What is the best way to shunt these carriages so that each train can continue its journey?
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?
Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?
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?
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?
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. . . .
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
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 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.
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?
Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.
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?
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 greatest number of counters you can place on the grid below without four of them lying at the corners of a square?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
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?
How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?
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?
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?
What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
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 ...
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
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,. . . .
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 see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
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 triangles can you make on a circular pegboard that has nine pegs?
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?
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 rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?
Exchange the positions of the two sets of counters in the least possible number of moves
Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?
Can you fit the tangram pieces into the outline of this plaque design?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Can you fit the tangram pieces into the outline of these rabbits?
Can you fit the tangram pieces into the outline of the rocket?
Can you fit the tangram pieces into the outline of this junk?
Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.
Can you fit the tangram pieces into the outline of this goat and giraffe?
Can you fit the tangram pieces into the outline of this sports car?
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
Can you fit the tangram pieces into the outline of these convex shapes?
Here's a simple way to make a Tangram without any measuring or ruling lines.
We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?
How can the same pieces of the tangram make this bowl before and after it was chipped? Use the interactivity to try and work out what is going on!
Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.