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?

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

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?

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

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?

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

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.

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

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.

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?

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

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.

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

The sums of the squares of three related numbers is also a perfect square - can you explain why?

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

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

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

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

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

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

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

Explore the effect of combining enlargements.

Explore the effect of reflecting in two parallel mirror lines.

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

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?

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

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?

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

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?

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?

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?

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

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.

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

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.

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?

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?

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

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

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

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

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

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

Show that is it impossible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units...

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?