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

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

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

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

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

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?

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?

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?

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

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

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

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?

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.

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

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

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?

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. . . .

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?

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

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

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

The clues for this Sudoku are the product of the numbers in adjacent squares.

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

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.

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

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

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

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?

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

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

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

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

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?

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

Is there an efficient way to work out how many factors a large number has?

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

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

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?

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

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?

Explore the effect of combining enlargements.

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?

Explore the effect of reflecting in two parallel mirror lines.

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

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?