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

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

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

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

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?

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.

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

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?

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.

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...

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?

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?

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

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

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.

Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?

Some people offer advice on how to win at games of chance, or how to influence probability in your favour. Can you decide whether advice is good or not?

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?

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 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

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

All CD Heaven stores were given the same number of a popular CD to sell for £24. In their two week sale each store reduces the price of the CD by 25% ... How many CDs did the store sell at. . . .

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?

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. . . .

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

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

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.

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?

There are lots of different methods to find out what the shapes are worth - how many can you find?

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

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

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

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. . . .

Use the differences to find the solution to this Sudoku.

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.

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

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

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

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

Explore the effect of combining enlargements.

Explore the effect of reflecting in two parallel mirror lines.

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

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

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 a sum invested gains 10% each year how long before it has doubled its value?