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Meanings of puzzle ratings
http://www.calcudoku.org/forum/viewtopic.php?f=2&t=854
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Author:  paulv66  [ Tue Jan 03, 2017 10:10 pm ]
Post subject:  Meanings of puzzle ratings

Most of the puzzles have a value assigned to them. For example, today's 9x9 has a rating value of 111.3 and has been allocated 6 stars. What does the rating value represent and what's the highest rating ever for a puzzle on this site?

PaulV

Author:  jpoos  [ Tue Jan 03, 2017 11:00 pm ]
Post subject:  Re: Meanings of puzzle ratings

Patrick wrote a paper (http://www.calcudoku.org/papers/choice_ ... _v1.01.pdf) in which he described what the rating means:
'The difficulty of a puzzle is estimated by averaging the natural logarithm of the number of possible permutations for each row and each column (after applying the restrictions imposed by each cage)'. In the paper he also gives a few examples.
I'm still a bit puzzled as to what this actually means: since the restrictions imposed by all the cages in the puzzle give you a unique solution, doesn't every row/column have only one possible permutation? (which would mean that every puzzle gets a rating of 0 [confused] )

Author:  paulv66  [ Wed Jan 04, 2017 12:34 am ]
Post subject:  Re: Meanings of puzzle ratings

Thanks jpoos. Interesting paper, but I couldn't find the reference to the rating system and how it is calculated.

Author:  paulv66  [ Wed Jan 04, 2017 12:41 am ]
Post subject:  Re: Meanings of puzzle ratings

paulv66 wrote:
Thanks jpoos. Interesting paper, but I couldn't find the reference to the rating system and how it is calculated.


I just did what I should have done initially and did a search on the document. I kind of agree with you as I'm not sure I understand the logic due to the fact that there is an unique solution to each puzzle, which seems to imply that each puzzle has only one possible solution and therefore all puzzles are equally difficult - which clearly makes no sense.

Author:  pnm  [ Wed Jan 04, 2017 12:52 am ]
Post subject:  Re: Meanings of puzzle ratings

This question comes up every now and then.

With some effort I tracked down one thread about it:

viewtopic.php?f=2&t=195

but I'm sure there must be more.

I'll do some more digging..


Edit:
> 'The difficulty of a puzzle is estimated by averaging the natural logarithm of the number of possible permutations
> for each row and each column (after applying the restrictions imposed by each cage)'.

The key here is that the number of options for a single cell is determined by _only_ looking at the cage.

Author:  jpoos  [ Wed Jan 04, 2017 2:10 pm ]
Post subject:  Re: Meanings of puzzle ratings

pnm wrote:
The key here is that the number of options for a single cell is determined by _only_ looking at the cage.


That still leaves it a bit unclear to me. Could you give an example?

I also just realised the following: if we take a n by n puzzle, the number of permutations for a row/column can at max only be n!, so the rating can at max be ln(n!), right? Well, many puzzles exceed this limit [confused] [confused] What am I doing wrong here?

Author:  jaek  [ Wed Jan 04, 2017 2:49 pm ]
Post subject:  Re: Meanings of puzzle ratings

jpoos wrote:
pnm wrote:
The key here is that the number of options for a single cell is determined by _only_ looking at the cage.


That still leaves it a bit unclear to me. Could you give an example?

I also just realised the following: if we take a n by n puzzle, the number of permutations for a row/column can at max only be n!, so the rating can at max be ln(n!), right? Well, many puzzles exceed this limit [confused] [confused] What am I doing wrong here?


If the cell's values are only determined by the cage it belongs to, then that 'history' won't carry over to the next cell/cage, right? So, is the max
ln(n) * n * n?

Author:  jpoos  [ Wed Jan 04, 2017 3:44 pm ]
Post subject:  Re: Meanings of puzzle ratings

jaek wrote:
jpoos wrote:
pnm wrote:
The key here is that the number of options for a single cell is determined by _only_ looking at the cage.


That still leaves it a bit unclear to me. Could you give an example?

I also just realised the following: if we take a n by n puzzle, the number of permutations for a row/column can at max only be n!, so the rating can at max be ln(n!), right? Well, many puzzles exceed this limit [confused] [confused] What am I doing wrong here?


If the cell's values are only determined by the cage it belongs to, then that 'history' won't carry over to the next cell/cage, right? So, is the max
ln(n) * n * n?


In my logic that would give a max of ln(n^n), which still leaves puzzles that exceed this rating...
I'm pretty sure I'm interpreting the averaging-part wrong...

Author:  paulv66  [ Wed Jan 04, 2017 3:51 pm ]
Post subject:  Re: Meanings of puzzle ratings

jpoos wrote:
pnm wrote:
The key here is that the number of options for a single cell is determined by _only_ looking at the cage.


That still leaves it a bit unclear to me. Could you give an example?

I also just realised the following: if we take a n by n puzzle, the number of permutations for a row/column can at max only be n!, so the rating can at max be ln(n!), right? Well, many puzzles exceed this limit [confused] [confused] What am I doing wrong here?


I think the maximum number of permutations would be n!^n. If this is correct, it would give the following maximum rating for each puzzle size from 4x4 to 12x12.

Size n! n!^n Rating
4 24 331776 12.71
5 120 24883200000 23.94
6 720 1.39314E+17 39.48
7 5040 8.26064E+25 59.68
8 40320 6.98496E+36 84.84
9 362880 1.09111E+50 115.22
10 3628800 3.95941E+65 151.04
11 39916800 4.09933E+83 192.53
12 479001600 1.459E+104 239.85

There must be something wrong with the above logic, as I know I've seen at least one 9x9 with a rating of 130+.

Paul

Author:  jpoos  [ Wed Jan 04, 2017 4:35 pm ]
Post subject:  Re: Meanings of puzzle ratings

paulv66 wrote:
Size n! n!^n Rating
...
7 5040 8.26064E+25 59.68
...

There must be something wrong with the above logic, as I know I've seen at least one 9x9 with a rating of 130+.


If you want to be sure, todays 7x7 difficult also has a rating of 64.7.

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