[math-fun] Counting problem
What are all the ways that 30-, 60-, and 90-degree angles can be arranged about the origin in the plane? Let two ways be equivalent if one can be carried into the other by either a rotation or a flip of the plane -- i.e., by an element of the group O(2). This appears to be a messy counting job but certainly doable by hand because of its small scale. As a first step, I found there are 19 partitions of 12 using only 1's, 2's, and 3's (appended). Can anyone suggest a smart way to go about this, or is there no good way to shorten the task? --Dan ----------------------- 3+3+3+3 2+2+2+3+3 2+2+2+2+2+2 1+2+3+3+3 1+2+2+2+2+3 1+1+2+2+3+3 1+1+2+2+2+2+2 1+1+1+3+3+3 1+1+1+2+2+2+3 1+1+1+1+2+3+3 1+1+1+1+2+2+2+2 1+1+1+1+1+2+2+3 1+1+1+1+1+1+3+3 1+1+1+1+1+1+2+2+2 1+1+1+1+1+1+1+2+3 1+1+1+1+1+1+1+1+2+2 1+1+1+1+1+1+1+1+1+3 1+1+1+1+1+1+1+1+1+1+2 1+1+1+1+1+1+1+1+1+1+1+1 ----------------------- ________________________________________________________________________________________ It goes without saying that .
This is a job for the Frobenius/Burnside/Polya/Messerschmitt/Merill/Lynch/Pierce/Fenner/Smith counting formula. On Tue, Dec 6, 2011 at 11:58 AM, Dan Asimov <dasimov@earthlink.net> wrote:
What are all the ways that 30-, 60-, and 90-degree angles can be arranged about the origin in the plane?
Let two ways be equivalent if one can be carried into the other by either a rotation or a flip of the plane -- i.e., by an element of the group O(2).
This appears to be a messy counting job but certainly doable by hand because of its small scale.
As a first step, I found there are 19 partitions of 12 using only 1's, 2's, and 3's (appended).
Can anyone suggest a smart way to go about this, or is there no good way to shorten the task?
--Dan
----------------------- 3+3+3+3
2+2+2+3+3
2+2+2+2+2+2
1+2+3+3+3
1+2+2+2+2+3
1+1+2+2+3+3
1+1+2+2+2+2+2
1+1+1+3+3+3
1+1+1+2+2+2+3
1+1+1+1+2+3+3
1+1+1+1+2+2+2+2
1+1+1+1+1+2+2+3
1+1+1+1+1+1+3+3
1+1+1+1+1+1+2+2+2
1+1+1+1+1+1+1+2+3
1+1+1+1+1+1+1+1+2+2
1+1+1+1+1+1+1+1+1+3
1+1+1+1+1+1+1+1+1+1+2
1+1+1+1+1+1+1+1+1+1+1+1 -----------------------
________________________________________________________________________________________ It goes without saying that .
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
This seems to be a particular case of the necklace problem which was solved by Hazel Perfect in Math Gaz, when? (more than half a century ago -- not in MR) R. On Tue, 6 Dec 2011, Dan Asimov wrote:
What are all the ways that 30-, 60-, and 90-degree angles can be arranged about the origin in the plane?
Let two ways be equivalent if one can be carried into the other by either a rotation or a flip of the plane -- i.e., by an element of the group O(2).
This appears to be a messy counting job but certainly doable by hand because of its small scale.
As a first step, I found there are 19 partitions of 12 using only 1's, 2's, and 3's (appended).
Can anyone suggest a smart way to go about this, or is there no good way to shorten the task?
--Dan
----------------------- 3+3+3+3
2+2+2+3+3
2+2+2+2+2+2
1+2+3+3+3
1+2+2+2+2+3
1+1+2+2+3+3
1+1+2+2+2+2+2
1+1+1+3+3+3
1+1+1+2+2+2+3
1+1+1+1+2+3+3
1+1+1+1+2+2+2+2
1+1+1+1+1+2+2+3
1+1+1+1+1+1+3+3
1+1+1+1+1+1+2+2+2
1+1+1+1+1+1+1+2+3
1+1+1+1+1+1+1+1+2+2
1+1+1+1+1+1+1+1+1+3
1+1+1+1+1+1+1+1+1+1+2
1+1+1+1+1+1+1+1+1+1+1+1 -----------------------
________________________________________________________________________________________ It goes without saying that .
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
As luck would have it, this afternoon I had a boring staff meeting in which I enumerated all of them ... I think. And the answer is 85 ... I think. On Tue, Dec 6, 2011 at 12:36 PM, Richard Guy <rkg@cpsc.ucalgary.ca> wrote:
This seems to be a particular case of the necklace problem which was solved by Hazel Perfect in Math Gaz, when? (more than half a century ago -- not in MR) R.
On Tue, 6 Dec 2011, Dan Asimov wrote:
What are all the ways that 30-, 60-, and 90-degree angles can be arranged
about the origin in the plane?
Let two ways be equivalent if one can be carried into the other by either a rotation or a flip of the plane -- i.e., by an element of the group O(2).
This appears to be a messy counting job but certainly doable by hand because of its small scale.
As a first step, I found there are 19 partitions of 12 using only 1's, 2's, and 3's (appended).
Can anyone suggest a smart way to go about this, or is there no good way to shorten the task?
--Dan
----------------------- 3+3+3+3
2+2+2+3+3
2+2+2+2+2+2
1+2+3+3+3
1+2+2+2+2+3
1+1+2+2+3+3
1+1+2+2+2+2+2
1+1+1+3+3+3
1+1+1+2+2+2+3
1+1+1+1+2+3+3
1+1+1+1+2+2+2+2
1+1+1+1+1+2+2+3
1+1+1+1+1+1+3+3
1+1+1+1+1+1+2+2+2
1+1+1+1+1+1+1+2+3
1+1+1+1+1+1+1+1+2+2
1+1+1+1+1+1+1+1+1+3
1+1+1+1+1+1+1+1+1+1+2
1+1+1+1+1+1+1+1+1+1+1+1 -----------------------
______________________________**______________________________** ____________________________ It goes without saying that .
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participants (3)
-
Allan Wechsler -
Dan Asimov -
Richard Guy