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.

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?

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

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

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?

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

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?

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?

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

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...

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

However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?

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

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?

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = nÂ² Use the diagram to show that any odd number is the difference of two squares.

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

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

The well known Fibonacci sequence is 1 ,1, 2, 3, 5, 8, 13, 21.... How many Fibonacci sequences can you find containing the number 196 as one of the terms?

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. . . .

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

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

Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.

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

Find the five distinct digits N, R, I, C and H in the following nomogram

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?

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

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

Can you find a rule which connects consecutive triangular numbers?

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?

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

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

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

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

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

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?

Use algebra to reason why 16 and 32 are impossible to create as the sum of consecutive numbers.

If a sum invested gains 10% each year how long before it has doubled its value?

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. . . .

The heptathlon is an athletics competition consisting of 7 events. Can you make sense of the scoring system in order to advise a heptathlete on the best way to reach her target?

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. . . .

The number 27 is special because it is three times the sum of its digits 27 = 3 (2 + 7). Find some two digit numbers that are SEVEN times the sum of their digits (seven-up numbers)?

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. . . .

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.

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?

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?

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?

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?