Taking 1 blue and 2 red beads there are 3 arrangements of the beads in a row. However, if you string the beads to make necklaces, all the arrangements give the same pattern when you allow beads to move over the knot. What if you make more complicated necklaces, for example using 1 blue, 1 white and 2 red beads? We shall explore a way of tackling such questions by using necklace symmetries. The method can also be applied to problems of colouring solids. For example, suppose you are supplied with a stock of identical cubes and 2 pots of paint. How many different objects can you make by painting the faces of the cubes? More generally we classify various structures by their symmetries and use some ideas from the theory of groups. Our pace will be gentle. Some memories of O level maths useful. Introduction to self I have taught maths from adult numeracy to university level. My interest in applications of symmetry has led me to introduce the topic at primary level upwards.