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.

Are these statements relating to odd and even numbers 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?

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

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

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

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

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.

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.

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

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

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.

Think of a number, square it and subtract your starting number. Is the number you’re left with odd or even? How do the images help to explain this?

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. . . .

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?

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

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.

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?

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

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

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

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

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

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

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?

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?”

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

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

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

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?

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

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

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

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. . . .

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. . . .

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.

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.

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

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

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. . . .

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

This task encourages you to investigate the number of edging pieces and panes in different sized windows.

Investigate the different ways that fifteen schools could have given money in a charity fundraiser.

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. . . .

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?

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?

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. . . .