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Posts:
202
Registered:
12/6/04
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Re: Tiling the plane with checkerboard patterns
Posted:
Jul 11, 2010 10:20 PM
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Rouben,
Thanks for the input and for the explanation of Avni's work.
I also thought my formulas were just estimates because of those
fractions.
However there is some logical justification in their creation.
Perhaps an explanation will help and also may help justify Avni's
solution if it is correct.
My starting point is the idea that if a formula is possible,
there must be a pattern in the results tied to N. Therefore
there should also be a pattern in the error generated by
(2^(n^2))/n^2. So I added an unknown X into the equation and
solved for it using the data you supplied. The formula I used was:
(2^(n^2) + X)/n^2 = a(n)
The result was:
N X
1 0
2 12
3 64
4 960
5 768
I had already thought that even and odd boards needed different
treatment since odd boards have a central square and even do not.
The fact that n=4 yields a greater error than N=5 reinforced that
belief.
Then I noticed what I think is a significant pattern in the X values.
In every case, when you break it down into smaller multiples,
you end up with a power of 2 multiplied by N-1 and N+1.
Example: 960 = (2^6)x3x5
This seems too coincidental to not be significant.
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