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

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 the area of a parallelogram defined by two vectors?

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.

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

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?

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

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.

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

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

Think of two whole numbers under 10. Take one of them and add 1. Multiply by 5. Add 1 again. Double your answer. Subract 1. Add your second number. Add 2. Double again. Subtract 8. Halve this. . . .

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

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

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. . . .

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?

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

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?

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

Can you arrange these numbers into 7 subsets, each of three numbers, so that when the numbers in each are added together, they make seven consecutive numbers?

A circle is inscribed in a triangle which has side lengths of 8, 15 and 17 cm. What is the radius of the circle?

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

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.

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.

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

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?

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.

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?

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

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?

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?

Two ladders are propped up against facing walls. The end of the first ladder is 10 metres above the foot of the first wall. The end of the second ladder is 5 metres above the foot of the second. . . .

Some 4 digit numbers can be written as the product of a 3 digit number and a 2 digit number using the digits 1 to 9 each once and only once. The number 4396 can be written as just such a product. Can. . . .

Chris and Jo put two red and four blue ribbons in a box. They each pick a ribbon from the box without looking. Jo wins if the two ribbons are the same colour. Is the game fair?

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

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

The number 2.525252525252.... can be written as a fraction. What is the sum of the denominator and numerator?

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

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

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?

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?

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. . . .