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?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
How will you go about finding all the jigsaw pieces that have one peg and one hole?
Think of a number, square it and subtract your starting number. Is the number you’re left with odd or even? How do the images help to explain this?
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?
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?
Can you find ways of joining cubes together so that 28 faces are visible?
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?
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?
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 many different triangles can you make on a circular pegboard that has nine pegs?
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?
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?
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?
How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?
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?
This article for teachers describes how modelling number properties involving multiplication using an array of objects not only allows children to represent their thinking with concrete materials,. . . .
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?
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?
Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time?
This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.
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?
A game for 1 or 2 people. Use the interactive version, or play with friends. Try to round up as many counters as 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?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?
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'.
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
Exploring and predicting folding, cutting and punching holes and making spirals.
Have a go at this 3D extension to the Pebbles problem.
What shape has Harry drawn on this clock face? Can you find its area? What is the largest number of square tiles that could cover this area?
Can you cut up a square in the way shown and make the pieces into a triangle?
A game has a special dice with a colour spot on each face. These three pictures show different views of the same dice. What colour is opposite blue?
How can you paint the faces of these eight cubes so they can be put together to make a 2 x 2 x 2 cube that is green all over AND a 2 x 2 x 2 cube that is yellow all over?
What is the greatest number of squares you can make by overlapping three squares?
Make a cube out of straws and have a go at this practical challenge.
What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?
This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?
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.
Reasoning about the number of matches needed to build squares that share their sides.
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.
Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?
A game for 2 players. Given a board of dots in a grid pattern, players take turns drawing a line by connecting 2 adjacent dots. Your goal is to complete more squares than your opponent.
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?
What happens when you turn these cogs? Investigate the differences between turning two cogs of different sizes and two cogs which are the same.
Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.