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

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

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?

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

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?

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?

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?

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

Find at least one way to put in some operation signs (+ - x ÷) to make these digits come to 100.

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

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?

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.

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

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?

Use the differences to find the solution to this Sudoku.

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...

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. . . .

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?

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?

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. . . .

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

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

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

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

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

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.

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?

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

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?

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.

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

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. . . .

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?

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".

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

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

Can you find rectangles where the value of the area is the same as the value of the perimeter?

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

Explore the effect of combining enlargements.

Explore the effect of reflecting in two parallel mirror lines.

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?

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?

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

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

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

Chris and Jo put two red and four blue ribbons in a box. They each pick a ribbon from the box without looking. Jo wins if the two ribbons are the same colour. Is the game fair?

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