Explaining, convincing and proving

Here are three 'tricks' to amaze your friends. But the really clever trick is explaining to them why these 'tricks' are maths not magic. Like all good magicians, you should practice by trying them. Can you explain how they work?

Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?

Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?

There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

Can you work out the number of chairs at a cafe from the number of legs?

Is it possible to arrange the numbers 1-6 on the nodes of this diagram, so that all the sums between numbers on adjacent nodes are different?

In this addition each letter stands for a different digit, with S standing for 3. What is the value of YxO?

Can you place the nine cards onto a 3×3 grid such that every row, column and diagonal has a product of 1?

Which digit replaces x in this crossnumber?