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.

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?

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

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?

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?

32 x 38 = 30 x 40 + 2 x 8; 34 x 36 = 30 x 40 + 4 x 6; 56 x 54 = 50 x 60 + 6 x 4; 73 x 77 = 70 x 80 + 3 x 7 Verify and generalise if possible.

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

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

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

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

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

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?

First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...

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

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

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.

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?

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?

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?

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?

Can you find a rule which connects consecutive triangular numbers?

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

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?

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.

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

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

Fifteen students had to travel 60 miles. They could use a car, which could only carry 5 students. As the car left with the first 5 (at 40 miles per hour), the remaining 10 commenced hiking along the. . . .

Can you find the lap times of the two cyclists travelling at constant speeds?

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 ?

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?

I added together the first 'n' positive integers and found that my answer was a 3 digit number in which all the digits were the same...

Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?

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?

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

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

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

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

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

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

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

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

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. . . .

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.

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. . . .

My train left London between 6 a.m. and 7 a.m. and arrived in Paris between 9 a.m. and 10 a.m. At the start and end of the journey the hands on my watch were in exactly the same positions but the. . . .

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