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?

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

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

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?

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. . . .

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

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

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

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

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

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?

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?

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

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

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

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

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

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

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

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

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

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

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.

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

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.

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

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

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?

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.

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

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

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?

Can all unit fractions be written as the sum of two unit fractions?

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

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

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

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

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

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

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

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

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