What does the overlap of these two shapes look like? Try picturing
it in your head and then use the interactivity to test your
Try to picture these buildings of cubes in your head. Can you make
them to check whether you had imagined them correctly?
Imagine a 3 by 3 by 3 cube. If you and a friend drill holes in some of the small cubes in the ways described, how many will have holes drilled through them?
Here are more buildings to picture in your mind's eye. Watch out -
they become quite complicated!
Imagine a 3 by 3 by 3 cube made of 9 small cubes. Each face of the large cube is painted a different colour. How many small cubes will have two painted faces? Where are they?
What shape is the overlap when you slide one of these shapes half
way across another? Can you picture it in your head? Use the
interactivity to check your visualisation.
Can you picture where this letter "F" will be on the grid if you
flip it in these different ways?
Imagine a pyramid which is built in square layers of small cubes. If we number the cubes from the top, starting with 1, can you picture which cubes are directly below this first cube?
Imagine a 4 by 4 by 4 cube. If you and a friend drill holes in some of the small cubes in the ways described, how many will not have holes drilled through them?
Bilbo goes on an adventure, before arriving back home. Using the
information given about his journey, can you work out where Bilbo
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?
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
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 ?
A visualisation problem in which you search for vectors which sum
to zero from a jumble of arrows. Will your eyes be quicker than
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?
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?
Investigate x to the power n plus 1 over x to the power n when x
plus 1 over x equals 1.
This problem provides training in visualisation and representation
of 3D shapes. You will need to imagine rotating cubes, squashing
cubes and even superimposing cubes!
In this problem we see how many pieces we can cut a cube of cheese
into using a limited number of slices. How many pieces will you be
able to make?
Takes you through the systematic way in which you can begin to
solve a mixed up Cubic Net. How close will you come to a solution?
Rowena has explained very clearly how she completed these Carroll
We received a number of different solutions to this puzzle.
Well done to Oliver for good problem-solving and reasoning.
Two by two matrices model the complex numbers and also the
quaternion number system.
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.
Use the differences to find the solution to this Sudoku.
Can you picture how to order the cards to reproduce Charlie's card trick for yourself?
The second in a series of articles on visualising and modelling shapes in the history of astronomy.
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!
The third installment in our series on the shape of astronomical systems, this article explores galaxies and the universe beyond our solar system.