There are three tables in a room with blocks of chocolate on each. Where would be the best place for each child in the class to sit if they came in one at a time?

How many different sets of numbers with at least four members can you find in the numbers in this box?

A description of some experiments in which you can make discoveries about triangles.

Numbers arranged in a square but some exceptional spatial awareness probably needed.

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.

This article (the first of two) contains ideas for investigations. Space-time, the curvature of space and topology are introduced with some fascinating problems to explore.

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!

Investigate the numbers that come up on a die as you roll it in the direction of north, south, east and west, without going over the path it's already made.

Follow the directions for circling numbers in the matrix. Add all the circled numbers together. Note your answer. Try again with a different starting number. What do you notice?

Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?

Here are many ideas for you to investigate - all linked with the number 2000.

Investigate the different ways these aliens count in this challenge. You could start by thinking about how each of them would write our number 7.

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

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.

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

Take ten sticks in heaps any way you like. Make a new heap using one from each of the heaps. By repeating that process could the arrangement 7 - 1 - 1 - 1 ever turn up, except by starting with it?

When Charlie asked his grandmother how old she is, he didn't get a straightforward reply! Can you work out how old she is?

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

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.

What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?

Can you find out how the 6-triangle shape is transformed in these tessellations? Will the tessellations go on for ever? Why or why not?

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

What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

All types of mathematical problems serve a useful purpose in mathematics teaching, but different types of problem will achieve different learning objectives. In generalmore open-ended problems have. . . .

In my local town there are three supermarkets which each has a special deal on some products. If you bought all your shopping in one shop, where would be the cheapest?

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

What is the largest cuboid you can wrap in an A3 sheet of paper?

This activity asks you to collect information about the birds you see in the garden. Are there patterns in the data or do the birds seem to visit randomly?

Can you make these equilateral triangles fit together to cover the paper without any gaps between them? Can you tessellate isosceles triangles?

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

How many different ways can you find of fitting five hexagons together? How will you know you have found all the 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?

I cut this square into two different shapes. What can you say about the relationship between them?

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

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"?

"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 ...?

Many natural systems appear to be in equilibrium until suddenly a critical point is reached, setting up a mudslide or an avalanche or an earthquake. In this project, students will use a simple. . . .

If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?

Formulate and investigate a simple mathematical model for the design of a table mat.

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?

How many ways can you find of tiling the square patio, using square tiles of different sizes?

Work with numbers big and small to estimate and calculate various quantities in biological contexts.

Investigate the number of faces you can see when you arrange three cubes in different ways.

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?

In how many ways can you stack these rods, following the rules?

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

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