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?

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?

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?

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

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.

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. . . .

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

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?

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

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

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

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.

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. . . .

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

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

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

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

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

A game that tests your understanding of remainders.

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

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

Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?

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

The five digit number A679B, in base ten, is divisible by 72. What are the values of A and B?

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.

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?

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.

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

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

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

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?

Can you complete this jigsaw of the multiplication square?

Twice a week I go swimming and swim the same number of lengths of the pool each time. As I swim, I count the lengths I've done so far, and make it into a fraction of the whole number of lengths I. . . .

An environment which simulates working with Cuisenaire rods.

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?

Find the frequency distribution for ordinary English, and use it to help you crack the code.

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?

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

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.

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?

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?

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...

Using your knowledge of the properties of numbers, can you fill all the squares on the board?

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