Fifteen students had to travel 60 miles. They could use a car, which could only carry 5 students. As the car left with the first 5 (at 40 miles per hour), the remaining 10 commenced hiking along the. . . .

Can you find the lap times of the two cyclists travelling at constant speeds?

Brian swims at twice the speed that a river is flowing, downstream from one moored boat to another and back again, taking 12 minutes altogether. How long would it have taken him in still water?

To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling...

A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .

My train left London between 6 a.m. and 7 a.m. and arrived in Paris between 9 a.m. and 10 a.m. At the start and end of the journey the hands on my watch were in exactly the same positions but the. . . .

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?

First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...

Is there a temperature at which Celsius and Fahrenheit readings are the same?

Can you make sense of these three proofs of Pythagoras' Theorem?

A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?

32 x 38 = 30 x 40 + 2 x 8; 34 x 36 = 30 x 40 + 4 x 6; 56 x 54 = 50 x 60 + 6 x 4; 73 x 77 = 70 x 80 + 3 x 7 Verify and generalise if possible.

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

What are the possible dimensions of a rectangular hallway if the number of tiles around the perimeter is exactly half the total number of tiles?

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

Can you see how to build a harmonic triangle? Can you work out the next two rows?

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

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?

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

Can you find a rule which connects consecutive triangular numbers?

Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?

Can you find a rule which relates triangular numbers to square numbers?

Show that all pentagonal numbers are one third of a triangular number.

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

The sums of the squares of three related numbers is also a perfect square - can you explain why?

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

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

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

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.

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

How good are you at finding the formula for a number pattern ?

A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .

Where should you start, if you want to finish back where you started?

Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.

Can you explain what is going on in these puzzling number tricks?

The Number Jumbler can always work out your chosen symbol. Can you work out how?

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

Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?