A cheap and simple toy with lots of mathematics. Can you interpret
the images that are produced? Can you predict the pattern that will
be produced using different wheels?
Can you maximise the area available to a grazing goat?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
In a right angled triangular field, three animals are tethered to posts at the midpoint of each side. Each rope is just long enough to allow the animal to reach two adjacent vertices. Only one animal. . . .
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
Find the ratio of the outer shaded area to the inner area for a six
pointed star and an eight pointed star.
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
A bicycle passes along a path and leaves some tracks. Is it
possible to say which track was made by the front wheel and which
by the back wheel?
How many different symmetrical shapes can you make by shading triangles or squares?
Here are four tiles. They can be arranged in a 2 by 2 square so that this large square has a green edge. If the tiles are moved around, we can make a 2 by 2 square with a blue edge... Now try to. . . .
A tilted square is a square with no horizontal sides. Can you
devise a general instruction for the construction of a square when
you are given just one of its sides?
Can you work out the dimensions of the three cubes?
A spider is sitting in the middle of one of the smallest walls in a
room and a fly is resting beside the window. What is the shortest
distance the spider would have to crawl to catch the fly?
How efficiently can you pack together disks?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
Imagine a large cube made from small red cubes being dropped into a
pot of yellow paint. How many of the small cubes will have yellow
paint on their faces?
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle
contains 20 squares. What size rectangle(s) contain(s) exactly 100
squares? Can you find them all?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
A 10x10x10 cube is made from 27 2x2 cubes with corridors between
them. Find the shortest route from one corner to the opposite
Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
A game for 2 people. Take turns joining two dots, until your opponent is unable to move.
A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be?
A rectangular field has two posts with a ring on top of each post.
There are two quarrelsome goats and plenty of ropes which you can
tie to their collars. How can you secure them so they can't. . . .
A blue coin rolls round two yellow coins which touch. The coins are
the same size. How many revolutions does the blue coin make when it
rolls all the way round the yellow coins? Investigate for a. . . .
Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?
You can move the 4 pieces of the jigsaw and fit them into both
outlines. Explain what has happened to the missing one unit of
See if you can anticipate successive 'generations' of the two
animals shown here.
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
Four rods are hinged at their ends to form a convex quadrilateral.
Investigate the different shapes that the quadrilateral can take.
Be patient this problem may be slow to load.
Glarsynost lives on a planet whose shape is that of a perfect
regular dodecahedron. Can you describe the shortest journey she can
make to ensure that she will see every part of the planet?
Two angles ABC and PQR are floating in a box so that AB//PQ and BC//QR. Prove that the two angles are equal.
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?
Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?
Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9,
12, 15... other squares? 8, 11, 14... other squares?
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
A half-cube is cut into two pieces by a plane through the long diagonal and at right angles to it. Can you draw a net of these pieces? Are they identical?
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?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.
Imagine starting with one yellow cube and covering it all over with
a single layer of red cubes, and then covering that cube with a
layer of blue cubes. How many red and blue cubes would you need?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle?
This is a simple version of an ancient game played all over the world. It is also called Mancala. What tactics will increase your chances of winning?
A and C are the opposite vertices of a square ABCD, and have
coordinates (a,b) and (c,d), respectively. What are the coordinates
of the vertices B and D? What is the area of the square?
P is a point on the circumference of a circle radius r which rolls,
without slipping, inside a circle of radius 2r. What is the locus
Have a go at this 3D extension to the Pebbles problem.
The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it. . . .
This task depends on groups working collaboratively, discussing and
reasoning to agree a final product.