A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

Can you make a right-angled triangle on this peg-board by joining up three points round the edge?

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

Can you recreate squares and rhombuses if you are only given a side or a diagonal?

A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?

I'm thinking of a rectangle with an area of 24. What could its perimeter be?

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

If you move the tiles around, can you make squares with different coloured edges?

Can you find rectangles where the value of the area is the same as the value of the perimeter?

Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

Can you find the values at the vertices when you know the values on the edges?

In this follow-up to the problem Odds and Evens, we invite you to analyse a probability situation in order to find the general solution for a fair game.

Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?

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

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

Can you work out which spinners were used to generate the frequency charts?

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

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

Alison and Charlie are playing a game. Charlie wants to go first so Alison lets him. Was that such a good idea?

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

Six balls are shaken. You win if at least one red ball ends in a corner. What is the probability of winning?

Seven balls are shaken. You win if the two blue balls end up touching. What is the probability of winning?

Play around with sets of five numbers and see what you can discover about different types of average...

An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?

Can you find any two-digit numbers that satisfy all of these statements?

You'll need to know your number properties to win a game of Statement Snap...

Imagine a room full of people who keep flipping coins until they get a tail. Will anyone get six heads in a row?

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

Who said that adding, subtracting, multiplying and dividing couldn't be fun?

There are nasty versions of this dice game but we'll start with the nice ones...

How many winning lines can you make in a three-dimensional version of noughts and crosses?

Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

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

Imagine you were given the chance to win some money... and imagine you had nothing to lose...

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

Where should you start, if you want to finish back where you started?

Think of a number and follow my instructions. Tell me your answer, and I'll tell you what you started with! Can you explain how I know?

There are lots of different methods to find out what the shapes are worth - how many can you find?

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

Interior angles can help us to work out which polygons will tessellate. Can we use similar ideas to predict which polygons combine to create semi-regular solids?