Challenge Level

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

Challenge Level

Kyle and his teacher disagree about his test score - who is right?

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

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

Can you find a rule which connects consecutive triangular numbers?

Challenge Level

Take a few whole numbers away from a triangle number. If you know the mean of the remaining numbers can you find the triangle number and which numbers were removed?

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

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

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

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

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

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

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

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

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

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

Challenge Level

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

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

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

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

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

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

How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?

Challenge Level

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.

Challenge Level

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Challenge Level

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

Challenge Level

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

Challenge Level

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

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

In this 100 square, look at the green square which contains the numbers 2, 3, 12 and 13. What is the sum of the numbers that are diagonally opposite each other? What do you notice?

Challenge Level

An algebra task which depends on members of the group noticing the needs of others and responding.

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?

Challenge Level

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

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

The heptathlon is an athletics competition consisting of 7 events. Can you make sense of the scoring system in order to advise a heptathlete on the best way to reach her target?

Challenge Level

What is special about the difference between squares of numbers adjacent to multiples of three?

Challenge Level

Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?

Challenge Level

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

Challenge Level

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

Challenge Level

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?