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?

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

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

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.

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

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

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...

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?

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?

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

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

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?

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?

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

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

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

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?

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

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

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

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.

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. . . .

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

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?

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

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?

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. . . .

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

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

Find the decimal equivalents of the fractions one ninth, one ninety ninth, one nine hundred and ninety ninth etc. Explain the pattern you get and generalise.

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. . . .

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

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

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.

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

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 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?

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?

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

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

Explore the effect of reflecting in two parallel mirror lines.

Explore the effect of combining enlargements.

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

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.