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?

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?

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

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

Here are four tiles. They can be arranged in a 2 by 2 square so that this large square has a green edge. If the tiles are moved around, we can make a 2 by 2 square with a blue edge... Now try to. . . .

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

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

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?

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

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

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

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.

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

Explore the effect of reflecting in two parallel mirror lines.

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

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?

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.

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?

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

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

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?

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?

If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?

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

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

Start with two numbers. This is the start of a sequence. The next number is the average of the last two numbers. Continue the sequence. What will happen if you carry on for ever?

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. . . .

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

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

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

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

Explore the effect of combining enlargements.

Triangle ABC is isosceles while triangle DEF is equilateral. Find one angle in terms of the other two.

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

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

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?

Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?

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?

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.

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?

Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...

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

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

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?

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

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

In a three-dimensional version of noughts and crosses, how many winning lines can you make?