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

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

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?

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

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

Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.

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?

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?

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

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

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

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

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?

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

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

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

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?

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?

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

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

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?

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

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

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

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

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 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 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.

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?

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

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

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

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

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

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

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?

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

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?

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?

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

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

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

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.