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<p>Consider a cube and paths
along the edges of the cube.</p>
<p>Mark one vertex red.</p>
<p style="text-align: center;"><mdo:image width="253" height="258" alt="wire" src="wireCube.jpg"></mdo:image></p>
<p>Colour other vertices <span style="text-decoration: underline; font-weight: bold;">red</span>
if they can be reached from a red vertex by travelling along an
<span style="text-decoration: underline; font-weight: bold;">even</span>
number of edges of the cube.</p>
<p>Colour vertices <span style="text-decoration: underline; font-weight: bold;">blue</span>
if they can be reached by travelling along an <span style="text-decoration: underline; font-weight: bold;">odd</span>
number of edges from a red vertex.</p>
<p>Is it possible to have
vertices which are both red and blue at the same time (call these
redblue vertices)?</p>
<p>Now do the same for a
tetrahedron.</p>
<p style="text-align: center;"><mdo:image width="148" height="133" src="wireTetra.jpg" alt=""></mdo:image></p>
<p>Do the same for other
solids, for example the octahedron, dodecahedron and icosahedron,
and prisms with different cross sections. Remember the paths must
be along the edges of the solids. Decide how to record what you
find. What property does the solid need if it is to have redblue
vertices?</p>
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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=761&part=">This activity</a> is very good to have pupils explore some special properties of solid shapes. It is particularly good in that the solutions do not appear very obviously.</div>
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<h3>Possible approach</h3>
<div>Introduce the pupils gradually to the recognition of what are vertices and what are edges. Most pupils will then be needing to handle the shapes unless they are gifted in visulaisation.</div>
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<h3>Key questions</h3>
<div>Tell me about your shape.</div>
<div>What vertices have you used?</div>
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<h3>Possible extension</h3>
<div>Extensions are included in the last part of the activity.</div>
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<h3>Possible support</h3>
<div>Shapes made from pipe-cleaners and straws can be a <a href="/content/01/02/six6/Pipe%20Cleaners%26Strawsnewlogo.doc">useful aid</a>.</div>
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<div style="text-align: center;"><mdo:image width="372" height="165" alt="shapes" src="shapes.gif"></mdo:image></div>
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<p class="editorial">To solve this tough nut, start by colouring
one vertex red. In the cube there are different paths from the red
to the blue vertex along the edges but all these paths have an odd
number of edges so this is a blue vertex and not a redblue one. By
contrast the vertex at the top of the tetrahedron is redblue
because it can be reached by going along one edge (an odd number)
and also by going along two edges (an even number).</p>
<p class="editorial">Simon and Nick sent in their results from
looking at several other solids.</p>
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<mdo:image width="680" height="140" alt="Solution" src="Redblue%20table%20solution.gif"></mdo:image><br></br>
"This shows that all shapes either have no redblue corners, or they
are all redblue. This is because if one of them is redblue then all
the ones next to it must be redblue too because they can't just be
red or blue. Alternate corners have to be red, then blue, then red
etc. If one vertex is redblue, then all the ones next to it are
both an odd number and an even number away from a red one, so they
are redblue too. That means they all must be, because all the
corners are connected to each other".<br></br>
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<p class="editorial">Unfortunately no-one has been able to spot the
key feature which determines whether a shape has all redblue, or no
redblue vertices. It is easier to explain if we look at the shape
of the faces of some of these solids. All of the ones with redblue
vertices have at least some faces that have an odd number of sides,
i.e. triangles, pentagons etc. This means that if you just look at
one face, you can reach any vertex by two different paths, say
clockwise or anticlockwise. One of these will always be odd, and
the other will always be even, because they have to add up to an
odd number of sides altogether.</p>
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<div style="text-align: center;"><mdo:image width="350" height="150" alt="redbluediag" src="Redblue%20diagram.gif"></mdo:image></div>
<div style="text-align: center;"></div>
<p><span class="editorial">The solid only has to have at least one
face with an odd number of edges, because as Simon and Nick say, as
long as one vertex is redblue then all the others must also be
redblue. If all of the faces have an even number of edges, then no
matter which route you take, each vertex can only be red or blue,
never both. This is because the two routes always have to add up to
an even number, so must be either both odd, or both even. Thank you
Simon and Nick!</span></p>
<div class="editorial"></div>
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2
5
0
1
0
0
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Redblue
Investigate the number of paths you can take from one vertex to
another in these 3D shapes. Is it possible to take an odd number
and an even number of paths to the same vertex?
3D Geometry, Shape and Space
Regular polyhedra
Decision Mathematics and Combinatorics
Networks/Graph Theory
Using, Applying and Reasoning about Mathematics
Investigations
Using, Applying and Reasoning about Mathematics
Visualising