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.

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)?

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...

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?

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. . . .

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

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?

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?

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

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

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. . . .

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?

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

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

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.

Can you explain why a sequence of operations always gives you perfect squares?

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

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?

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. . . .

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?

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 more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?

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

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.

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. . . .

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?

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

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

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.

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

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

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

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

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 explain what is going on in these puzzling number tricks?

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

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

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

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. . . .

Play around with the Fibonacci sequence and discover some surprising results!

The sums of the squares of three related numbers is also a perfect square - can you explain why?

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

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

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

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?