Is there an efficient way to work out how many factors a large number has?

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?

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

Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?

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

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?

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.

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?

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...

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

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

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

How many winning lines can you make in a three-dimensional version of noughts and crosses?

Can you arrange these numbers into 7 subsets, each of three numbers, so that when the numbers in each are added together, they make seven consecutive numbers?

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

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

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

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?

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?

How many more miles must the car travel before the numbers on the milometer and the trip meter 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?

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

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

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

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?

Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?

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

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.

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.

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?

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.

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?

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.

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?

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?

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 napkin is folded so that a corner coincides with the midpoint of an opposite edge . Investigate the three triangles formed .

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

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?

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.

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. . . .

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

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?

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