How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

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?

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

Make some loops out of regular hexagons. What rules can you discover?

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

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?

Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I. . . .

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

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

If you take a three by three square on a 1-10 addition square and multiply the diagonally opposite numbers together, what is the difference between these products. Why?

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .

A box has faces with areas 3, 12 and 25 square centimetres. What is the volume of the box?

Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.

How good are you at finding the formula for a number pattern ?

Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?

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 find a rule which connects consecutive triangular numbers?

This article explains how to make your own magic square to mark a special occasion with the special date of your choice on the top line.

The Number Jumbler can always work out your chosen symbol. Can you work out how?

Create some shapes by combining two or more rectangles. What can you say about the areas and perimeters of the shapes you can make?

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...

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

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

Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?

Think of a number and follow my instructions. Tell me your answer, and I'll tell you what you started with! Can you explain how I know?

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

Show that all pentagonal numbers are one third of a triangular number.

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

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?

Can you find a rule which relates triangular numbers to square numbers?

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. . . .

If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?