Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

It starts quite simple but great opportunities for number discoveries and patterns!

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

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

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

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

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

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?

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

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. . . .

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. . . .

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.

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

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.

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

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

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

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

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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

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

Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?

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?

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

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?

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?

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?

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

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

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

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

Find out what a "fault-free" rectangle is and try to make some of your own.

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.

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

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?

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

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.

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?

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

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.

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

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?

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?