Make one big triangle so the numbers that touch on the small triangles add to 10. You could use the interactivity to help you.

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

What is the least number of moves you can take to rearrange the bears so that no bear is next to a bear of the same colour?

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?

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?

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

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.

In how many ways can you fit all three pieces together to make shapes with line symmetry?

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

A picture is made by joining five small quadrilaterals together to make a large quadrilateral. Is it possible to draw a similar picture if all the small quadrilaterals are cyclic?

A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?

It is believed that weaker snooker players have a better chance of winning matches over eleven frames (i.e. first to win 6 frames) than they do over fifteen frames. Is this true?

A circular plate rolls inside a rectangular tray making five circuits and rotating about its centre seven times. Find the dimensions of the tray.

How can we decide what the 'best' solution to this problem is? Take a look at these and make up your own mind!

This article is based on some of the ideas that emerged during the production of a book which takes visualising as its focus. We began to identify problems which helped us to take a structured view of the purposes and skills of visualising.

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.

Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?

Sharon Walter, an NRICH teacher fellow, talks about her experiences of trying to embed NRICH tasks into her everyday practice.

Two sudokus in one. Challenge yourself to make the necessary connections.

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.