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

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. . . .

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 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?

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?

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?

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. . . .

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. . . .

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?

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

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?

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

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

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?

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 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?

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?

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?

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?

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?

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?

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

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

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

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

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

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

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

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

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

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?

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 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?

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.

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?

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

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

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

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

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. . . .