A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

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

What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?

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

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

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

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.

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

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. . . .

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.

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. . . .

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?

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

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

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

Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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?

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

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

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?

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

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

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?

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?

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

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

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. . . .

There are lots of different methods to find out what the shapes are worth - how many can you find?

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

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.

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...

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.

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

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

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

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?

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.

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.

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

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. . . .

Which set of numbers that add to 10 have the largest product?