Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .

In how many ways can you arrange three dice side by side on a surface so that the sum of the numbers on each of the four faces (top, bottom, front and back) is equal?

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

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?

Blue Flibbins are so jealous of their red partners that they will not leave them on their own with any other bue Flibbin. What is the quickest way of getting the five pairs of Flibbins safely to. . . .

Prove Pythagoras' Theorem using enlargements and scale factors.

Which set of numbers that add to 10 have the largest product?

A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . .

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

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. . . .

A huge wheel is rolling past your window. What do you see?

Can you make sense of the three methods to work out the area of the kite in the square?

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

ABC is an equilateral triangle and P is a point in the interior of the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP must be less than 10 cm.

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

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?

Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?

Write down a three-digit number Change the order of the digits to get a different number Find the difference between the two three digit numbers Follow the rest of the instructions then try. . . .

If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest. . . .

A blue coin rolls round two yellow coins which touch. The coins are the same size. How many revolutions does the blue coin make when it rolls all the way round the yellow coins? Investigate for a. . . .

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .

Carry out cyclic permutations of nine digit numbers containing the digits from 1 to 9 (until you get back to the first number). Prove that whatever number you choose, they will add to the same total.

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

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

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

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?

When is it impossible to make number sandwiches?

Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

Can you make sense of these three proofs of Pythagoras' Theorem?

What fractions can you divide the diagonal of a square into by simple folding?

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.