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/www/nrich/html/content/03/02/penta2/
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2011-02-01T00:00:01
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<p>We had interlocking cubes (all the same size) in $10$ different
colours, up to $1000$ of each colour. We started with one yellow
cube. This was covered all over with a single layer of red
cubes:</p>
<mdo:image width="425" height="186" alt="diagram of one yellow cube being covered with a layer of red cubes with a pile of blue cubes to the side, not yet used" src="CubeLayers.gif"></mdo:image>
<p>This was then covered with a layer of blue cubes. Then came a
layer of green, followed by black, brown, white, orange, pink and
purple for as long as there were enough cubes of that colour to
cover the layer that came before.</p>
<mdo:image width="415" height="98" alt="piles of different coloured cubes" src="CubeLayer2.gif"></mdo:image>
<p>The unused cubes were put away. The many-layered cube was then
broken up and each colour made into cubes. These were just of the
one colour and the largest cubes possible made. For example, the
red layer made three $2\times 2\times2$ cubes with two $1\times
1\times1$ cubes left over, whereas the larger layers made much
larger cubes as well as smaller ones.</p>
<p>What colour was the largest cube that was made?</p>
<p>Which colour made into cubes had no $1\times 1\times1$
cubes?</p>
<p>Which colour was made into the most cubes including the $1\times
1\times1$ cubes?</p>
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<p>This problem was very well answered by Martha from Tattingstone
School:</p>
<p>We worked out this problem by finding out how many cubes would
cover 1 (yellow) cube, which was $3\times 3\times3$
(27). Then we found that a $5\times 5\times5$ cube
would be the next one and so on. The trouble was that the $3\times
3\times3$ cube was a totally different cube than the $1\times
1\times1$ cube. So with the 27 we had to take away 1 as the
$3\times 3\times3$ was only the skin. So in order to get the skin
each time it would be $5\times 5\times5$ - $3\times 3\times3$ etc.
Here are my results:</p>
<mdo:image width="300" height="279" src="marthasol1.gif" alt="yellow 1, red 26, blue 98, green 218, black 386, brown 602, white 866, orange 1178, pink 1538, purple 1946"></mdo:image>
<p>To answer the second part of question, when this large cube is
broken up, Martha drew another table which clearly shows how each
colour was made into the largest possible cubes:</p>
<mdo:image width="300" height="227" src="Marthasol2%20.gif" alt="ways of makin gsmaller cubes with each colour"></mdo:image>
<p>Martha continues:</p>
<p>The only problem with our results is that from the orange cubes
downwards (in the table) they are all over 1000 and it said in the
question "up to 1000 of each colour".</p>
<p>So, in answer to the three questions that were posed, Martha
says:</p>
<p>The colour of the largest cube that was made was white.<br></br>
The colour with no $1\times 1\times1$ cubes in it was black.<br></br>
The colour with the most cubes made out of it including the
$1\times 1\times1$ cubes was blue.</p>
<p>Bronya, also from Tattingstone sent in a good solution too -
well done.</p>
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<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
This is a tough <a href="http://nrich.maths.org/public/viewer.php?obj_id=1155&part=index">problem</a> for learners who relish the challenge of working with large and difficult numbers. It would become more accessible if calculators were used.<br></br>
<h3>Key questions</h3>
<div>Have you found out how cubes are needed to cover the single cube?</div>
<br></br>
<div>Have you remembered that there is only "up to $1000$ of each colour"?</div>
<br></br>
<div>What is the cube root of $1000$?</div>
<br></br>
<div>Have you made list of the cubes up to $1000$?</div>
<br></br>
<h3>Possible support</h3>
Suggest starting by finding how many cubes are needed to cover one (yellow) cube. This can be done practically with interlocking cubes.<br></br>
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$3\times 3\times3$ covers a single cube and a $5\times 5\times$
covers that.
<p>The cubes you need are:</p>
<p>$5^3=125$, $6^3=216$, $7^3=343$, $8^3=512$, $9^3=729$,
$11^3=1331$, $13^3=2197$, $15^3=3375$.</p>
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Cubes within Cubes
We start with one yellow cube and build around it to make a 3x3x3
cube with red cubes. Then we build around that red cube with blue
cubes and so on. How many cubes of each colour have we used?
Numbers and the Number System
Cube numbers
Calculations and Numerical Methods
Addition & subtraction
Using, Applying and Reasoning about Mathematics
Visualising
Mathematics Tools
Interlocking cubes