in reply to Re^2: Pattern enumeration. (KISS)
in thread Pattern enumeration.

You could get an approximate answer by generating a large number of boards at random

Generating random boards and then just checking for pass/fail very quickly converges to a figure of 8.9%; giving a approximation of 5.657e+48 legal boards.

I don't understand your "calculating the average number of constraints violated by a random illegal board" bit.

I do realise that for any given legal board, there are 720 "symmetries", where the arrangement of tokens is the same, but the actual tokens are different. Um. Not a good description.

Ie. The following are symmetries because the pattern of the tokens remains the same, though the values of the tokens are different: 

1 2 3    2 3 1    3 2 1
2 3 1    3 1 2    2 1 3            
3 2 1    1 3 2    1 2 3

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For my purpose, reflection and rotational symmetries are different.

So, I think that once I've calculated the total number of legal arrangements, I divide by 720 to determine the number of patterns?

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Re^4: Pattern enumeration. (KISS)
by ikegami (Patriarch) on Jul 27, 2010 at 18:39 UTC

    So, I think that once I've calculated the total number of legal arrangements, I divide by 720 to determine the number of patterns?

    A 3x3 with 4 symbols has 95,340178,068 solutions, a number that's not even divisible by 4!. This is because some solutions don't use all the available symbols.

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      Good point!

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