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

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

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

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?

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.

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.

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?

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?

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

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.

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?

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?

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

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

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

Explore the effect of combining enlargements.

Explore the effect of reflecting in two parallel mirror lines.

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

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

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

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

Can you see how to build a harmonic triangle? Can you work out the next two rows?

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

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. . . .

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

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?

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?

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

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?

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?

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

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

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

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?

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

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

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

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?

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?

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

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

Show that is it impossible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units...

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

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

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?

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