Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

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

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

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?

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.

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.

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . .

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

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

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?

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. . . .

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. . . .

You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance?

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. . . .

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.

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

This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with. . . .

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

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

You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?

L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?

I am exactly n times my daughter's age. In m years I shall be ... How old am I?

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

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

Four jewellers share their stock. Can you work out the relative values of their gems?

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

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. . . .

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

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

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?

When is it impossible to make number sandwiches?

What is the largest number of intersection points that a triangle and a quadrilateral can have?

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

Explore what happens when you draw graphs of quadratic equations with coefficients based on a geometric sequence.

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

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

What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.

The sums of the squares of three related numbers is also a perfect square - can you explain why?

An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.

Prove Pythagoras' Theorem using enlargements and scale factors.

Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation.

A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .

Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places?