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

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

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

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

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

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?

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.

If you have a large supply of 3kg and 8kg weights, how many of each would you need for the average (mean) of the weights to be 6kg?

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

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

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

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?”

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. . . .

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

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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?

Can you find the values at the vertices when you know the values on the edges?

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?

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

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.

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

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.

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

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?

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

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

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

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

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

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

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

Nim-7 game for an adult and child. Who will be the one to take the last counter?

Got It game for an adult and child. How can you play so that you know you will always win?

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

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

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.

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

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

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

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

Try out this number trick. What happens with different starting numbers? What do you notice?

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

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

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

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

Can you figure out how sequences of beach huts are generated?

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.

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?