A118887
Number of ways to place n objects with weights 1,2,...,n evenly spaced around the circumference of a circular disk so that the disk will exactly balance on the center point.
Original entry on oeis.org
0, 0, 0, 0, 0, 2, 0, 0, 0, 24, 0, 732, 0, 720, 48, 0, 0
Offset: 1
The smallest n for which a solution exists is n=6 with 4 solutions:
...........Weight
......1..2..3..4..5..6
.Count...at.position
..1...1..4..5..2..3..6
..2...1..5..3..4..2..6
..3...1..6..2..4..3..5
..4...1..6..3..2..5..4
Configurations 1 is the mirror image of configuration 4, ditto for configurations 2 and 3. Therefore a(6)=2.
- Andrew Bernoff, Bernoff's Puzzler, MuddMath Newsletter Volume 4, No. 1, Page 10, Spring 2005
- Marius Munteanu and Laura Munteanu, Rational equiangular polygons, Applied Math., 4 (2013), 1460-1465.
- Hugo Pfoertner, Balanced weights on circle (Tables of configurations)
- G. J. Woeginger, Nothing new about equiangular polygons, Amer. Math. Monthly, 120 (2013), 849-850.
Cf.
A118888 (configurations with minimum imbalance),
A063697 (positions of positive coefficients in cyclotomic polynomial in binary),
A063699 (positions of negative coefficients in cyclotomic polynomial in binary),
A326921.
-
Needs["DiscreteMath`Combinatorica`"]; Table[eLst=E^(2.*Pi*I*Range[n]/n); Count[(Permutations[Range[n]]), q_List/;Chop[q.eLst]===0]/(2n), {n,10}] (* very slow for n>10 *) (* T. D. Noe, May 05 2006 *)
A118888
Number of ways to place n objects with weights 1,2,...,n evenly spaced around the circumference of a circular disk such that the remaining imbalance is minimized.
Original entry on oeis.org
1, 1, 1, 1, 1, 2, 1, 2, 3, 24, 1, 732, 1, 720, 48, 144, 2
Offset: 1
a(5)=1: The configuration minimizing the remaining imbalance with respect to the center of the circle is [1 4 3 2 5] (and its mirror image).
Examples of minimum imbalance configurations not in A118887:
a(7)=1: [1 4 7 2 3 5 6];
a(8)=2: [1 4 7 3 6 2 5 8], [1 7 4 3 6 5 2 8];
a(9)=3: [1 5 9 2 7 3 4 8 6], [1 5 9 4 2 6 7 3 8], [1 6 5 4 9 2 3 7 8];
a(11)=1: [1 8 9 5 2 6 10 7 3 4 11];
a(13)=1: [1 2 7 12 13 4 5 3 8 6 11 9 10];
a(16)=144: lexicographically earliest [1 3 5 13 16 7 10 2 14 4 6 9 12 8 11 15];
a(17)=2: [1 7 3 17 10 9 15 2 14 6 5 4 16 8 13 12 11],
[1 8 9 3 16 4 12 13 14 2 10 5 6 7 17 11 15] and their mirror configurations (e.g. [1 11 12 13 8 ...]) both produce a center of gravity with distance 2.1884*10^(-7) from the center of a circle with radius 1. All other configurations produce greater distances, e.g. [1 3 11 16 9 5 7 12 14 4 10 8 2 15 13 6 17] -> 2.5126*10^(-7). - _Hugo Pfoertner_, Oct 24 2019
Showing 1-2 of 2 results.
Comments