Weird coloured necklaces

Have a look at the necklace in the picture on the left. It consists of 13 beads of two colours, 9 black and 4 yellow. Do you note anything special about the actual sequence of colours used ?

Choose any two of the yellow beads in this necklace and count the number of beads (of any colour) between them. Now choose two different beads and count again. You end up with a different number. In fact, for every different pair of yellow beads you care to choose, you find a different number of beads that are strung between them, and this is even true when you start with the same pair as before, but now count the other way round.

What is happening ?

You have just met an object with fascinating mathematical characteristics, called a `perfect difference set'. This page, and the pages that follow, will try to introduce you to the mathematical properties of these sets and teach you how to construct them.

This text is intended for people with a minimal mathematical background (say high school mathematics, but then again, high schools are not the same around the world), but with a healthy mathematical appetite.

Do these necklaces have any use ?

Of course not ! This is mathematics !

However, quite a lot of people seem to be interested in this subject. Difference sets are somewhat related to so-called Golomb rulers to which many mathematicians (both amateur and professional) have devoted their attention.

If you already know about Golomb rulers and do not immediately recognize the connection with `weird coloured necklaces', then take a pair of scissors, cut the necklace and straighten it out.

Problems

Before you turn to the next page, try to solve the following problems :
• Can you construct a necklace with 4 yellow beads, but with fewer black beads than in the example above ? Take care: you need to count every pair of yellow beads twice, once for each direction, and ALL possible counts must be different!
• Can you construct a necklace with 5 yellow beads ? How many black beads did you need ?
• Are you sure you cannot solve the 5-bead problem with fewer black beads ? (Prove it !)
Answers to these problems may be found on the next pages.
Necklaces and numbers.

96/12/30 - Kris Coolsaet