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?

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. . . .

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.

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?

Explore the effect of reflecting in two parallel mirror lines.

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?

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

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.

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

Explore the effect of combining enlargements.

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

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?

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

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

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?

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?

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

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

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

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

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. . . .

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

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?

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?

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

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

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

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

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 number. . . .

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

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

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

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?

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 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?

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.

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

Use the differences to find the solution to this Sudoku.

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

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.

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 clues for this Sudoku are the product of the numbers in adjacent 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?

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?

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

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?