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...

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}.

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

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

Can you find a rule which connects consecutive triangular numbers?

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?

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?

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. . . .

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

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

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

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?

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

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.

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

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?

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

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 diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

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

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

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

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

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

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?

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

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

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. . . .

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. . . .

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

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

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

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. . . .

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

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

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

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?

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. . . .

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

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

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

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

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

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

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?