Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
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?
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....?
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?
What is the best way to shunt these carriages so that each train can continue its journey?
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?
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?
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?
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?
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?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
A student in a maths class was trying to get some information from her teacher. She was given some clues and then the teacher ended by saying, "Well, how old are they?"
Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.
Given the products of adjacent cells, can you complete this Sudoku?
An activity making various patterns with 2 x 1 rectangular tiles.
How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?
Find the smallest whole number which, when mutiplied by 7, gives a product consisting entirely of ones.
Make your own double-sided magic square. But can you complete both sides once you've made the pieces?
If you take a three by three square on a 1-10 addition square and multiply the diagonally opposite numbers together, what is the difference between these products. Why?
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?
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.
These practical challenges are all about making a 'tray' and covering it with paper.
Find the values of the nine letters in the sum: FOOT + BALL = GAME
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?
Each clue in this Sudoku is the product of the two numbers in adjacent cells.
Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?
When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?
Whenever a monkey has peaches, he always keeps a fraction of them each day, gives the rest away, and then eats one. How long could he make his peaches last for?
Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
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?
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?
A cinema has 100 seats. Show how it is possible to sell exactly 100 tickets and take exactly £100 if the prices are £10 for adults, 50p for pensioners and 10p for children.
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. . . .
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.
These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.
Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.
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?
How many different triangles can you make on a circular pegboard that has nine pegs?
If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?
Use the clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.
I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?
Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?
In this matching game, you have to decide how long different events take.
The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.
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 how many ways can you stack these rods, following the rules?