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

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.

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...

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?

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?

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?

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

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

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?

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

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?

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time?

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

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 smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

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

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

48 is called an abundant number because it is less than the sum of its factors (without itself). Can you find some more abundant numbers?

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

If you have only four weights, where could you place them in order to balance this equaliser?

Use the interactivities to complete these Venn diagrams.

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

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

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

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?

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

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

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

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.

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

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

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.

Factors and Multiples game for an adult and child. How can you make sure you win this game?

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?

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

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

Can you complete this jigsaw of the multiplication square?

Use the interactivity to create some steady rhythms. How could you create a rhythm which sounds the same forwards as it does backwards?

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?

Starting with the number 180, take away 9 again and again, joining up the dots as you go. Watch out - don't join all the dots!

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

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.

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?