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?
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?
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?
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?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
What is the best way to shunt these carriages so that each train can continue its journey?
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?
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?
My cousin was 24 years old on Friday April 5th in 1974. On what day of the week was she born?
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?
Make your own double-sided magic square. But can you complete both sides once you've made the pieces?
How many different triangles can you make on a circular pegboard that has nine pegs?
Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?
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?
Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?
On a digital 24 hour clock, at certain times, all the digits are consecutive. How many times like this are there between midnight and 7 a.m.?
Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.
Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?
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. . . .
Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?
Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.
An activity making various patterns with 2 x 1 rectangular tiles.
These practical challenges are all about making a 'tray' and covering it with paper.
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?
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?
Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?
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?
In this matching game, you have to decide how long different events take.
Can you find all the different ways of lining up these Cuisenaire rods?
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?
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper.
Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the rules.
You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.
Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?
A Sudoku with clues as ratios or fractions.
Find out what a "fault-free" rectangle is and try to make some of your own.
Stuart's watch loses two minutes every hour. Adam's watch gains one minute every hour. Use the information to work out what time (the real time) they arrived at the airport.
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?
Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?
Two sudokus in one. Challenge yourself to make the necessary connections.
This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.
Can you find all the different triangles on these peg boards, and find their angles?