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

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

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.

I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

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

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

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

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

Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?

A game for 2 players with similarities to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up the last counter.

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

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?

Make some loops out of regular hexagons. What rules can you discover?

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

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?

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.

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

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

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.

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.

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

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

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

What's the largest volume of box you can make from a square of paper?

How many moves does it take to swap over some red and blue frogs? Do you have a method?

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?

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

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

These tasks give learners chance to generalise, which involves identifying an underlying structure.

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.

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?

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?

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

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?

Can you describe this route to infinity? Where will the arrows take you next?

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

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?

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

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

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

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

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.

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

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

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

This challenge asks you to imagine a snake coiling on itself.

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

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