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

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?

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

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter.

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

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

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

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

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

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

Delight your friends with this cunning trick! Can you explain how it works?

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

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. . . .

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

What happens when you round these three-digit numbers to the nearest 100?

What happens when you round these numbers to the nearest whole number?

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.

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

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

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.

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

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

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.

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

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?

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. . . .

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?

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

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?”

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

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

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

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

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?

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

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

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

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

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?

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

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

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?