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?

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

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?

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. . . .

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?

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...

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

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...

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

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.

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.

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)?

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?

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?

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. . . .

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

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?

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. . . .

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.

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

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?

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?

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.

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?

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. . . .

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?

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

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

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.

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

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 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. . . .

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

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?

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

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

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

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

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

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

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

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?

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

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

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.

Use algebra to reason why 16 and 32 are impossible to create as the sum of consecutive numbers.

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. . . .

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

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