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

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?

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}.

Choose any four consecutive even numbers. Multiply the two middle numbers together. Multiply the first and last numbers. Now subtract your second answer from the first. Try it with your own. . . .

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

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

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. . . .

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

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

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.

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

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

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.

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?

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 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?

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

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?”

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

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?

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

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. . . .

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. . . .

Surprising numerical patterns can be explained using algebra and diagrams...

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

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

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

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

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.

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

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?

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 rectangles where the value of the area is the same as the value of the perimeter?

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

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 ?

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?

Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.

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

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

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?

Can you find a rule which connects consecutive triangular numbers?

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

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

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?

In this 100 square, look at the green square which contains the numbers 2, 3, 12 and 13. What is the sum of the numbers that are diagonally opposite each other? What do you notice?

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...