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

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

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.

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

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

Are these statements relating to calculation and properties of shapes always true, sometimes true or never true?

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

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

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?

This problem challenges you to find out how many odd numbers there are between pairs of numbers. Can you find a pair of numbers that has four odds between them?

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

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

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

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?

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

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

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?

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

If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?

Two children made up a game as they walked along the garden paths. Can you find out their scores? Can you find some paths of your own?

This is a game for two players. Can you find out how to be the first to get to 12 o'clock?

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

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

Can you find all the ways to get 15 at the top of this triangle of numbers?

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

How many centimetres of rope will I need to make another mat just like the one I have here?

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

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

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.

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

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

Got It game for an adult and child. How can you play so that you know you will always win?

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

Stop the Clock game for an adult and child. How can you make sure you always win this game?

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?

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

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

In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case?

This challenge is about finding the difference between numbers which have the same tens digit.

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

Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?

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

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?

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.