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. . . .

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

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?

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

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.

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

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.

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?

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

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

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

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?

Can you explain the strategy for winning this game with any target?

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.

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

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

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

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.

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?

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

Which pairs of cogs let the coloured tooth touch every tooth on the other cog? Which pairs do not let this happen? Why?

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

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?

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

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

Substitution and Transposition all in one! How fiendish can these codes get?

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

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

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

Is there an efficient way to work out how many factors a large number has?

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

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

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?

Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.

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?

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

Given the products of adjacent cells, can you complete this Sudoku?

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

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

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

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.

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...

Use the interactivities to complete these Venn diagrams.

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

Factor track is not a race but a game of skill. The idea is to go round the track in as few moves as possible, keeping to the rules.

Can you order the digits from 1-3 to make a number which is divisible by 3 so when the last digit is removed it becomes a 2-figure number divisible by 2, and so on?

Each light in this interactivity turns on according to a rule. What happens when you enter different numbers? Can you find the smallest number that lights up all four lights?

Can you complete this calculation by filling in the missing numbers? In how many different ways can you do it?