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?

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

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

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.

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

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

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

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

Explore the relationships between different paper sizes.

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

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

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

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?

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?

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

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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

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?

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?

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

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?

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

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

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

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?

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

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

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

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?

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

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

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

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

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

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 a way to identify times tables after they have been shifted up or down?

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?

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?

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

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

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

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 any two-digit numbers that satisfy all of these statements?

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?

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

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?

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".