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

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

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".

Explore the relationships between different paper sizes.

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?

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

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

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

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

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?

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

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

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?

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?

Which countries have the most naturally athletic populations?

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

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

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

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 develop a strategy to work out the rules controlling each light?

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?

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

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

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

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?

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?

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

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

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

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

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

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

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

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

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

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

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?

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?