In a three-dimensional version of noughts and crosses, how many winning lines can you make?
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?
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. . . .
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 net of a cube is to be cut from a sheet of card 100 cm square.
What is the maximum volume cube that can be made from a single
piece of card?
Jo made a cube from some smaller cubes, painted some of the faces
of the large cube, and then took it apart again. 45 small cubes had
no paint on them at all. How many small cubes did Jo use?
What is the surface area of the tetrahedron with one vertex at O
the vertex of a unit cube and the other vertices at the centres of
the faces of the cube not containing O?
P is the midpoint of an edge of a cube and Q divides another edge
in the ratio 1 to 4. Find the ratio of the volumes of the two
pieces of the cube cut by a plane through PQ and a vertex.
There are thirteen axes of rotational symmetry of a unit cube. Describe them all. What is the average length of the parts of the axes of symmetry which lie inside the cube?
Can you work out the dimensions of the three cubes?
How can you change the surface area of a cuboid but keep its volume the same? How can you change the volume but keep the surface area the same?
Toni Beardon has chosen this article introducing a rich area for
practical exploration and discovery in 3D geometry
You have 27 small cubes, 3 each of nine colours. Use the small cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour.
This problem provides training in visualisation and representation
of 3D shapes. You will need to imagine rotating cubes, squashing
cubes and even superimposing cubes!
A description of how to make the five Platonic solids out of paper.