Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?

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?

Try out this number trick. What happens with different starting numbers? What do you notice?

Find the sum of all three-digit numbers each of whose digits is odd.

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

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.

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

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

Ben’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?

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

This activity involves rounding four-digit numbers to the nearest thousand.

Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?

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

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

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?

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

Does this 'trick' for calculating multiples of 11 always work? Why or why not?

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .

Here are two kinds of spirals for you to explore. What do you notice?

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

Think of a number, square it and subtract your starting number. Is the number you’re left with odd or even? How do the images help to explain this?

In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

Are these statements relating to odd and even numbers always true, sometimes true or never true?

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

This task follows on from Build it Up and takes the ideas into three dimensions!

Great Granddad is very proud of his telegram from the Queen congratulating him on his hundredth birthday and he has friends who are even older than he is... When was he born?

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

These tasks give learners chance to generalise, which involves identifying an underlying structure.

Investigate the different ways that fifteen schools could have given money in a charity fundraiser.

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

Take a look at the video of this trick. Can you perform it yourself? Why is this maths and not magic?

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

Surprise your friends with this magic square trick.

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?

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

This article for primary teachers discusses how we can help learners generalise and prove, using NRICH tasks as examples.

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?