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

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

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

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

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?

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?

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

Explore the effect of combining enlargements.

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?

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

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?

Water freezes at 0°Celsius (32°Fahrenheit) and boils at 100°C (212°Fahrenheit). Is there a temperature at which Celsius and Fahrenheit readings are the same?

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?

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.

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

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

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

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

Explore the effect of reflecting in two parallel mirror lines.

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

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

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

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?

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

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

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?

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.

If you move the tiles around, can you make squares with different coloured edges?

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?

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

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?

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

Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

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

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

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. . . .

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

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?

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

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?

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. . . .

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

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

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

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

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

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

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