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/www/nrich/html/content/98/04/bbprob2/
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2011-02-01T00:00:01
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<p>I don't know whether you've had the same problem as me. I've
found that I have loads of CDs and DVDs and they are not very tidy.
I decided to collect them all together in their boxes and after
putting them all the same way up I started putting them together
into cuboid-type shapes. I found when I started that just two could
be put together to give three different cuboids. Here's what they
looked like:-</p>
<p style="text-align: center;"><mdo:image width="325" height="306" alt="pic1" src="Picture%201.jpg"></mdo:image></p>
<br></br>
<p>I've used different colours to show each box, just to make it
clearer to see, but we're going to imagine each box is identical.
</p>
<p>I then tried it with an extra box making three altogether. I
moved them around and found that again I could make three different
cuboids. These are the ones that I found:-</p>
<p style="text-align: center;"><mdo:image width="443" height="346" alt="pic2" src="Picture%202.jpg"></mdo:image></p>
<br></br>
<p>But, perhaps like you, I've got lots of these boxes of CDs and
DVDs. So I moved on to using four. Again I found three but realised
that there were more that I could find. Here are the three:-</p>
<p style="text-align: center;"><mdo:image width="552" height="305" alt="pic3" src="Picture%203.jpg"></mdo:image></p>
<br></br>
<p>Well, you could try the same ideas. If you have not got CD/DVD
boxes it does not matter. You could use old match boxes or books
that are all the same size. In fact you can use any objects as long
as they are cuboid shaped and you have plenty of them that are
exactly the same. All you have to do is to find out how many
different cuboids you can make when you use fourof them, then five
of them, then six <em>etc.</em></p>
<p>But notice that when you've put them together it does not matter
which way up they are, they count as being the same. For example
these three would count as being the same:-</p>
<p style="text-align: center;"><mdo:image width="513" height="235" alt="pic4" src="Picture%204.jpg"></mdo:image></p>
<br></br>
<p>Now that you've tried some, how about getting other friends and
people at home to have a go and see what they all come up with -
maybe you could work together?</p>
<p>Well, Good Luck.</p>
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<p class="editorial">No real ideas came forward as to what people
did get from this exploration. We are interested in any work that
proceeds from looking at the challenges - so, let us know if you
can.</p>
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<h2>Little Boxes</h2>
<br></br>
<p>I don't know whether you've had the same problem as me. I've found that I have loads of CDs and DVDs and they are not very tidy. I decided to collect them all together in their boxes and after putting them all the same way up I started putting them together into cuboid-type shapes. I found when I started that just two could be put together to give three different cuboids. Here's what they
looked like:-</p>
<p style="text-align: center;"><mdo:image alt="pic1" height="306" src="Picture%201.jpg" width="325"></mdo:image></p>
<br></br>
<p>I've used different colours to show each box, just to make it clearer to see, but we're going to imagine each box is identical.</p>
<p>I then tried it with an extra box making three altogether. I moved them around and found that again I could make three different cuboids. These are the ones that I found:-</p>
<p style="text-align: center;"><mdo:image alt="pic2" height="346" src="Picture%202.jpg" width="443"></mdo:image></p>
<br></br>
<p>But, perhaps like you, I've got lots of these boxes of CDs and DVDs. So I moved on to using four. Again I found three but realised that there were more that I could find. Here are the three:-</p>
<p style="text-align: center;"><mdo:image alt="pic3" height="305" src="Picture%203.jpg" width="552"></mdo:image></p>
<br></br>
<p>Well, you could try the same ideas. If you have not got CD/DVD boxes it does not matter. You could use old match boxes or books that are all the same size. In fact you can use any objects as long as they are cuboid shaped and you have plenty of them that are exactly the same. All you have to do is to find out how many different cuboids you can make when you use fourof them, then five of them,
then six <em>etc.</em></p>
<p>But notice that when you've put them together it does not matter which way up they are, they count as being the same. For example these three would count as being the same:-</p>
<p style="text-align: center;"><mdo:image alt="pic4" height="235" src="Picture%204.jpg" width="513"></mdo:image></p>
<br></br>
<p>Now that you've tried some, how about getting other friends and people at home to have a go and see what they all come up with - maybe you could work together?</p>
<p>Well, Good Luck.</p>
</div>
<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=41&part=index">This enjoyable activity</a> helps children develop their spatial awareness and their concepts surrounding cuboids.</div>
<br></br>
<h3>Possible approach</h3>
<div>This activity is probably best done in groups of two or three in the classroom, for the discussion forms a very vital part of the work. The resource that you use does not matter too much but it is best to avoid cuboids that have three measurements that are related in some multiple/factor way. So a 2 by 4 by 10 or even a 3 by 5 by 9 would not be so good really.</div>
<br></br>
<div>When the children get to the four cuboids which can be arranged in more than three ways as I have shown, then the interest grows. If the children are encouraged to tabulate their results in some way then there are lots of fascinating things that crop up which are not too difficult to get to the bottom of, usually.</div>
<br></br>
<h3>Key questions</h3>
<div>Do you know if you have found them all?</div>
<div>How do you know you have found them all?</div>
<div>What else could you find out?</div>
<br></br>
<h3>Possible extension</h3>
<div>For those pupils who have confidently completed the activity with maybe up to 7 or 8 boxes then the time could be right for looking at the patterns of numbers generated and to see whether they can come up with any predictions.</div>
<br></br>
<h3>For the exceptionally mathematically able</h3>
<div>These pupils could be challenged to consider the activity in which the dimensions of the boxes relate. For example suppose they were $2$x$4$x$8$ cm?What difference does it make? What sequence of numbers does it now generate as you increase the number of boxes? Why?</div>
<br></br>
<h3>Possible support</h3>
<div>Some children might find it useful to have an adult to help organise the way they go about finding all the possibilities.</div>
<br></br>
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How will you know you have found all the different ways?<br></br>
It might help to describe how you have put the boxes together to
someone else.<br></br>
Are there any faces which you haven't joined together?<br></br></mdoxml>
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<mdoxml version="1.0"><mdo:image width="548" height="310" src="AnsID41.jpg" alt=""></mdo:image><br></br>
<p class="editorial">Here are two solutions to this problem, one
from Sarah at Withington Girls' School, Manchester and the other
from John in West Flegg.</p>
<p class="editorial"></p>
<ul>
<li>
<p>Sarah's written:-</p>
<p>Arranging a cuboid is based on making the number by multiplying
three numbers together as the three sides of the cuboid: length,
breadth and height.</p>
<p>I carried out the investigation for up to 12 cassette boxes and
compared the results to the possible ways of making the
number:-</p>
<center>
<table>
<tbody>
<tr>
<th>Number of Boxes</th>
<th>Ways of Making Number</th>
<th>No. of Cuboids</th>
</tr>
<tr>
<td>1</td>
<td>1x1x1</td>
<td>1</td>
</tr>
<tr>
<td>2</td>
<td>1x1x2</td>
<td>3</td>
</tr>
<tr>
<td>3</td>
<td>1x1x3</td>
<td>3</td>
</tr>
<tr>
<td>4</td>
<td>1x1x4, 1x2x2</td>
<td>6</td>
</tr>
<tr>
<td>5</td>
<td>1x1x5</td>
<td>3</td>
</tr>
<tr>
<td>6</td>
<td>1x1x6, 1x2x3</td>
<td>9</td>
</tr>
<tr>
<td>7</td>
<td>1x1x7</td>
<td>3</td>
</tr>
<tr>
<td>8</td>
<td>1x1x8, 1x2x4, 2x2x2</td>
<td>10</td>
</tr>
<tr>
<td>9</td>
<td>1x1x9, 1x3x3</td>
<td>6</td>
</tr>
<tr>
<td>10</td>
<td>1x1x10, 1x2x5</td>
<td>9</td>
</tr>
<tr>
<td>11</td>
<td>1x1x11</td>
<td>3</td>
</tr>
<tr>
<td>12</td>
<td>1x1x12, 1x2x6, 1x3x4, 2x2x3</td>
<td>18</td>
</tr>
</tbody>
</table>
</center>
<p>For each way of making the number with three different numbers
(AxBxC) there are six possible ways of arranging the boxes as a
cuboid (AxBxC, AxCxB, BxAxC, BxCxA, CxAxB, CxBxA). For each way of
making the number with two different numbers and the third number
the same as one of the previous numbers (AxAxB) there are 3
different ways of arranging the boxes (AxAxB, AxBxA, BxAxA). For
each way of arranging the boxes by cubing one number (AxAxA) there
is just one way of arranging the boxes (AxAxA).</p>
<p>For example, 8 can be made as 1x1x8, 1x2x4 and 2x2x2. The first
way is AxAxB so the cuboid can be made in three ways. The second is
AxBxC so this cuboid can be arranged in six ways. The third is
AxAxA and can only be arranged in one way. The total of these is 10
cuboids for 8 cassettes.</p>
<p>This will work for any number of cassettes. Prime numbers will
always have 3 arrangements because the only way of making the
number is by multiplying itself by one twice and so it is
AxAxB.</p>
<p class="editorial">Well Done Sarah!</p>
</li>
<li>
<p class="editorial">From John in West Flegg we have:-</p>
<p>When I started with four boxes I found four solutions so I
started thinking that the number of boxes you had was the number of
solutions there was,</p>
<center><mdo:image src="drawoneA.gif"></mdo:image></center>
<center><mdo:image src="drawoneB.gif"></mdo:image></center>
<center><mdo:image src="drawoneC.gif"></mdo:image></center>
<center><mdo:image src="drawoneD.gif"></mdo:image></center>
<hr></hr>
<center><mdo:image src="drawtwoA.gif"></mdo:image></center>
<center><mdo:image src="drawtwoB.gif"></mdo:image></center>
<center><mdo:image src="drawtwoC.gif"></mdo:image></center>
<p>After doing this with five boxes I started to think the same as
I did with four boxes except that on odd numbers it was three
solutions.</p>
<center><mdo:image src="drawthrA.gif"></mdo:image></center>
<center><mdo:image src="drawthrB.gif"></mdo:image></center>
<center><mdo:image src="drawthrC.gif"></mdo:image></center>
<center><mdo:image src="drawthrD.gif"></mdo:image></center>
<center><mdo:image src="drawthrE.gif"></mdo:image></center>
<center><mdo:image src="drawthrF.gif"></mdo:image></center>
<p>After doing this I think that my thoughts for five boxes were
correct.''</p>
</li>
</ul>
<p><span class="editorial">These are the possibilities with 2,3
& 4 boxes;</span></p>
<div class="editorial"><mdo:image width="548" height="310" alt="Ans ID41" src="AnsID41.jpg"></mdo:image></div>
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Little Boxes
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
Using, Applying and Reasoning about Mathematics
Investigations
Decision Mathematics and Combinatorics
Combinations
3D Geometry, Shape and Space
Cuboids
Using, Applying and Reasoning about Mathematics
Practical Activity
Using, Applying and Reasoning about Mathematics
Visualising