47
/www/nrich/html/content/98/07/bbprob2/
1
2011-02-01T00:00:01
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<p>Some time ago I was walking past a garage. Down on the ground
was the big sign that they have that tells motorists how much the
petrol will cost. You've probably seen them yourselves and you may
have been asked to look out for the cheapest petrol around. The
signs usually tell you the price for the different kinds of petrol.
Since we measure in litres it's the price for one litre.</p>
<p>Well, this sign was being prepared to be put up by the side of
the garage. I went over and looked at the part that shows the
price. I was interested in how the numbers were shown and how they
altered when the price changed. This particular one, like so many,
showed the numbers like they are on a calculator display.</p>
<p>The little lines on a calculator display can be called
'light bars'. This is how they generally look for the
figures 0 through to 9:-</p>
<p style="text-align: center;"><mdo:image height="254" width="211" alt="pic1" src="Picture%201.jpg"></mdo:image></p>
<br></br>
<p>In this sign they were brightly coloured flaps which on one side
showed the colour and on the other side were blank. As I walked
away from the garage I got thinking. This is the challenge that
came to my mind.</p>
<p>If we had 16 light bars we could only make certain numbers. For
example:-</p>
<p style="text-align: center;"><mdo:image height="202" width="223" alt="pic2" src="Picture%202.jpg"></mdo:image></p>
<br></br>
<p>So, my challenge to you is to find all the numbers you can make,
using 16 light bars all the time and forming the figures in the
same way as I did them for 0 to 9.</p>
<br></br>
<p>When you've done a few you may be able to think of a method or
system for helping you along the way. When you do, do write and let
us know what it was, as well as sending us your solutions.</p>
<p>The last word, as usual, is to say when you are happy with what
you have got, "I wonder what would happen if ...?''</p>
<br></br></mdoxml>
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<p class="editorial">Here is a solution to Calcunos from Ned who
has just left Christ Church Cathedral School in Oxford and is about
to go to Abingdon School. Adam of Swavesey Village College,
Cambridgeshire also sent in some good work on this
investigation.</p>
<p>Dear Bernard</p>
<p>I have solved your question from the July challenges, CalcuNos,
as there being 1,374 methods. The numbers of lightbars for each
digit are:</p>
<table>
<tbody>
<tr>
<td>1</td>
<td>2</td>
<td>3</td>
<td>4</td>
<td>5</td>
<td>6</td>
<td>7</td>
<td>8</td>
<td>9</td>
<td>0</td>
</tr>
<tr>
<td>2</td>
<td>5</td>
<td>5</td>
<td>4</td>
<td>5</td>
<td>6</td>
<td>3</td>
<td>7</td>
<td>6</td>
<td>6</td>
</tr>
</tbody>
</table>
<p>So we only need to consider combinations which add up to 16
using the numbers 2 to 7 and no others.</p>
<p>There are 32 ways of making 16.</p>
<table>
<tbody>
<tr>
<td>7,7,2</td>
<td>6,6,2,2</td>
<td>5,5,4,2</td>
<td>4,4,3,3,2</td>
</tr>
<tr>
<td>7,6,3</td>
<td>6,5,5</td>
<td>5,5,3,3</td>
<td>4,4,2,2,2,2</td>
</tr>
<tr>
<td>7,5,4</td>
<td>6,5,3,2</td>
<td>5,5,2,2,2</td>
<td>4,3,3,3,3</td>
</tr>
<tr>
<td>7,5,2,2</td>
<td>6,4,4,2</td>
<td>5,4,4,3</td>
<td>4,3,3,2,2,2</td>
</tr>
<tr>
<td>7,4,3,2</td>
<td>6,4,3,3</td>
<td>5,4,3,2,2</td>
<td>4,2,2,2,2,2,2</td>
</tr>
<tr>
<td>7,3,3,3</td>
<td>6,4,2,2,2</td>
<td>5,3,2,2,2,2</td>
<td>3,3,3,3,2,2</td>
</tr>
<tr>
<td>7,3,2,2,2</td>
<td>6,3,3,2,2</td>
<td>4,4,4,4</td>
<td>3,3,2,2,2,2,2</td>
</tr>
<tr>
<td>6,6,4</td>
<td>6,2,2,2,2,2</td>
<td>4,4,4,2,2</td>
<td>2,2,2,2,2,2,2,2</td>
</tr>
</tbody>
</table>
<p>The number of lightbars is unique except for three numbers which
have 5 bars and three which have 6, so it is necessary to work out
the number of different ways of arranging each set of numbers and
then multiply by three for each of the 5's or 6's involved in the
set.</p>
<p>For example:</p>
<p>7, 6, 3: the 3 could go in one of 3 places, the 7 in one of 2
(one has been taken up by the 3) and the 6 in one of 1; this makes
6 combinations.</p>
<table>
<tbody>
<tr>
<td>7,6,3</td>
<td>7,3,6</td>
<td>6,7,3</td>
<td>6,3,7</td>
<td>3,7,6</td>
<td>3,6,7</td>
</tr>
</tbody>
</table>
<p>However, as the 6 can represent any one of three numbers, one
must multiply by 3, making a total of 18 combinations for numbers
whose digits contain 7, 6 and 3 lightbars.</p>
<p>For combinations like 6,2,2,2,2,2 one sees that, as the 2's must
be all the same, only 6 combinations exist (622222, 262222, 226222,
222622, 222262, 222226) times three (for the six), making 18 for
this example.</p>
<p>The numbers of combinations for each set of numbers are:</p>
<table>
<tbody>
<tr>
<td>7,7,2</td>
<td>3</td>
</tr>
<tr>
<td>7,6,3</td>
<td>18</td>
</tr>
<tr>
<td>7,5,4</td>
<td>18</td>
</tr>
<tr>
<td>7,5,2,2</td>
<td>36</td>
</tr>
<tr>
<td>7,4,3,2</td>
<td>24</td>
</tr>
<tr>
<td>7,3,3,3</td>
<td>4</td>
</tr>
<tr>
<td>7,3,2,2,2</td>
<td>20</td>
</tr>
<tr>
<td>6,6,4</td>
<td>27</td>
</tr>
<tr>
<td>6,6,2,2</td>
<td>36</td>
</tr>
<tr>
<td>6,5,5</td>
<td>81</td>
</tr>
<tr>
<td>6,5,3,2</td>
<td>216</td>
</tr>
<tr>
<td>6,4,4,2</td>
<td>36</td>
</tr>
<tr>
<td>6,4,3,3</td>
<td>36</td>
</tr>
<tr>
<td>6,4,2,2,2</td>
<td>60</td>
</tr>
<tr>
<td>6,3,3,2,2</td>
<td>90</td>
</tr>
<tr>
<td>6,2,2,2,2,2</td>
<td>18</td>
</tr>
<tr>
<td>5,5,4,2</td>
<td>54</td>
</tr>
<tr>
<td>5,5,3,3</td>
<td>36</td>
</tr>
<tr>
<td>5,5,2,2,2</td>
<td>90</td>
</tr>
<tr>
<td>5,4,4,3</td>
<td>36</td>
</tr>
<tr>
<td>5,4,3,2,2</td>
<td>180</td>
</tr>
<tr>
<td>5,3,2,2,2,2</td>
<td>90</td>
</tr>
<tr>
<td>4,4,4,4</td>
<td>1</td>
</tr>
<tr>
<td>4,4,4,2,2</td>
<td>10</td>
</tr>
<tr>
<td>4,4,3,3,2</td>
<td>30</td>
</tr>
<tr>
<td>4,4,2,2,2,2</td>
<td>15</td>
</tr>
<tr>
<td>4,3,3,3,3</td>
<td>5</td>
</tr>
<tr>
<td>4,3,3,2,2,2</td>
<td>60</td>
</tr>
<tr>
<td>4,2,2,2,2,2,2</td>
<td>7</td>
</tr>
<tr>
<td>3,3,3,3,2,2</td>
<td>15</td>
</tr>
<tr>
<td>3,3,2,2,2,2,2</td>
<td>21</td>
</tr>
<tr>
<td>2,2,2,2,2,2,2,2</td>
<td>1</td>
</tr>
</tbody>
</table>
<p>This gives a grand total of 1,374 numbers which, on a
calculator, have 16 light bars.</p>
<br></br></mdoxml>
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<mdoxml version="1.0"><div class="embed">
<h2>Calcunos</h2>
<br></br>
<br></br>
<p>Some time ago I was walking past a garage. Down on the ground was the big sign that they have that tells motorists how much the petrol will cost. You've probably seen them yourselves and you may have been asked to look out for the cheapest petrol around. The signs usually tell you the price for the different kinds of petrol. Since we measure in litres it's the price for one litre.</p>
<p>Well, this sign was being prepared to be put up by the side of the garage. I went over and looked at the part that shows the price. I was interested in how the numbers were shown and how they altered when the price changed. This particular one, like so many, showed the numbers like they are on a calculator display.</p>
<p>The little lines on a calculator display can be called 'light bars'. This is how they generally look for the figures 0 through to 9:-</p>
<p style="text-align: center;"><mdo:image alt="pic1" height="254" src="Picture%201.jpg" width="211"></mdo:image></p>
<br></br>
<p>In this sign they were brightly coloured flaps which on one side showed the colour and on the other side were blank. As I walked away from the garage I got thinking. This is the challenge that came to my mind.</p>
<p>If we had 16 light bars we could only make certain numbers. For example:-</p>
<p style="text-align: center;"><mdo:image alt="pic2" height="202" src="Picture%202.jpg" width="223"></mdo:image></p>
<br></br>
<p>So, my challenge to you is to find all the numbers you can make, using 16 light bars all the time and forming the figures in the same way as I did them for 0 to 9.</p>
<br></br>
<p>When you've done a few you may be able to think of a method or system for helping you along the way. When you do, do write and let us know what it was, as well as sending us your solutions.</p>
<p>The last word, as usual, is to say when you are happy with what you have got, "I wonder what would happen if ...?''</p>
</div>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div>This <a href="http://nrich.maths.org/public/viewer.php?obj_id=47&part=">number exploration</a> offers an opportunity for pupils to investigate different aspects of our number system.</div>
<br></br>
<h3>Possible approach</h3>
<div>Some younger pupils will like to have 16 little sticks or rods to see what they can do. Other pupils might work on squared paper drawing the numerals along the sides of the squares.</div>
<br></br>
<div>It is handy to have some calculators available for them to check how numbers are written (particularly the 4 and the 7 - 7 varies from calculator to calculator so look carefully).</div>
<br></br>
<h3>Key questions</h3>
<div>Tell me about the numbers you've found.</div>
<div>How did you find these?</div>
<div>It looks as if you have a kind of system for finding more, can you tell me about it?</div>
<br></br>
<h3>Possible extension</h3>
<div>If some of the pupils have been looking at taking a number like 565 and writing also 556 and 655; then taking different kinds of numbers and looking at all the possibilities just using those numerals, then this can be extended. They could look at the number of ways you can rearrange 3, 4, 5, etc different numerals and/or 3, 4, 5 numerals in which two are the same.</div>
<div>Searching for the largest and smallest numbers and maybe allowing a decimal point to be used where ever they wish is another possibility.</div>
<br></br>
<h3>Possible support</h3>
<div>Pupils who find it hard to make a start may need an adult to work alongside them in helping to construct the right shape for each digit.</div>
<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">You could use sixteen sticks or headless matches to
make the numbers.<br></br>
How will you record what you've done?<br></br>
How will you know when you have got all the possibilities?<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">This particular activity has had surprising outcomes that are often
not anticipated, so be brave and be ready for almost anything.
Taking the example of 4152 I have found that some children start to
explore, often for the first time making arrangements like:-
1245,1254,1425,1452,1524,1542, etc . On other occasions a lot has
been done, talked about and reflected upon when considering the
size of the numbers obtained. This has often been started when
children are looking for the largest number. When considering the
smallest number decisions hace to be made about whether a decimal
point is allowed. It also makes a difference if you ask them to use
a calculator alongside the work for when a decimal point is used
you have to have a zero in the units column. From a practical point
of view I've found that lower ability years 3/4 can do quite a lot
if given some practical help, like using quarters of straws to
represent the light bars. Squared paper should be made available as
some pupils find that it can be very helpful.<br></br></mdoxml>
2
5
0
1
0
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Calcunos
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
Using, Applying and Reasoning about Mathematics
Investigations
Decision Mathematics and Combinatorics
Combinations
Using, Applying and Reasoning about Mathematics
Working systematically