Explore the effect of reflecting in two parallel mirror lines.

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?

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

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.

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

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?

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

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

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?

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

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?

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?

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?

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?

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?

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

Explore the effect of combining enlargements.

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?

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

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

Can you maximise the area available to a grazing goat?

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

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?

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

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?

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

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.

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

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.

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

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

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

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.

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

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?

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

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?

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

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

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

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

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. . . .

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.

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?

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?

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

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