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

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

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

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.

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?

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

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?

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

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?

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.

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

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

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

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

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?

Explore the effect of reflecting in two parallel mirror lines.

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. . . .

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

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 difference of two perfect squares. What do you notice about the numbers you CANNOT make?

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?

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

Explore the effect of combining enlargements.

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?

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

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

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

Do you know a quick way to check if a number is a multiple of two? How about three, four or six?

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

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

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?

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

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

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

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

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

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. . . .

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?

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?

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.

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

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?

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

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

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?

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

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

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?