Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some. . . .

Can you find any perfect numbers? Read this article to find out more...

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

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.

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?

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?

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?

Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2?

Find the words hidden inside each of the circles by counting around a certain number of spaces to find each letter in turn.

An investigation that gives you the opportunity to make and justify predictions.

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku.

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

Is it possible to draw a 5-pointed star without taking your pencil off the paper? Is it possible to draw a 6-pointed star in the same way without taking your pen off?

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

Given the products of diagonally opposite cells - can you complete this Sudoku?

Complete the following expressions so that each one gives a four digit number as the product of two two digit numbers and uses the digits 1 to 8 once and only once.

Do you know a quick way to check if a number is a multiple of two? How about three, four or six?

Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number?

What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?

Can you work out what size grid you need to read our secret message?

Look at three 'next door neighbours' amongst the counting numbers. Add them together. What do you notice?

Are these statements always true, sometimes true or never true?

Follow this recipe for sieving numbers and see what interesting patterns emerge.

The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?

The number 8888...88M9999...99 is divisible by 7 and it starts with the digit 8 repeated 50 times and ends with the digit 9 repeated 50 times. What is the value of the digit M?

Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?

The sum of the first 'n' natural numbers is a 3 digit number in which all the digits are the same. How many numbers have been summed?

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

This article for teachers describes how number arrays can be a useful reprentation for many number concepts.

Benâ€™s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

A game for 2 or more people. Starting with 100, subratct a number from 1 to 9 from the total. You score for making an odd number, a number ending in 0 or a multiple of 6.

Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?

How can you use just one weighing to find out which box contains the lighter ten coins out of the ten boxes?

Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why?

Find the highest power of 11 that will divide into 1000! exactly.

How many integers between 1 and 1200 are NOT multiples of any of the numbers 2, 3 or 5?

In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time?