we only have 5 different shapes for the 4x4 cages, 12 for the 5x5's, 35 for the 6x6's and 107 for the 7x7's (excluding that "strange cage" in these 7x7's) (thread "The shape of the cages (structure and names)" in the section "Solving strategies and tips") I would suggest as an easy way just to prepare a "library" with those shapes (like numbering them as in the post) and then using any random number for the "pointer" just to select any of them each time. This process is really fast and insures that all shapes are randomly used (the size and number of them in each type of puzzle to be programmed).

Author:  picklepep  [ Mon Apr 30, 2012 1:29 am ]
Post subject:  Re: cage shape

clm wrote:
picklepep wrote:
Have you ever thought about using a mandelbrot equation to generate cage shapes?


I assume you are thinking of developing a Mandelbrot set and after that getting the "boundary" of the set to create the cage shape (by similarity)?.

The idea is very interesting though I think it would require a very powerful computer (and graphics) and a lot of time (due to the huge number of iterations) to generate each cage; when the module of the "seed", the complex "c" in the equation, is < 1 or even << 1 then we usually arrive to very small (in module) vectors which require calculations that quickly go beyond the limits of the normal software and computers (we generally need supercomputers), and the same happens to the graphics in order to represent the points (the "tips of the arrows" of the vectors).



Actually wasn't thinking much beyond how cool mandelbrot sets are. After Patrick asked me how I was thinking of implementing said idea, I was a little stumped [biggrin] i did thinkof something kind of cool though... The best part of mandelbrots is that the space they infect never get bigger. i was thinking about the calkudoku doesn't neccesarily have to fill up the square, though that could make the single soloution condition hard to fulfill..

Author:  beaker  [ Wed May 02, 2012 12:22 am ]
Post subject:  Re: cage shape

Thank you for introducing the "T" shaped cages in the twin puzzles.....they certainly put a different "spin" on the way I solved the twins today.....am hoping you will continue this trend with the twins instead of placing the "T" cage in the general puzzles as I seem to have diffculty with them as proven in one of the book puzzles (b4v2-page 121 of the "101 Advance Puzzles" with only a 2 star rating)...........Ken

Author:  jake4  [ Wed May 02, 2012 5:38 am ]
Post subject:  Re: cage shape

beaker wrote:
Thank you for introducing the "T" shaped cages in the twin puzzles.....they certainly put a different "spin" on the way I solved the twins today.....am hoping you will continue this trend with the twins instead of placing the "T" cage in the general puzzles as I seem to have diffculty with them as proven in one of the book puzzles (b4v2-page 121 of the "101 Advance Puzzles" with only a 2 star rating)...........Ken

Interesting. I noticed while doing the 5x5 difficult today that the T shape is actually easier than the "S" (that is, something like a1-b1-b2-c2). The T allows for only one repeated number, while the S allows two, so the T cuts down on the possible solutions for the cage.

On that same puzzle, I was able to solve just by looking at the left-hand puzzle and leaving the T until almost the end. I went back and looked just at the right-hand side and was able to use the 9+ T strategically after working some of the other cages.

Thanks for including these, Patrick! I can't wait to see how difficult (or not?) a cross shaped cage will be.

Author:  clm  [ Wed May 02, 2012 1:29 pm ]
Post subject:  Re: cage shape

jake4 wrote:
... I noticed while doing the 5x5 difficult today that the T shape is actually easier than the "S" (that is, something like a1-b1-b2-c2). The T allows for only one repeated number, while the S allows two, so the T cuts down on the possible solutions for the cage. ...


That's an interesting observation. My favourite cage anyway is the 5-cell W-shape which also permits to repeat 3 times a number and twice another, for instance.

I am proposing here new formats for the 9x9's. In the first graphic (I am not sure if it is doable with a unique solution) there are not individual cells nor 2-cell cages so if the puzzle is possible it should be very difficult; also the big rectangles (9x3 cells) are interleaved making difficult to apply the "addition" or the "multiplication" rules, etc.

An alternative to make it easier (second graphic) is to create 5 individual cells by breaking the long "T"s and dividing the 3-in line cage into 3 individual cells to cooperate in the solution.

The other format has four 5-cell "in-cross" cages, four 5-cell "W-shape" cages and four 5-cell "boxes", but it has 9 individual cells.

All formats contain enough 3-cell L-shape cages to include "-" or ":" cages, for instance.

Image

Image

Image

Author:  jomapil  [ Wed May 02, 2012 2:35 pm ]
Post subject:  Re: cage shape

Clm, these eventual new formats ( patterns ) are interesting to go out from the routine but they look like equally very difficult. We like puzzles not easy, with some work, but within limits. The value and the operations of the cages must be controlled what turns to the question of the other day: The diagrams are automatic or has Patrick an additional work to change personally some cages at the end of the process?

Author:  clm  [ Wed May 02, 2012 8:39 pm ]
Post subject:  Re: cage shape

jomapil wrote:
Clm, these eventual new formats ( patterns ) are interesting to go out from the routine but they look like equally very difficult. We like puzzles not easy, with some work, but within limits. The value and the operations of the cages must be controlled what turns to the question of the other day: The diagrams are automatic or has Patrick an additional work to change personally some cages at the end of the process?


In this moment I am not very sure about what is more interesting if having larger size puzzles (13x13, 16x16, 17x17, 19x19, ... welcome anyway) or sometimes increasing the level of difficulty of the actual sizes, i.e., introducing 8x8 subtractions only, no-op or "killer" calcudokus, or new functions/operators (apart of those logical AND, XOR, etc., but "unfortunately" all new operators would necessarily be more complex than the actual we are using). Anyway as the number of puzzlers increase (actually in the range of 3200 "all time" and soon in the 5000 [thumbup]) it would be convenient to introduce some more difficult puzzles to motivate a wider "population" (i.e., the "official" kenken.com page, and the corresponding New York Times kenkens, are tedious in my opinion, regardless of the actual options to select the level of difficulty in the Kenken page, ... , because all are arithmetic operations and conventional cages; in the case of the New York Times, regardless of the number of visitors, perhaps 100,000, because by definition [smile] a newspaper page is not expected to be very imaginative; but this is a specialized page and that's why it's clearly better than the rest).

With respect to your question perhaps Patrick has both options, a "switch" to intervene at any moment, and the possibility for the software to run alone for some long time. [biggrin]

Author:  pnm  [ Wed May 02, 2012 10:04 pm ]
Post subject:  Re: cage shape

jomapil wrote:
The diagrams are automatic or has Patrick an additional work to change personally some cages at the end of the process?

There are 3 ways in which cage layouts can be generated:
    - randomly with a maximum cage size (e.g. 3)
    - according to a distribution of cage sizes (e.g. 4 cages of size 1, 4 of size 2, 3 of size 3, 3 of size 5, for a 6x6 puzzle)
    - according to an exact layout
I don't have any mechanism for tweaking a layout once it's been generated.

The file format of exact layouts:

- cells are numbered row by row, column by column, starting from zero.
A 4x4 example:
Code:
0  1  2  3
4  5  6  7
8  9 10 11
12 13 14 15

- each cage is specified on a separate line, with commas between the cell numbers

An example 4x4 layout specification:
Code:
0,1
2,3
4,5
6,7
8,9
10,11
12,13
14,15

(so a pretty boring one, all 2-cell cages, two horizontal ones on each row)

Patrick

Author:  clm  [ Thu May 03, 2012 12:08 am ]
Post subject:  Re: cage shape

pnm wrote:
... There are 3 ways in which cage layouts can be generated:
    - randomly with a maximum cage size (e.g. 3)
    - according to a distribution of cage sizes (e.g. 4 cages of size 1, 4 of size 2, 3 of size 3, 3 of size 5, for a 6x6 puzzle)
    - according to an exact layout
I don't have any mechanism for tweaking a layout once it's been generated.

The file format of exact layouts:

...

- each cage is specified on a separate line, with commas between the cell numbers

An example 4x4 layout specification:
Code:
0,1
2,3
4,5
6,7
8,9
10,11
12,13
14,15

(so a pretty boring one, all 2-cell cages, two horizontal ones on each row)

Patrick


I am curious, then, Patrick, do you think a puzzle like this 9x9 could be generated? And what would be the solver rating?:

Code:
0,9,10,19,20
1,2,11,12,21
3,4,5,13
6,7,14,15,23
8,16,17,24,25
18,27,28
22,30,31,32
26,34,35
29,36,37,38,47
39,40,41
33,42,43,44,51
45,46,54
48,49,50,58
52,53,62
55,56,63,64,72
57,65,66,73,74
59,68,69,78,79
60,61,70,71,80
67,75,76,77

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