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You have 27 small cubes, 3 each of nine colours.<br></br>
<br></br>
Use the small cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour.<br></br>
<br></br>
<a href="/content/01/04/six1/squareso.swf">Full Screen Version.</a><br></br>
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<p><span style="font-style: italic;">This problem features in Maths Trails - Working Systematically, one of the books in the Maths Trails series written by members of the NRICH Team and published by Cambridge University Press. For more details, please see our</span> <a href="http://nrich.maths.org/public/viewer.php?obj_id=4833&part=index" style="font-style: italic;">publications page</a>
<span style="font-style: italic;">.</span></p>
<p> </p>
<a href="http://nrich.maths.org/7207&part=">Click here for a poster of this problem</a>.<br></br></mdoxml>
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<p class="editorial">We have received a very elegant strategy for
solving this problem from Tom and Adam, of Alcester Grammar School
in England:</p>
<p>We have found a solution to this puzzle. Our solution works,
because we selected all the nine colours for the bottom face. Their
position did not matter as long as there was one of each on the
base.</p>
<p>For each colour, we then moved up one level, right one space (if
they were on the far right already, they moved to the left row),
and forward once (if they were at the back already, they moved to
the front row).</p>
<p>We repeated the same procedure to fill the top level.</p>
<p>This ensured that there was one of each colour in each row,
column, and on each level. Because of this, each of the faces could
only contain one of each colour, as when we consider a whole face,
all of the colours will just shift one place, inwards, and move off
the row and level that they were on before. Also, as every colour
was on the initial face, there would have to be every single colour
on all the other faces too.</p>
<p>Thank you Tom and Adam for such a neat solution.</p>
<p>Anybody who would like to check that the procedure works can
return to the problem page and try the strategy out for
themselves.</p>
<div class="editorial">Chris from Alcester Grammar School sent us
his solution:</div>
<br></br>
<div><mdo:image width="420" height="263" src="Chris%20nine%20colours.gif" alt="ccchris' solution"></mdo:image></div>
<br></br>
<div class="editorial">And so did Giancarlo from Haut-Lac
School:</div>
<br></br>
<div><mdo:image width="401" height="266" src="Giancarlo%20nine%20colours.gif" alt="Giancarlo's solution"></mdo:image></div>
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<h3>Why do this problem?</h3>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=768&part=">
This is an engaging problem</a> that challenges students to work in
3 dimensions and to use different representations of the cube. It
can be used to encourage students to persevere, collaborate, work
systematically and reason logically.</div>
<br></br>
<h3>Possible approach</h3>
<div>Introduce the task by
challenging the students to create an anti-rubik's cube.</div>
<div>Offer them multilink
cubes (plastic coloured cubes that fix together), pencil and paper
and the computer interactivity so that they have a choice of ways
in which to approach the problem.</div>
<br></br>
Let students pursue their own attempts to orientate themselves
within this context, but attention may be drawn, at well-judged
moments, to the number of faces that cubes in individual positions
will have 'visible'.<br></br>
<h3>Key questions</h3>
<ul>
<li>Some of the 27 cubes have faces that are invisible from the
'outside' of the large cube. How many cubes have no 'visible'
faces? One face visible? Two faces visible? Three faces
visible?<br></br>
</li>
<li>If one colour appears in a corner, where will the other two
cubes of the same colour need to appear?<br></br>
</li>
<li>There will be a cube of some colour at the centre. Where else
will cubes of that colour need to be positioned?</li>
</ul>
<h3>Possible extension</h3>
<div>If students have chosen
how to solve the problem from a range of possibilities (ie.
multilink cubes, pencil and paper or the computer interactivity)
challenge them to solve the puzzle again from scratch using a
different approach.</div>
<br></br>
<h3>Possible support</h3>
<div>Students could attempt <a href="http://nrich.maths.org/public/viewer.php?obj_id=2322&part=">
Painted Cube</a> <a href="http://nrich.maths.org/public/viewer.php?obj_id=895&part=">
</a> before trying this problem.</div>
<div> </div>
<div> </div>
<div>Handouts for teachers are
available here (<a href="/content/01/04/six1/Nine%20Colours.doc">word document</a>, <a href="/content/01/04/six1/Nine%20Colours.pdf">pdf document</a>),
with the problem on one side and the notes on the
other. </div>
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Each colour must appear on all six faces of the larger cube.<br></br>
<br></br>
Small cubes can be placed <br></br>
<ul>
<li>at the corners of the large cube,</li>
</ul>
<ul>
<li>on the edges of the large cube,</li>
</ul>
<ul>
<li>in the middle of the faces of the large cube,</li>
</ul>
<ul>
<li>or at the very centre of the large cube.</li>
</ul>
How many faces of the small cube will be visible in each of these
different positions?<br></br>
<br></br>
A small cube will need to go in at the very centre of the larger
cube. <br></br>
Where will the other two small cubes of the same colour go?<br></br>
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Nine Colours
You have 27 small cubes, 3 each of nine colours. Use the small
cubes to make a 3 by 3 by 3 cube so that each face of the bigger
cube contains one of every colour.
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