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?

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

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?

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

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

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

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. . . .

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

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. . . .

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. . . .

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?

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?

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?

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

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.

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

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?

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

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

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?

Write down a three-digit number Change the order of the digits to get a different number Find the difference between the two three digit numbers Follow the rest of the instructions then try. . . .

Here are three 'tricks' to amaze your friends. But the really clever trick is explaining to them why these 'tricks' are maths not magic. Like all good magicians, you should practice by trying. . . .

Visitors to Earth from the distant planet of Zub-Zorna were amazed when they found out that when the digits in this multiplication were reversed, the answer was the same! Find a way to explain. . . .

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. . . .

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

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?

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?

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

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...

Can you find a rule which connects consecutive triangular numbers?

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 you find a rule which relates triangular numbers to square 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.

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.

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

Brian swims at twice the speed that a river is flowing, downstream from one moored boat to another and back again, taking 12 minutes altogether. How long would it have taken him in still water?

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

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

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

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

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?

An algebra task which depends on members of the group noticing the needs of others and responding.

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?

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?

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