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All in the Mind
Imagine you are suspending a cube from one vertex and allowing it to hang freely. What shape does the surface of the water make around the cube?
Instant Insanity
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.
Is There a Theorem?
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?
Age 11 to 14
Challenge Level
Teachers may want to provide students with squared paper for this problem.
One way of introducing this problem is to challenge students to first find all the different ways of arranging two squares (dominoes), three squares (triominoes), four squares (tetrominoes), five squares (pentominoes) and six squares (hexominoes) - all the squares must touch at least one other square along one of its edges (with the edges lining up exactly).
| Number of Squares | Number of Arrangements |
| 2 | 1 |
| 3 | 2 |
| 4 | 5 |
| 5 | 12 |
| 6 | 35 |
This assumes that rotating or reflecting an arrangement does not produce a new arrangement.
Students could then be asked to look at the pentominoes to decide which will make cubes without lids.
They could then be asked to look at the hexominoes to decide which are the nets of cubes.
This problem could be followed up with Christmas Presents which asks students to consider cuboids.
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NRICH is part of the family of activities in the Millennium Mathematics Project.
NRICH is part of the family of activities in the Millennium Mathematics Project.