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

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?

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?

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

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

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

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

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

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

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

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?

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

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?

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

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?

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

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

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

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.

This article for primary teachers discusses how we can help learners generalise and prove, using NRICH tasks as examples.

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?

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

Can you find a way of counting the spheres in these arrangements?

Watch this animation. What do you see? Can you explain why this happens?

Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?

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

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.

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

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?

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

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.

A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.

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

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

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?

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

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

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 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?

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?

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?

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 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.

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

Can you explain the strategy for winning this game with any target?

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

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

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

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