More of Carrie's cubes

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Purpose

This problem solving activity has a geometry focus.

Achievement Objectives
GM4-5: Identify classes of two- and three-dimensional shapes by their geometric properties.
Student Activity

Four small cubes.Carrie makes a cube by sticking together 27 small cubes.

Can she colour the exposed faces of the original cubes red, white, blue and yellow, so that no faces with a common edge have the same colour?

Does she need the yellow?

Can she get away with just red and white?

 

Specific Learning Outcomes
  • Apply the properties of 3-D objects to solve a problem.
  • Visualise 3-D objects.
  • Devise and use problem solving strategies to explore situations mathematically (be systematic).
Description of Mathematics

This problem explores the shape of a cube and ways that parts of it can be coloured.

For a similar problem see Carrie’s First Cubes, Geometry, Level 4.

There are links here to a famous problem called the Four Colour Problem that was solved in 1976 after having been around for about 126 years. This problem asked how many colours were needed to colour any map. The restriction here was that no two countries with a common border were allowed to be coloured in the same colour. Two mathematicians in America, Appel and Haken, were able to prove that four colours were all that were needed.

Questions such as the Four Colour Problem have important applications. Using them we are able to see when certain routing problems (for buses or milk trucks, for instance) can be solved. They are an important part of topics called graph theory and operations research.

Activity

The Problem

Carrie makes a cube by sticking together 27 small cubes. Can she colour the exposed faces of the original cubes red, white, blue and yellow, so that no faces with a common edge have the same colour?

Does she need the yellow?

Can she get away with just red and white?

Teaching Sequence

  1. Introduce the problem by looking at Rubik’s cube and discussing the colours used and reasons for this.
  2. Pose the problem to the class. Discuss the shape of the cube and how they could easily see all faces of the cube at the same time (e.g. displaying the net of the cube).
  3. Give the students time to think before asking them to share their ideas for proceeding with the problem.
  4. As the students work on the problem ask them to share the strategy they are using and their reasons for the approach taken.
  5. Share solutions.

Solution

Here is a 3-colouring. There are lots of ways of using 4 colours. Can you find any other ways of 3-colouring the cube?

      313   
      232   
      313   
121212121212
212121212121
121212121212
      313   
      232   
      313   

Here 1 = red, 2 = white and 3 = blue.

To find a 3-colouring it was easy first of all to just use two colours across the middle band of the net above. The difficulty was fixing up the top and bottom faces to "match" the middle band of faces.

It’s not possible to just use two colours. Look at one of the small cubes at a corner of the big cube. This will have three faces all touching each other. That means that they must all be coloured differently.

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Level Four