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

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?

A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?

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?

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.

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

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

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

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

Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.

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

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?

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?

Create some shapes by combining two or more rectangles. What can you say about the areas and perimeters of the shapes you can make?

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

Find the five distinct digits N, R, I, C and H in the following nomogram

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

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?

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

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. . . .

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

This article explains how to make your own magic square to mark a special occasion with the special date of your choice on the top line.

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. . . .

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

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

Use algebra to reason why 16 and 32 are impossible to create as the sum of consecutive numbers.

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

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

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?

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

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

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

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

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

Attach weights of 1, 2, 4, and 8 units to the four attachment points on the bar. Move the bar from side to side until you find a balance point. Is it possible to predict that position?

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

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

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. . . .

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

Can you find a rule which connects consecutive triangular numbers?

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?

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?

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?

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 the area of a parallelogram defined by two vectors?

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