Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

If: A + C = A; F x D = F; B - G = G; A + H = E; B / H = G; E - G = F and A-H represent the numbers from 0 to 7 Find the values of A, B, C, D, E, F and H.

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?

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

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

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.

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

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?

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

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

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?

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

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?

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 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?

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

The sums of the squares of three related numbers is also a perfect square - can you explain why?

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?

If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?

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

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

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

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

Explore the effect of combining enlargements.

Explore the effect of reflecting in two parallel mirror lines.

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

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

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?

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

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

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

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

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?

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?

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

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?

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?

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

A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?

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

Can you work out how to produce different shades of pink paint?

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?

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

The Egyptians expressed all fractions as the sum of different unit fractions. The Greedy Algorithm might provide us with an efficient way of doing this.

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

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?