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 could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
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?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
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 huge wheel is rolling past your window. What do you see?
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?
In how many ways can you fit all three pieces together to make
shapes with line symmetry?
Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
How many different symmetrical shapes can you make by shading triangles or squares?
Starting with four different triangles, imagine you have an
unlimited number of each type. How many different tetrahedra can
you make? Convince us you have found them all.
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
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. . . .
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 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.
A bus route has a total duration of 40 minutes. Every 10 minutes,
two buses set out, one from each end. How many buses will one bus
meet on its way from one end to the other end?
Here is a solitaire type environment for you to experiment with. Which targets can you reach?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Can you describe this route to infinity? Where will the arrows take you next?
A Hamiltonian circuit is a continuous path in a graph that passes through each of the vertices exactly once and returns to the start.
How many Hamiltonian circuits can you find in these graphs?
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?
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 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?
Which hexagons tessellate?
Four rods, two of length a and two of length b, are linked to form
a kite. The linkage is moveable so that the angles change. What is
the maximum area of the kite?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Use the interactivity to play two of the bells in a pattern. How do
you know when it is your turn to ring, and how do you know which
bell to ring?
Show that among the interior angles of a convex polygon there
cannot be more than three acute angles.
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?
Use the interactivity to listen to the bells ringing a pattern. Now
it's your turn! Play one of the bells yourself. How do you know
when it is your turn to ring?
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
Mathematics is the study of patterns. Studying pattern is an
opportunity to observe, hypothesise, experiment, discover and
This article for teachers discusses examples of problems in which
there is no obvious method but in which children can be encouraged
to think deeply about the context and extend their ability to. . . .
The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?
The aim of the game is to slide the green square from the top right
hand corner to the bottom left hand corner in the least number of
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?
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
Lyndon Baker describes how the Mobius strip and Euler's law can
introduce pupils to the idea of topology.
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. . . .
Can you maximise the area available to a grazing goat?
These are pictures of the sea defences at New Brighton. Can you
work out what a basic shape might be in both images of the sea wall
and work out a way they might fit together?
A game for 2 players
Can you mark 4 points on a flat surface so that there are only two
different distances between them?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Can you mentally fit the 7 SOMA pieces together to make a cube? Can
you do it in more than one way?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
What is the shape of wrapping paper that you would need to completely wrap this model?