In this article for teachers, Bernard gives an example of taking an initial activity and getting questions going that lead to other explorations.

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

In this investigation, we look at Pascal's Triangle in a slightly different way - rotated and with the top line of ones taken off.

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?

These caterpillars have 16 parts. What different shapes do they make if each part lies in the small squares of a 4 by 4 square?

Bernard Bagnall describes how to get more out of some favourite NRICH investigations.

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

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

Why does the tower look a different size in each of these pictures?

Investigate what happens when you add house numbers along a street in different ways.

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

Well now, what would happen if we lost all the nines in our number system? Have a go at writing the numbers out in this way and have a look at the multiplications table.

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?

In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.

Complete these two jigsaws then put one on top of the other. What happens when you add the 'touching' numbers? What happens when you change the position of the jigsaws?

Explore Alex's number plumber. What questions would you like to ask? Don't forget to keep visiting NRICH projects site for the latest developments and questions.

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?

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

What happens when you add the digits of a number then multiply the result by 2 and you keep doing this? You could try for different numbers and different rules.

A group of children are discussing the height of a tall tree. How would you go about finding out its height?

Bernard Bagnall looks at what 'problem solving' might really mean in the context of primary classrooms.

48 is called an abundant number because it is less than the sum of its factors (without itself). Can you find some more abundant numbers?

Which times on a digital clock have a line of symmetry? Which look the same upside-down? You might like to try this investigation and find out!

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

Start with four numbers at the corners of a square and put the total of two corners in the middle of that side. Keep going... Can you estimate what the size of the last four numbers will be?

Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?

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

In this section from a calendar, put a square box around the 1st, 2nd, 8th and 9th. Add all the pairs of numbers. What do you notice about the answers?

The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.

A challenging activity focusing on finding all possible ways of stacking rods.

This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?

If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

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

This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?

There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?

Investigate this balance which is marked in halves. If you had a weight on the left-hand 7, where could you hang two weights on the right to make it balance?

Can you design a new shape for the twenty-eight squares and arrange the numbers in a logical way? What patterns do you notice?

A follow-up activity to Tiles in the Garden.

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?