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Path: utzoo!linus!decvax!harpo!eagle!mhuxt!mhuxi!mhuxa!ulysses!kpv
From: kpv@ulysses.UUCP
Newsgroups: net.math
Subject: Perfect rings
Message-ID: <588@ulysses.UUCP>
Date: Tue, 30-Aug-83 09:54:29 EDT
Article-I.D.: ulysses.588
Posted: Tue Aug 30 09:54:29 1983
Date-Received: Wed, 31-Aug-83 21:32:57 EDT
Organization: Bell Labs, Murray Hill
Lines: 30

Here's an interesting problem that I discovered about a year ago
after buying a toy for my son:

	Let C = {1,2,3,...n} be a set of colors (read color 1, color 2 ...)
	Suppose that you have Bi beads of color i  for 1 <= i <= n.

	The problem is to arrange the beads on a ring so that beads with
	same color are equidistant on the ring.
	I call such a ring a PERFECT RING.

Example:
	Say we have beads: a,a,a, b,b,b, c,c, d,d, e,e
	They can be arranged perfectly as in the following
	circular permutation:

		a d b e a c b d a e b c

Questions:
   1.	For what combinations of the number Bi's do perfect rings exist?

   2.	For such a combination, find an algorithm that will produce a ring.

   3.	If one ring exists, how many are there in all, ie, is there a formula
	based on the Bi's for the number of perfect rings?

I have a few partial results and conjectures based on least common multiples.
But the full solution seems hard.

	Kiem-Phong Vo, Bell Labs, Murray Hill, New Jersey
	ulysses!kpv