There are **208** NRICH Mathematical resources connected to **Investigations**, you may find related items under Mathematical Thinking.

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

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

Take a look at these data collected by children in 1986 as part of the Domesday Project. What do they tell you? What do you think about the way they are presented?

This problem is intended to get children to look really hard at something they will see many times in the next few months.

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

We think this 3x3 version of the game is often harder than the 5x5 version. Do you agree? If so, why do you think that might be?

Where should runners start the 200m race so that they have all run the same distance by the finish?

In this investigation we are going to count the number of 1s, 2s, 3s etc in numbers. Can you predict what will happen?

These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.

What statements can you make about the car that passes the school gates at 11am on Monday? How will you come up with statements and test your ideas?

What can you say about the child who will be first on the playground tomorrow morning at breaktime in your school?

"Ip dip sky blue! Who's 'it'? It's you!" Where would you position yourself so that you are 'it' if there are two players? Three players ...?

This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.

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.

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.

Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.

How fast would you have to throw a ball upwards so that it would never land?

Try continuing these patterns made from triangles. Can you create your own repeating pattern?

Look at the advanced way of viewing sin and cos through their power series.

By exploring the concept of scale invariance, find the probability that a random piece of real data begins with a 1.

Given the equation for the path followed by the back wheel of a bike, can you solve to find the equation followed by the front wheel?

Which parts of these framework bridges are in tension and which parts are in compression?

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?

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?

Use your mouse to move the red and green parts of this disc. Can you make images which show the turnings described?

The red ring is inside the blue ring in this picture. Can you rearrange the rings in different ways? Perhaps you can overlap them or put one outside another?

Sort the houses in my street into different groups. Can you do it in any other ways?

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

Draw three equal line segments in a unit circle to divide the circle into four parts of equal area.

These pictures show squares split into halves. Can you find other ways?

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?

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?

Explore ways of colouring this set of triangles. Can you make symmetrical patterns?

Ben has five coins in his pocket. How much money might he have?

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?

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

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.

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

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?

Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possible answers?

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

How many faces can you see when you arrange these three cubes in different ways?

Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?