Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

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?

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

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?”

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

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

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.

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.

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

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?

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

Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I. . . .

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

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

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

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.

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 � 1 [1/3]. What other numbers have the sum equal to the product and can this be so. . . .

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. . . .

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

Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?

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

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?

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

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.

Can you find sets of sloping lines that enclose a square?

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?

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

It would be nice to have a strategy for disentangling any tangled ropes...

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

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.

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

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

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

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

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

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

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

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

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.

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

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

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

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?