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

The Egyptians expressed all fractions as the sum of different unit fractions. The Greedy Algorithm might provide us with an efficient way of doing this.

Is it always possible to combine two paints made up in the ratios 1:x and 1:y and turn them into paint made up in the ratio a:b ? Can you find an efficent way of doing this?

A jigsaw where pieces only go together if the fractions are equivalent.

Can all unit fractions be written as the sum of two unit fractions?

A decorator can buy pink paint from two manufacturers. What is the least number he would need of each type in order to produce different shades of pink.

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

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?

A game for 2 or more people, based on the traditional card game Rummy. Players aim to make two `tricks', where each trick has to consist of a picture of a shape, a name that describes that shape, and. . . .

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

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

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?

Do you know a quick way to check if a number is a multiple of two? How about three, four or six?

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

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

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

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

Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?

Can you find the area of a parallelogram defined by two vectors?

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?

A napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

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?

If: A + C = A; F x D = F; B - G = G; A + H = E; B / H = G; E - G = F and A-H represent the numbers from 0 to 7 Find the values of A, B, C, D, E, F and H.

Different combinations of the weights available allow you to make different totals. Which totals can you make?

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

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

What is the same and what is different about these circle questions? What connections can you make?

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

How many different symmetrical shapes can you make by shading triangles or squares?

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?

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

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?

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

Some people offer advice on how to win at games of chance, or how to influence probability in your favour. Can you decide whether advice is good or not?

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

Can you describe this route to infinity? Where will the arrows take you next?

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.

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

Five children went into the sweet shop after school. There were choco bars, chews, mini eggs and lollypops, all costing under 50p. Suggest a way in which Nathan could spend all his money.

Explore the effect of reflecting in two parallel mirror lines.

Explore the effect of combining enlargements.

The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?

There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?

Take any four digit number. Move the first digit to the 'back of the queue' and move the rest along. Now add your two numbers. What properties do your answers always have?

How many solutions can you find to this sum? Each of the different letters stands for a different number.