Challenge Level

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

Challenge Level

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?

Challenge Level

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

Challenge Level

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?

Challenge Level

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?

Challenge Level

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

Challenge Level

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. . . .

Challenge Level

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. . . .

Challenge Level

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

Challenge Level

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?

Challenge Level

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?

Challenge Level

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.

Challenge Level

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. . . .

Challenge Level

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.

Challenge Level

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

Challenge Level

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

Challenge Level

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

Challenge Level

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

Challenge Level

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?

Challenge Level

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

Challenge Level

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

Challenge Level

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. . . .

Challenge Level

In a snooker game the brown ball was on the lip of the pocket but it could not be hit directly as the black ball was in the way. How could it be potted by playing the white ball off a cushion?

Challenge Level

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 � 1 [1/3]. What other numbers have the sum equal to the product and can this be so. . . .

Challenge Level

Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.

Challenge Level

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.

Challenge Level

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.

Challenge Level

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

Challenge Level

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

Challenge Level

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}.

Challenge Level

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

Challenge Level

Can you find a rule which connects consecutive triangular numbers?

Challenge Level

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

Challenge Level

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?

Challenge Level

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

Challenge Level

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. . . .

Challenge Level

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

Challenge Level

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

Challenge Level

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

Challenge Level

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

Challenge Level

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. . . .

Challenge Level

There are unexpected discoveries to be made about square numbers...

Challenge Level

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

Challenge Level

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

Challenge Level

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

Challenge Level

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

Challenge Level

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

Challenge Level

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