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
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
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
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.
Before you turn to the next page, try to solve the following
Answers to these problems may be found on the next
- 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 !)
Necklaces and numbers.