Attach weights of 1, 2, 4, and 8 units to the four attachment points on the bar. Move the bar from side to side until you find a balance point. Is it possible to predict that position?

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

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

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

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.

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?

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

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

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

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

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?

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.

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

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. . . .

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?

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.

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

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

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

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 ?

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

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

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

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

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

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

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?

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?

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

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.

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...

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?

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

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

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

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. . . .

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

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}.

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...

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

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?

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. . . .

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...