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

Can you find a way to identify times tables after they have been shifted up?

The clues for this Sudoku are the product of the numbers in adjacent squares.

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

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 a large number of 1s, 4s, 7s and 10s. What numbers can we make?

Some 4 digit numbers can be written as the product of a 3 digit number and a 2 digit number using the digits 1 to 9 each once and only once. The number 4396 can be written as just such a product. Can. . . .

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?

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

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?

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...

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?

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?

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

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?

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 game that tests your understanding of remainders.

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

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?

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

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.

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

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?

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

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?

A game in which players take it in turns to choose a number. Can you block your opponent?

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

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

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

Katie and Will have some balloons. Will's balloon burst at exactly the same size as Katie's at the beginning of a puff. How many puffs had Will done before his balloon burst?

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.

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.

A collection of resources to support work on Factors and Multiples at Secondary level.

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

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?

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

Can you complete this jigsaw of the multiplication square?

An environment which simulates working with Cuisenaire rods.

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

Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.

What is the smallest number of answers you need to reveal in order to work out the missing headers?

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

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

Becky created a number plumber which multiplies by 5 and subtracts 4. What do you notice about the numbers that it produces? Can you explain your findings?

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.

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

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

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