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

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?

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

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. 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. . . .

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.

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?

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.

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

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

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.

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?

Use the interactivities to complete these Venn diagrams.

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

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

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

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?

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?

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?

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

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...

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!

Can you complete this jigsaw of the multiplication square?

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?

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

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

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

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.

A game that tests your understanding of remainders.

Given the products of diagonally opposite 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?

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.

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

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

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

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. . . .

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

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

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?

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?

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

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

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

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

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?

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

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

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