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?

Explore the relationships between different paper sizes.

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".

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

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?

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?

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

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?

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

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

7 balls are shaken in a container. You win if the two blue balls touch. What is the probability of winning?

Six balls of various colours are randomly shaken into a trianglular arrangement. What is the probability of having at least one red in the corner?

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

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

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?

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?

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

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

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.

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

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?

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

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

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?

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...

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

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?

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.

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

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

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

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

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

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

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.

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

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