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?

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?

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

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.

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

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.

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 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.

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?

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

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

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 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?

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

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

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?

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

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. . . .

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

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

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

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

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?

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

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.

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?

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

Explore the effect of reflecting in two parallel mirror lines.

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

Explore the effect of combining enlargements.

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

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

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

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. . . .

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?

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

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

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

A car's milometer reads 4631 miles and the trip meter has 173.3 on it. How many more miles must the car travel before the two numbers contain the same digits in the same order?

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?

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

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

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?

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