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2011-02-01T00:00:01
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<p>Here is a picture of the five Platonic solids:</p>
<p><mdo:image alt="platonic solids" height="393" src="platonic.gif" width="395"></mdo:image></p>
<p>Imagine you want to make each of the five Platonic solids and colour the faces so that, in every case, no two faces which meet along an edge have the same colour.</p>
<p>Can you find the least number of colours for which this is possible for each polyhedron.</p>
<p>How did you go about finding your solutions?<br></br>
<br></br>
If you'd like to make these solids out of paper, have a look at <a href="http://nrich.maths.org/5480">Ian Short's article</a>.</p>
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<p class="editorial">We received this solution from someone who
didn't give their name:</p>
<p>In a tetrahedron any two faces have a common edge so no two
faces can be the same colour. A tetrahedron needs 4 colours. If we
start by colouring one face, then the 3 faces adjoining it need 3
more colours.</p>
<p>A cube needs at least 3 colours because 3 faces meet at a point.
Three colours are sufficient because each pair of opposite faces
can be painted in one of the 3 colours.</p>
<p>An octahedron needs 2 colours. At each vertex 4 faces meet and
they can be painted in alternate colours.</p>
<p>A dodecahedron needs at least 4 colours because if we start by
colouring one face then we have to use 3 more colours to paint the
faces around it. The net shows how the dodecahedron can be painted
with 3 faces of each colour so 4 colours are sufficient.</p>
<mdo:image width="550" height="267" alt="dodecahedron net coloured with four colours" src="facepainting.gif"></mdo:image><br></br>
An icosahedron needs at least 3 colours because we have to use 3
colours to paint the 5 faces around each vertex. Three colours are
sufficient as shown in the net. <mdo:image width="500" height="250" src="icos.gif" alt="Net of icosahedron"></mdo:image><br></br>
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<h3><span style="font-weight: bold;">Why do this
problem?</span></h3>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=523&part=index">
This problem</a> is a good way to explore properties of the
Platonic solids and gives children opportunities to visualise 3D
shapes.</div>
<br></br>
<h3>Possible approach</h3>
<div>You might like to get children making their own Platonic
solids using sheets of paper before trying this
activity. <a href="http://nrich.maths.org/public/viewer.php?obj_id=5480&part=index">
This article</a> shows you how to go about it.</div>
<br></br>
<div>Alternatively (or in addition) it would be useful to have some
Polydron available for children to build their own solids as they
work.</div>
<br></br>
<div>You could introduce the problem by asking children to
visualise the cube, asking for justifications of the minimum number
of colours needed before they are able to physically make it to
check.</div>
<br></br>
<h3>Key questions</h3>
<div>How many faces/edges/vertices does this shape have?</div>
<div>How many other faces does each face touch?</div>
<br></br>
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<mdoxml version="1.0">How many faces/edges/vertices does this shape have? <br></br>
How many other faces does each face touch?<br></br>
Have you tried drawing a net of the shape?<br></br>
It might help to make the shape out of Polydron, or perhaps you can
find some solid shapes to work with?<br></br></mdoxml>
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Face Painting
You want to make each of the 5 Platonic solids and colour the faces
so that, in every case, no two faces which meet along an edge have
the same colour.
Using, Applying and Reasoning about Mathematics
Visualising
Mathematics Tools
Sets of shapes
3D Geometry, Shape and Space
Regular polyhedra
3D Geometry, Shape and Space
Nets
Mathematics Tools
Polydron