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?
It is possible to dissect any square into smaller squares. What is
the minimum number of squares a 13 by 13 square can be dissected
Can you maximise the area available to a grazing goat?
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
A right-angled isosceles triangle is rotated about the centre point
of a square. What can you say about the area of the part of the
square covered by the triangle as it rotates?
ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.
Seven small rectangular pictures have one inch wide frames. The
frames are removed and the pictures are fitted together like a
jigsaw to make a rectangle of length 12 inches. Find the dimensions
of. . . .
Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3. . . .
Find the ratio of the outer shaded area to the inner area for a six
pointed star and an eight pointed star.
ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP
: PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED.
What is the area of the triangle PQR?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
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. . . .
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?
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?
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?
How efficiently can you pack together disks?
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. . . .
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
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?
A 10x10x10 cube is made from 27 2x2 cubes with corridors between
them. Find the shortest route from one corner to the opposite
A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
What would be the smallest number of moves needed to move a Knight
from a chess set from one corner to the opposite corner of a 99 by
99 square board?
Square It game for an adult and child. Can you come up with a way of always winning this game?
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. . . .
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9,
12, 15... other squares? 8, 11, 14... other squares?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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.
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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. . . .
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?
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?
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.
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?
Can you work out the dimensions of the three cubes?
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?
Imagine you are suspending a cube from one vertex (corner) and
allowing it to hang freely. Now imagine you are lowering it into
water until it is exactly half submerged. What shape does the
surface. . . .
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?
There are 27 small cubes in a 3 x 3 x 3 cube, 54 faces being
visible at any one time. Is it possible to reorganise these cubes
so that by dipping the large cube into a pot of paint three times
you. . . .
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?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
To avoid losing think of another very well known game where the
patterns of play are similar.
Place a red counter in the top left corner of a 4x4 array, which is
covered by 14 other smaller counters, leaving a gap in the bottom
right hand corner (HOME). What is the smallest number of moves. . . .
A square of area 3 square units cannot be drawn on a 2D grid so that each of its vertices have integer coordinates, but can it be drawn on a 3D grid? Investigate squares that can be drawn.
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?
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.
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. . . .
Two motorboats travelling up and down a lake at constant speeds
leave opposite ends A and B at the same instant, passing each
other, for the first time 600 metres from A, and on their return,
400. . . .
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?