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?

There are many different methods to solve this geometrical problem - how many can you find?

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

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".

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?

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

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

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

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

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?

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

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

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.

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.

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?

Explore the relationships between different paper sizes.

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

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

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

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

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

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?

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?

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

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

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

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.