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?

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?

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...

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?

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.

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...

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. . . .

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. . . .

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

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.

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. . . .

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...

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

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

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. . . .

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

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?

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?

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

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.

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?

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 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. . . .

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.

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

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

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?

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

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?

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

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}.

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

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

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?

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

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

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?

First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...

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

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

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. . . .

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)?