[math-fun] poly-anygons; count of chess positions
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
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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o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o q. how many sides in a pollyannagon? a. the best of all possible numbers! ja o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o inquiry e-lab: http://stderr.org/pipermail/inquiry/ o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
i think the two sequences diverge eventually, once there is overlap. erich
This is evidently A001004 in OEIS. R.
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. Rich
I would be quite surprised if this is in fact A001004, although it might possibly match for a few more terms. Franklin T. Adams-Watters 16 W. Michigan Ave. Palatine, IL 60067 847-776-7645 -----Original Message----- From: Richard Guy <rkg@cpsc.ucalgary.ca> To: math-fun <math-fun@mailman.xmission.com> Cc: Richard Schroeppel <rcs@cs.arizona.edu>; seqfan@ext.jussieu.fr Sent: Tue, 18 Oct 2005 16:08:02 -0600 (MDT) Subject: Re: [math-fun] poly-anygons; count of chess positions 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. ...
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participants (5)
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Erich Friedman -
franktaw＠netscape.net -
Jon Awbrey -
Richard Guy -
Schroeppel, Richard