This challenge asks you to imagine a snake coiling on itself.

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

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

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

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. . . .

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

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 see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

Strike it Out game for an adult and child. Can you stop your partner from being able to go?

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

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

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.

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

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

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?

Can all unit fractions be written as the sum of two unit fractions?

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .

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 for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

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

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?

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

It would be nice to have a strategy for disentangling any tangled ropes...

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

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?

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.

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

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

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?

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

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

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

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

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?

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

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?

Nim-7 game for an adult and child. Who will be the one to take the last counter?

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

This task encourages you to investigate the number of edging pieces and panes in different sized windows.

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

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

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

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

Can you work out how to win this game of Nim? Does it matter if you go first or second?

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