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 The solution of a 6x6 and a 9x9 puzzles with "strange cages" 
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Posted on: Thu Nov 17, 2011 12:33 am




Posts: 690
Joined: Fri May 13, 2011 6:51 pm
Post Re: The solution of a 6x6 and a 9x9 puzzles with "strange ca
pnm wrote:
That last one ("version e") had 9 solutions, computed in 0.03 seconds.

You see how hard it is to create puzzles with a single solution: especially
for the larger puzzles I like to add more large cages, but they
(1) increase solving time, and (2) make it more likely there is more than one
solution. Also, generally, puzzles with more symmetry are more likely
to have more than one solution (can't prove this, only intuition confirmed).

Another cheap way of reducing the number of solutions is by adding single cell cages
(e.g. split a double cell, or remove one from a larger cell).


version a (8 cages): 42672 solutions
version b (9 cages): 4280
version c (10 cages): 512
version d (11 cages): 66
version e (12 cages): 9

Looking to the curve I think that one more restriction would finally give the unique solution, maybe we can try it tomorrow, each number is obtained dividing the previous one by 8, more or less.

Yes, it is hard to create puzzles with the unique solution. When we add single cell cages we are limiting the "liberty" of those numbers to fly over the grid, this is what apparently happens when we have many cages, we are reducing the degrees of "freedom" of the numbers to move elsewhere (in the extreme case 36 cages in a 6x6... solved!!!), so a big proportion of cages... easy puzze, less cages... difficult puzzle... your intuition is good. It looks like to create a very difficult puzzle it is necessary to find the best compromise between the size of the puzzle and the size of the wider cage. This "crossing" point seems to be in 5 (in the very difficult 9x9's on tuesdays, which curiously are symmetric with respect to the horizontal and to the vertical axis). Sometimes we see 7-cell or 6-cell (big cages) in the 10x10's or the 12x12's but by increasing the total number of cages the difficulty of the puzzle decreases, so a big number of cages looks to compensate for the presence of a few "big cages" (I do not know if the "solver rating" would be related in some way with this "proportionality" but intuitively I would say that yes).


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Posted on: Thu Nov 17, 2011 7:48 pm




Posts: 690
Joined: Fri May 13, 2011 6:51 pm
Post Re: The solution of a 6x6 and a 9x9 puzzles with "strange ca
pnm wrote:
That last one ("version e") had 9 solutions, computed in 0.03 seconds.

... Another cheap way of reducing the number of solutions is by adding single cell cages
(e.g. split a double cell, or remove one from a larger cell).


Here goes a new split (but for two cells).
Version f:

6
11,+,a1b1c1
2,+,b2
16,+,a2a3b3
4,+,c2c3
150,x,d1d2d3e3f3
4,+,e2
2,-,e1f1f2
450,x,a4a5a6b6c6
1,-,b4c4c5
3,+,b5
19,+,d4e4f4f5f6
3,-,d5d6e6
5,+,e5

Image


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Posted on: Thu Nov 17, 2011 8:57 pm




Posts: 2193
Joined: Thu May 12, 2011 11:58 pm
Post Re: The solution of a 6x6 and a 9x9 puzzles with "strange ca
version f: 4 solutions, computed in 0.0029 seconds.

now we have only four, here they are (will help in determining
how to make it have a unique solution):

2 4 5 3 6 1
6 2 1 5 4 3
4 6 3 1 2 5
5 1 2 4 3 6
1 3 4 6 5 2
3 5 6 2 1 4
---
solution 2:
2 4 5 3 6 1
6 2 1 5 4 3
4 6 3 2 1 5
5 1 2 4 3 6
1 3 4 6 5 2
3 5 6 1 2 4
---
solution 3:
2 4 5 1 3 6
6 2 3 5 4 1
4 6 1 3 2 5
5 1 2 4 6 3
1 3 4 6 5 2
3 5 6 2 1 4
---
solution 4:
2 4 5 1 6 3
6 2 3 5 4 1
4 6 1 3 2 5
5 1 2 4 3 6
1 3 4 6 5 2
3 5 6 2 1 4


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Posted on: Fri Nov 18, 2011 11:23 pm




Posts: 690
Joined: Fri May 13, 2011 6:51 pm
Post Re: The solution of a 6x6 and a 9x9 puzzles with "strange ca
pnm wrote:
version f: 4 solutions, computed in 0.0029 seconds.

now we have only four, here they are (will help in determining
how to make it have a unique solution):
...
solution 4:
2 4 5 1 6 3
6 2 3 5 4 1
4 6 1 3 2 5
5 1 2 4 3 6
1 3 4 6 5 2
3 5 6 2 1 4


Hi, Patrick, the solution 4 is the one we have being working with, let's try a new split (14 cages in total) version g, and see this time, thanks:

6
11,+,a1b1c1
2,+,b2
16,+,a2a3b3
4,+,c2c3
180,x,d1d2d3e1e3
4,+,e2
3,+,f1
5,x,f2f3
450,x,a4a5a6b6c6
1,-,b4c4c5
3,+,b5
19,+,d4e4f4f5f6
3,-,d5d6e6
5,+,e5

Image


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Posted on: Sat Nov 19, 2011 12:31 am




Posts: 2193
Joined: Thu May 12, 2011 11:58 pm
Post Re: The solution of a 6x6 and a 9x9 puzzles with "strange ca
2 solutions this time (0.0007 sec):

2 4 5 1 6 3
6 2 1 3 4 5
4 6 3 5 2 1
5 1 2 4 3 6
1 3 4 6 5 2
3 5 6 2 1 4
---
2 4 5 1 6 3
6 2 3 5 4 1
4 6 1 3 2 5
5 1 2 4 3 6
1 3 4 6 5 2
3 5 6 2 1 4


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Posted on: Sat Nov 19, 2011 2:46 pm




Posts: 690
Joined: Fri May 13, 2011 6:51 pm
Post Re: The solution of a 6x6 and a 9x9 puzzles with "strange ca
pnm wrote:
2 solutions this time (0.0007 sec):

2 4 5 1 6 3
6 2 1 3 4 5
4 6 3 5 2 1
5 1 2 4 3 6
1 3 4 6 5 2
3 5 6 2 1 4
---
2 4 5 1 6 3
6 2 3 5 4 1
4 6 1 3 2 5
5 1 2 4 3 6
1 3 4 6 5 2
3 5 6 2 1 4


Hi, Patrick, I have moved the "cage limitting line" from the left of cell d3 to the top of this cell d3, then keeping fourteen cages for this version h:

6
11,+,a1b1c1
2,+,b2
16,+,a2a3b3
9,+,c2c3d3e3
30,x,d1d2e1
4,+,e2
3,+,f1
5,x,f2f3
450,x,a4a5a6b6c6
1,-,b4c4c5
3,+,b5
19,+,d4e4f4f5f6
3,-,d5d6e6
5,+,e5

Image


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Posted on: Sat Nov 19, 2011 3:03 pm




Posts: 2193
Joined: Thu May 12, 2011 11:58 pm
Post Re: The solution of a 6x6 and a 9x9 puzzles with "strange ca
Still two solutions (version g)

2 4 5 1 6 3
6 2 1 3 4 5
4 6 3 5 2 1
5 1 2 4 3 6
1 3 4 6 5 2
3 5 6 2 1 4
---
2 4 5 1 6 3
6 2 3 5 4 1
4 6 1 3 2 5
5 1 2 4 3 6
1 3 4 6 5 2
3 5 6 2 1 4


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Posted on: Sat Nov 19, 2011 3:33 pm




Posts: 690
Joined: Fri May 13, 2011 6:51 pm
Post Re: The solution of a 6x6 and a 9x9 puzzles with "strange ca
pnm wrote:
Still two solutions (version g)

2 4 5 1 6 3
6 2 1 3 4 5
4 6 3 5 2 1
5 1 2 4 3 6
1 3 4 6 5 2
3 5 6 2 1 4
---
2 4 5 1 6 3
6 2 3 5 4 1
4 6 1 3 2 5
5 1 2 4 3 6
1 3 4 6 5 2
3 5 6 2 1 4


No, something is wrong, the version h is the one to be tested, not the g (already tested), see that the first of those two solutions is not valid since d1 x e1 x d2 = 1 x 6 x 3 = 18 and not 30, which is the code I sent (graphic), thanks


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Posted on: Sat Nov 19, 2011 3:48 pm




Posts: 2193
Joined: Thu May 12, 2011 11:58 pm
Post Re: The solution of a 6x6 and a 9x9 puzzles with "strange ca
Ah, yes, sorry: 1 solution (0.00036 seconds):

2 4 5 1 6 3
6 2 3 5 4 1
4 6 1 3 2 5
5 1 2 4 3 6
1 3 4 6 5 2
3 5 6 2 1 4

Difficulty factor for this puzzle would be 21 (so it would rate somewhere
between the medium and difficult puzzles of the site)

Patrick


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Posted on: Sat Nov 19, 2011 4:12 pm




Posts: 690
Joined: Fri May 13, 2011 6:51 pm
Post Re: The solution of a 6x6 and a 9x9 puzzles with "strange ca
pnm wrote:
Ah, yes, sorry: 1 solution (0.00036 seconds):

2 4 5 1 6 3
6 2 3 5 4 1
4 6 1 3 2 5
5 1 2 4 3 6
1 3 4 6 5 2
3 5 6 2 1 4

Difficulty factor for this puzzle would be 21 (so it would rate somewhere
between the medium and difficult puzzles of the site)

Patrick


Yes, it has finally become really easy, but it has been very instructive the full process to insure the unique solution starting from 42672 solutions. To do something similar with the 9x9 is an utopia due to the huge number of initial solutions in case of finding all (probably many hours!... or days!). The conclusion is that it's better not using so big cages or "strange cages" and to find always that compromise we were talking about between the size of the puzzle and the maximum size of the cages.
Also, your intuition with respect to the introduction of single-cell cages has resulted good. In fact, it can be observed that those single cells many times are the key for the solution of the big puzzles (10x10's and 12x12's) and probably are, as you say, not only the cheap but the only way to guarantee a unique solution.
Thank you very much (I will take soon the liberty of "bombarding" your solver with new challenges...).


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