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

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

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

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.

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

Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?

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

A job needs three men but in fact six people do it. When it is finished they are all paid the same. How much was paid in total, and much does each man get if the money is shared as Fred suggests?

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.

Here are three 'tricks' to amaze your friends. But the really clever trick is explaining to them why these 'tricks' are maths not magic. Like all good magicians, you should practice by trying. . . .

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. . . .

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. . . .

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?

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

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

A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?

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?

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

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?

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?

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

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?

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

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

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

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

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

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?

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

Brian swims at twice the speed that a river is flowing, downstream from one moored boat to another and back again, taking 12 minutes altogether. How long would it have taken him in still water?

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

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

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. . . .

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

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?

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?

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 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?

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

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}.