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?

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

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?

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

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

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.

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

Which countries have the most naturally athletic populations?

How well can you estimate 10 seconds? Investigate with our timing tool.

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?

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

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

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

Explore the relationships between different paper sizes.

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

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?

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

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

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?

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?

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?

If everyone in your class picked a number from 1 to 225, do you think any two people would pick the same number?

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

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

Can you work out the probability of winning the Mathsland National Lottery? Try our simulator to test out your ideas.

If you have a large supply of 3kg and 8kg weights, how many of each would you need for the average (mean) of the weights to be 6kg?

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...

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

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?

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...

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?

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

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

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?

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