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?

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?

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?

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

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.

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

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".

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

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

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.

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?

Which countries have the most naturally athletic populations?

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

Explore the relationships between different paper sizes.

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?

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

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

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

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

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

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

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?

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

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

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?

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

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?

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

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?

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

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?

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

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?

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?

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

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

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

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

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

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?

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

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

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

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?