Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
Can you maximise the area available to a grazing goat?
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?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
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. . . .
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
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. . . .
In the game of Noughts and Crosses there are 8 distinct winning
lines. How many distinct winning lines are there in a game played
on a 3 by 3 by 3 board, with 27 cells?
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. . . .
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?
A huge wheel is rolling past your window. What do you see?
A useful visualising exercise which offers opportunities for
discussion and generalising, and which could be used for thinking
about the formulae needed for generating the results on a
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?
Can you mark 4 points on a flat surface so that there are only two
different distances between them?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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?
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?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
A package contains a set of resources designed to develop pupils'
mathematical thinking. This package places a particular emphasis on
“visualising” and is designed to meet the needs. . . .
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
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?
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?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
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?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable.
Decide which of these diagrams are traversable.
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
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?
Draw a square. A second square of the same size slides around the
first always maintaining contact and keeping the same orientation.
How far does the dot travel?
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. . . .
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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 size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Triangle numbers can be represented by a triangular array of
squares. What do you notice about the sum of identical triangle
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?
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?
I found these clocks in the Arts Centre at the University of
Warwick intriguing - do they really need four clocks and what times
would be ambiguous with only two or three of them?
Is it true that any convex hexagon will tessellate if it has a pair
of opposite sides that are equal, and three adjacent angles that
add up to 360 degrees?
How many different ways can I lay 10 paving slabs, each 2 foot by 1
foot, to make a path 2 foot wide and 10 foot long from my back door
into my garden, without cutting any of the paving slabs?
ABC is an equilateral triangle and P is a point in the interior of
the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP
must be less than 10 cm.
Bilbo goes on an adventure, before arriving back home. Using the
information given about his journey, can you work out where Bilbo
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. . . .
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?
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.
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?
What can you see? What do you notice? What questions can you ask?