991 /www/nrich/html/content/00/03/penta3/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>There are eight small cubes. Each face of each cube is to be painted either green or yellow but each cube must use each of the two colours somehow.</p> <p><mdo:image alt="GrYell" height="126" src="GrYell991.jpg" width="521"></mdo:image></p> <p> </p> <p>Work out how to paint the faces so that the cubes can be put together to make a $2$ by $2$ cube that is green all over AND can be rearranged to make a $2$ by $2$ cube that is yellow all over.</p> <div><mdo:image alt="Two Cubes" height="" src="fig2.gif" width=""></mdo:image></div> <br></br> <br></br> <p>Now work out how to paint the faces to make a $3$ by $3$ green cube and a $3$ by $3$ yellow cube.</p> <p> </p> <mdo:image alt="Cube" height="141" src="fig3.gif" width="145"></mdo:image><br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <strong>For the $2$ by $2$ cube</strong><br></br> <p><span class="editorial">Idil (Private IRMAK Primary &amp; Secondary School, Istanbul, Turkey) says:</span></p> <div>All of them must be the same. Three faces with one corner in common will be yellow, the other three faces with the diagonal corner will be green.</div> <div><mdo:image height="84" width="232" src="fig4.gif" alt=""></mdo:image></div> <br></br> <p><span class="editorial">Jason (Priory School UK), Christina (Malborough Primary School, UK), and Dogan</span> <span class="editorial">(Private IRMAK Primary &amp; Secondary School, Istanbul, Turkey) drew a net of a cube to show the colour of each face.</span></p> <mdo:image height="126" width="95" src="fig5.gif" alt=""></mdo:image><br></br> <p><strong>For the $3$ by $3$ cube</strong></p> <p><span class="editorial">Dogan Private IRMAK Primary &amp; Secondary School, Istanbul, Turkey) explains that:</span></p> We can paint corners the same as we did before. Then we can paint cubes in the middle like two sides yellow, two sides green - you don't need to paint the other sides. The following figure shows an opened middle cube and how it should be painted:<br></br> <br></br> <mdo:image height="167" width="364" src="fig6.gif" alt=""></mdo:image><br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <div><a href="http://nrich.maths.org/public/viewer.php?obj_id=991&amp;part=index"> This activity</a> is very good for giving pupils the opportunity to explore spatial properties, particularly properties of cubes. It is also an opportunity to develop their skills in perseverance and extending challenges for themselves.</div> <br></br> <h3>Possible approach</h3> <div>For some pupils there will be a need to show how a cube can be coloured in different ways. So, you may find it useful to have a large cube made out of cardboard as an example, or a blank dice and coloured stickers. </div> <br></br> <h3>Key questions</h3> <div>Tell me about the cubes you've got.</div> <div>What colours are the faces of this cube?</div> <div>How many yellows here?</div> <br></br> <h3>Possible extension</h3> <div>Children could change the 'rules' slightly and decide on some simple patterns of colours that should show on each face. Like this one:</div> <div style="text-align: center;"><mdo:image height="133" width="123" src="GrYell991%20Ext.jpg" alt="ext"></mdo:image></div> <br></br> <h3>Possible support</h3> <div>Little cubes with small coloured squares available for pupils to choose to use may be helpful for some.</div> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> For the 2 x 2 cube <br></br> Idil (Private IRMAK Primary &amp; Secondary School, Istanbul, Turkey) says, &quot;All of them must be the same. Three faces with one corner in common will be yellow, the other three faces with the diagonal corner will be green&quot;. <br></br> <br></br> <mdo:image width="232" height="84" src="fig4.gif" alt="fig4"></mdo:image><br></br> Jason (Priory School UK), Christina (Malborough Primary School, UK), and Dogan (Private IRMAK Primary &amp; Secondary School, Istanbul, Turkey) drew a net of a cube to show the colour of each face. <br></br> <br></br> <mdo:image width="95" height="126" src="fig5.gif" alt="fig5"></mdo:image><br></br> For the 3 x 3 cube <br></br> Dogan (Private IRMAK Primary &amp; Secondary School, Istanbul, Turkey) explains that &quot;We can paint corners the same as we did before. Then we can paint cubes in the middle like 2 sides yellow, 2 sides green - you don't need to paint the other sides. The following figure shows an opened middle cube and how it should be painted&quot;. <br></br> <br></br> <mdo:image width="364" height="167" src="fig6.gif" alt="fig6"></mdo:image><br></br></mdoxml> 2 4 0 1 0 0 0 How can you paint the faces of these eight cubes so they can be put together to make a 2 x 2 cube that is green all over AND a 2 x 2 cube that is yellow all over? 3D Geometry, Shape and Space Cubes Using, Applying and Reasoning about Mathematics Visualising Mathematics Tools Interlocking cubes