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

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

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

What does this number mean ? Which order of 1, 2, 3 and 4 makes the highest value ? Which makes the lowest ?

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.

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

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?

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.

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

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

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

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?

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?

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

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

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?

Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?

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

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.

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.

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?

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

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. . . .

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

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

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?

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?

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...

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

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. . . .

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

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?

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?

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 the area of a parallelogram defined by two vectors?

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

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

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

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

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?

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?

Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".

Explore the effect of reflecting in two parallel mirror lines.