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

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. . . .

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

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. . . .

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?

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

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?

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. . . .

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?

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?”

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.

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

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

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

Can you figure out how sequences of beach huts are generated?

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

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

Can you see how to build a harmonic triangle? Can you work out the next two rows?

Can you find a rule which connects consecutive triangular numbers?

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.

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

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. . . .

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

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

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

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

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

There are unexpected discoveries to be made about square numbers...

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.

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?

Where should you start, if you want to finish back where you started?

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?

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

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?

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?

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

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

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

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. . . .

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

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?

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

If you know the perimeter of a right angled triangle, what can you say about the area?

What is special about the difference between squares of numbers adjacent to multiples of three?

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?

Can you explain what is going on in these puzzling number tricks?

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