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.

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

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

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

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

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

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

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

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

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?

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

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

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?

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

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

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?

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

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

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 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?

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

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

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

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.

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

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

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?

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?

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

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.

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.

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

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

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

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

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.

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

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?

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

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?

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.

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.