[math-fun] poly-anygons; count of chess positions

[Richard Guy]

This is evidently  A001004  in  OEIS.   R.

On Tue, 18 Oct 2005, Schroeppel, Richard wrote:

> I decided to count polyanygons.
> These are kind of like polyominos, except you can use
> any regular polygons of side = 1, as long as there's no
> overlapping.
>
> A regular polygon of N sides has a score of N-2:
> triangles score 1, squares 2, etc.
> The score for a polyanygon is the total of its components.
>
> My counts (by hand):  1 2 3 9 20 75.
> These are with all symmetries (rotations, reflections) removed.
> (Confirmations wanted!)
>
> The 3 polyanygons of score 3 are
> the triamond  (three regular triangles),
> a square stuck to a triangle  (a house?),
> and the pentagon.
>
> A computer program to do this will be challenging.
> I think it will require working exactly with fairly high degree roots of
> 1.
> Like LCM(1...8) = 840.
>
> I made three decisions that affect the counting.
> (a) scoring: I might have used total area, but didn't.
> This would have split things into many more groups.
> (b) I might have used perimeter as the score, but didn't.
> My way of scoring is easier.  This doesn't affect the
> counts much (so far).  Note that higher scoring polyanygons
> can have partially offset edges, so the perimeter might
> not always be an integer.
> (c ) I regard the regular hexagon as different from the
> same shape made out of six triangles.  It's also possible
> to make a dodecagon by adding a fringe or alternating
> triangles and squares to a regular hexagon.  I think these
> are the only ambiguous cases.  Again, the effect on the
> count is minor for small scores.
>
> I'm guessing that the limiting ratio Polyanygons(N+1)/
> Polyanygons(N) is unbounded, but I have no good argument.
>
> -----
> count of chess positions.
> The note from Guy Haworth points to Tim Krabbe's chess blog.
> http://www.xs4all.nl/~timkr/chess2/diary.htm
> Item 283 mentions
> "a serious approximation (see link
> <http://www.chessbox.de/Compu/schachzahl1b_e.html> ) of the number of
> possible
> chess positions is 2.28*10^46. "
>
> This is a little higher than our group's informal result from a few
> years ago; I think we got around 10^43.  Unfortunately, the link
> seems broken.
>
> Rich
>
>
>
>
>
>
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