Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".

An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?

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

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

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.

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

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?

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

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

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

Charlie likes to go for walks around a square park, while Alison likes to cut across diagonally. Can you find relationships between the vectors they walk along?

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

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?

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

What's the largest volume of box you can make from a square of paper?

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?

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

Which countries have the most naturally athletic populations?

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?

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

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

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

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

Use your skill and judgement to match the sets of random data.

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

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?

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

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

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?

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?

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

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?

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

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?

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

Can you find a way to identify times tables after they have been shifted up or down?

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

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

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?

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

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?

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

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

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

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

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