
Re: Tiling the plane with checkerboard patterns
Posted:
Jul 12, 2010 5:44 PM


On Mon, Jul 12, 2010 at 12:27:24PM +0000, mark wrote: > > 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 N1 and N+1. > Example: 960 = (2^6)x3x5 > This seems too coincidental to not be significant.
I see your reasoning and there may be some significance to the patterns that you have observed. If we knew the true value of a(6), it would have helped to settle the issue. If we extend your table of N and X to N=6 according to Avni's formula, we get:
N X 6 6020
Since 6020 = 2^2 . 5 . 7. 43, the X value deviates from your pattern of multiples of (n1)*(n+1). Having the true value of a(6) would tell us if your formula, or Avni's, or neither, is correct. Unfortunately, the direct computation of a(6) is impossible with my computer algorithm which searches all possible patterns, because the number of patterns is too large.
Rouben

