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

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.

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. . . .

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. . . .

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

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

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

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?

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?

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...

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

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

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

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?

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?

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

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

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?

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. . . .

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

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

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?

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. . . .

A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .

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. . . .

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

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?

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

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?

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

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?

Kyle and his teacher disagree about his test score - who is right?

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?

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

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

What are the possible dimensions of a rectangular hallway if the number of tiles around the perimeter is exactly half the total number of tiles?

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?

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

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. . . .

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

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