In this article for primary teachers, Fran describes her passion for paper folding as a springboard for mathematics.

When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'.

Can you find ways of joining cubes together so that 28 faces are visible?

Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?

Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

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?

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

What is the best way to shunt these carriages so that each train can continue its journey?

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

How will you go about finding all the jigsaw pieces that have one peg and one hole?

A hundred square has been printed on both sides of a piece of paper. What is on the back of 100? 58? 23? 19?

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

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

This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.

Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

What happens when you try and fit the triomino pieces into these two grids?

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

Design an arrangement of display boards in the school hall which fits the requirements of different people.

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

How many different triangles can you make on a circular pegboard that has nine pegs?

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

A toy has a regular tetrahedron, a cube and a base with triangular and square hollows. If you fit a shape into the correct hollow a bell rings. How many times does the bell ring in a complete game?

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

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?

One face of a regular tetrahedron is painted blue and each of the remaining faces are painted using one of the colours red, green or yellow. How many different possibilities are there?

For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

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.

Find your way through the grid starting at 2 and following these operations. What number do you end on?

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

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

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

Have a go at making a few of these shapes from paper in different sizes. What patterns can you create?

An extension of noughts and crosses in which the grid is enlarged and the length of the winning line can to altered to 3, 4 or 5.

Can you work out what shape is made when this piece of paper is folded up using the crease pattern shown?

Reasoning about the number of matches needed to build squares that share their sides.

Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?

You have been given three shapes made out of sponge: a sphere, a cylinder and a cone. Your challenge is to find out how to cut them to make different shapes for printing.

This second article in the series refers to research about levels of development of spatial thinking and the possible influence of instruction.