Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
What is the shape of wrapping paper that you would need to completely wrap this model?
A cube is made from smaller cubes, 5 by 5 by 5, then some of those
cubes are removed. Can you make the specified shapes, and what is
the most and least number of cubes required ?
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?
Lyndon Baker describes how the Mobius strip and Euler's law can
introduce pupils to the idea of topology.
Find the point whose sum of distances from the vertices (corners)
of a given triangle is a minimum.
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?
The second in a series of articles on visualising and modelling shapes in the history of astronomy.
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
The image in this problem is part of a piece of equipment found in the playground of a school. How would you describe it to someone over the phone?
A game for 2 people. Take turns joining two dots, until your opponent is unable to move.
This article explores ths history of theories about the shape of our planet. It is the first in a series of articles looking at the significance of geometric shapes in the history of astronomy.
Find all the ways to cut out a 'net' of six squares that can be
folded into a cube.
What 3D shapes occur in nature. How efficiently can you pack these shapes together?
Imagine a stack of numbered cards with one on top. Discard the top,
put the next card to the bottom and repeat continuously. Can you
predict the last card?
An introduction to bond angle geometry.
The reader is invited to investigate changes (or permutations) in the ringing of church bells, illustrated by braid diagrams showing the order in which the bells are rung.
A 3x3x3 cube may be reduced to unit cubes in six saw cuts. If after
every cut you can rearrange the pieces before cutting straight
through, can you do it in fewer?
A visualisation problem in which you search for vectors which sum
to zero from a jumble of arrows. Will your eyes be quicker than
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?
Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.
Can you make a tetrahedron whose faces all have the same perimeter?
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. . . .
This problem is about investigating whether it is possible to start at one vertex of a platonic solid and visit every other vertex once only returning to the vertex you started at.
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.
How efficiently can you pack together disks?
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?
Discover a way to sum square numbers by building cuboids from small
cubes. Can you picture how the sequence will grow?
Two boats travel up and down a lake. Can you picture where they
will cross if you know how fast each boat is travelling?
This task depends on groups working collaboratively, discussing and
reasoning to agree a final product.
This is the first article in a series which aim to provide some insight into the way spatial thinking develops in children, and draw on a range of reported research. The focus of this article is the. . . .
This is an interactive net of a Rubik's cube. Twists of the 3D cube become mixes of the squares on the 2D net. Have a play and see how many scrambles you can undo!
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?
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. . . .
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.
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. . . .
Mike and Monisha meet at the race track, which is 400m round. Just to make a point, Mike runs anticlockwise whilst Monisha runs clockwise. Where will they meet on their way around and will they ever. . . .
Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?
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?
In how many different ways can I colour the five edges of a
pentagon red, blue and green so that no two adjacent edges are the
Can you mentally fit the 7 SOMA pieces together to make a cube? Can
you do it in more than one way?
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?
A circular plate rolls in contact with the sides of a rectangular
tray. How much of its circumference comes into contact with the
sides of the tray when it rolls around one circuit?
This article outlines the underlying axioms of spherical geometry giving a simple proof that the sum of the angles of a triangle on the surface of a unit sphere is equal to pi plus the area of the. . . .
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?
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.
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?
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
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
On the 3D grid a strange (and deadly) animal is lurking. Using the tracking system can you locate this creature as quickly as possible?