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

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

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

An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?

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.

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.

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

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?

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?

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. . . .

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. . . .

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

Water freezes at 0°Celsius (32°Fahrenheit) and boils at 100°C (212°Fahrenheit). Is there a temperature at which Celsius and Fahrenheit readings are the same?

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

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

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

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?

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?

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.

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

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?

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?

Explore the effect of combining enlargements.

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

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

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?

Explore the effect of reflecting in two parallel mirror lines.

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

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.

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

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?

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?

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

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

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

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?

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

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

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

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?

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 ?

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

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?

What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?

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

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?