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?

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?

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

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

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?

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?

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?

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?

Can you work out how to balance this equaliser? You can put more than one weight on a hook.

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

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

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

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

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

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?

An activity making various patterns with 2 x 1 rectangular tiles.

Hover your mouse over the counters to see which ones will be removed. Click to remover 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?

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?

Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.

Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.

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?

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

These practical challenges are all about making a 'tray' and covering it with paper.

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

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?

In how many ways could Mrs Beeswax put ten coins into her three puddings so that each pudding ended up with at least two coins?

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. . . .

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?

In this problem it is not the squares that jump, you do the jumping! The idea is to go round the track in as few jumps as possible.

How many different rhythms can you make by putting two drums on the wheel?

How many models can you find which obey these rules?

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!

My briefcase has a three-number combination lock, but I have forgotten the combination. I remember that there's a 3, a 5 and an 8. How many possible combinations are there to try?

There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs.

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?

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

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.

Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99 How many ways can you do it?

I was in my car when I noticed a line of four cars on the lane next to me with number plates starting and ending with J, K, L and M. What order were they in?

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

Using the cards 2, 4, 6, 8, +, - and =, what number statements can you make?

Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.

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?

This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!