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MAA: Math Horizons September 2004. Solution.

Biography of a Contest Problem 
Steven R. Dunbar & David Hankin
 September 2004 page 12

The Problem: Define a regular n-pointed star to be the union of n line segments P1P2, P2P3, . . . , PnP1, such that
  • the points P1, P2, . . . , Pn are coplanar and no three of them are collinear.
  • each of the n line segments intersects at least one of the other line segments at a point other than an endpoint,
  • all of the angles at P1, P2, . . . , Pn are congruent,
  • all of the n line segments P1P2,  P2P3. . . , PnP1are congruent, and
  • the path P1 P2 . . .  Pn P1 turns counterclockwise at an angle of less than 180o at each vertex.

There are no regular 3-pointed, 4-pointed, or 6-pointed stars. All regular 5-pointed stars are similar, but there are two non-similar regular 7-pointed stars. How many non-similar regular 1000-pointed stars are there?

Solution: The vertices of an n-pointed star are the vertices of a regular n-gon, numbered 0 through n-1 in clockwise order. The star is determined by choosing a vertex m and drawing the line segments from 0 to m, from m to 2m, from 2m to 3m, and (n-1)m to 0, where all numbers are reduced modulo m. In order for the figure to satisfy our conditions, m must be relatively prime to n and not equal to 1 or m-1. For example, the two 7-pointed stars are presented below.

There are 400 positive numbers below 1000 that are relatively prime to 1000. Since the same star results from choosing the first edge to go from 0 to k as when it goes from 0 to n-k, there are (400-2)/2 = 199 1000-pointed stars.

Last updated 03 April 2006