2645
/www/nrich/html/content/id/2645/
1
2011-02-01T00:00:01
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<p>Tim had nine cards, each with a different number from 1 to 9 on it.<br></br>
He put the cards into three piles so that the total in each pile was 15.<br></br>
How could he have done this?<br></br>
<br></br>
Can you find <span style="font-style: italic;">all</span> the different ways Tim could have done this?<br></br>
<br></br>
You may like to print off and cut out these <a href="/content/id/2645/digitcards.doc">digit cards</a> to help you.</p></mdoxml>
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<p class="editorial">Thank you for the many solutions to this
problem. It was interesting to see that some of you presumed there
had to be three cards in each pile which totalled 15. In fact, the
question simply said there had to be three PILES, which makes the
problem a little trickier than it looks at first.</p>
<p class="editorial">Jessica and Ruby from Aldermaston C of E
Primary School told us how they went about tackling the
problem:</p>
<div>We tried to be systematic by looking for other ways to make a
number, for example 7. So we made 6+1, 3+4, 5+2 and 7 on its own
and put 8 with it each time. It was good because then we could get
more solutions to make 15. Here are our solutions to make 15
(reading across the table for the three piles):</div>
<table border="1">
<tbody>
<tr>
<td>2, 3, 9, 1</td>
<td>6, 5, 4</td>
<td>7, 8</td>
</tr>
<tr>
<td>3, 8, 4</td>
<td>6, 7, 2</td>
<td>9, 1, 5</td>
</tr>
<tr>
<td>5, 2, 8</td>
<td>1, 3, 4, 7</td>
<td>9, 6</td>
</tr>
<tr>
<td>6, 1, 8</td>
<td>5, 7, 3</td>
<td>2, 4, 9</td>
</tr>
</tbody>
</table>
<br></br>
<p><span class="editorial">Wilbury Primary School Mathletics Club
also got the idea. Some of the solutions they found were the same
as Jessica's and Ruby's, but here are their different
solutions:</span></p>
<br></br>
<br></br>
<table border="1">
<tbody>
<tr>
<td>
<p>1, 2, 3, 4, 5</p>
</td>
<td>9, 6</td>
<td>7, 8</td>
</tr>
<tr>
<td>9, 1,5</td>
<td>7, 8</td>
<td>4, 3, 6, 2</td>
</tr>
<tr>
<td>9, 6</td>
<td>3, 5, 7</td>
<td>1, 8, 2, 4</td>
</tr>
</tbody>
</table>
<br></br>
<span class="editorial">So, in total Jessica, Ruby and the
Mathletics Club at Wilbury have found seven different ways of
putting the cards into three piles.</span> <br></br>
<br></br>
<div style="clear: both;"><span class="editorial">Then we asked you
if these were all the combinations possible. Alicia and William of
St Hugh's Woodhall Spa rose to the challenge, finding one more
combination:</span></div>
<br></br>
<table border="1">
<tbody>
<tr>
<td>9, 6</td>
<td>8, 4, 3</td>
<td>7, 5, 2, 1</td>
</tr>
</tbody>
</table>
<br></br>
<p class="editorial">and telling us why they thought they'd found
them all:</p>
<div>We worked systematically swapping combinations of the same
value, ie 1 and 5 with 2 and 4. That way we took piles like (9, 1,
5), (8, 7), (6, 3, 2, 4) and got the new bunch of piles (9, 2, 4),
(8, 7), (6, 3, 1, 5). We did this for all the pairs, ones and
threes that made up the same value, and then got rid of the ones
we'd counted twice.</div>
<br></br>
<p class="editorial">That makes eight ways altogether. Well done,
Alicia and William! I think there might be one more to
find ...</p>
<br></br></mdoxml>
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<mdoxml version="1.0"><div class="embed">
<h2>Finding Fifteen</h2>
<br></br>
<p>Tim had nine cards, each with a different number from 1 to 9 on it.<br></br>
He put the cards into three piles so that the total in each pile was 15.<br></br>
How could he have done this?<br></br>
<br></br>
Can you find <span style="font-style: italic;">all</span> the different ways Tim could have done this?<br></br>
<br></br>
You may like to print off and cut out these <a href="/content/id/2645/digitcards.doc">digit cards</a> to help you.</p>
</div>
<h3>Why do this problem?</h3>
<a href="/2645">This task</a> will encourage children to develop a systematic approach as well as giving them opportunities to practise simple addition.<br></br>
<br></br>
<h3>Possible approach</h3>
Introduce the problem without saying too much more and then give children chance to have a go in pairs. Having digit cards will help them try out their ideas without feeling inhibited. Suggest that they record each solution on a different piece of paper, large enough so that it could be seen from some distance away.<br></br>
<br></br>
After a while, bring the whole group together and invite several pupils to come up holding one of their solutions. Keep adding to those standing at the front until the group doesn't have any more different solutions. How do we know that there aren't any other solutions? If no-one offers an idea, suggest to the children that they arrange the solutions in some kind of order or
pattern which will then reveal any that are missing. In this way, a system is imposed afterwards. This will help them to see the value of working systematically on this kind of problem.<br></br>
<br></br>
<h3>Key questions</h3>
How do you know you haven't got that solution already?<br></br>
How will you know when you have found them all?<br></br>
Can you convince me that you haven't left any out?<br></br>
<br></br>
<h3>Possible extension</h3>
Investigating magic squares is a nice follow-on activity. The <a href="/1223">Fifteen</a> game also links well.<br></br>
<br></br>
<h3>Possible support</h3>
Having digit cards available will make this activity accessible for most children.<br></br></mdoxml>
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<mdoxml version="1.0"><br></br>Try starting with one number, for example 1. Which other two numbers can you add to 1 to make 15? Is there more than one way of doing it?<br></br>How many ways can you make 15?<br></br>
<br></br></mdoxml>
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Ways to make 15 cards:<br></br>
<table border="1">
<tbody>
<tr>
<td>9, 6</td>
<td>8, 7</td>
<td>1, 2, 3, 4, 5</td>
</tr>
<tr>
<td>9, 5, 1</td>
<td>8, 7</td>
<td>6, 4, 3, 2</td>
</tr>
<tr>
<td>9, 4, 2</td>
<td>8, 7</td>
<td>6, 5, 3, 1</td>
</tr>
<tr>
<td>9, 1, 2, 3</td>
<td>8, 7</td>
<td>6, 5, 4</td>
</tr>
<tr>
<td>9, 6</td>
<td>8, 5, 2</td>
<td>7, 4, 3, 1</td>
</tr>
<tr>
<td>9, 6</td>
<td>8,3, 4</td>
<td>7, 5, 2, 1</td>
</tr>
<tr>
<td>9, 5, 1</td>
<td>8, 3, 4</td>
<td>7, 6, 2</td>
</tr>
<tr>
<td>9, 4, 2</td>
<td>8, 1, 6</td>
<td>7, 5, 3</td>
</tr>
<tr>
<td>9, 6</td>
<td>8, 4, 2, 1</td>
<td>7, 5, 3</td>
</tr>
</tbody>
</table>
<br></br>
<br></br>
Still to find: 942, 87, 1356 <br></br>
<br></br></mdoxml>
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Finding Fifteen
Tim had nine cards each with a different number from 1 to 9 on it.
How could he have put them into three piles so that the total in
each pile was 15?
Decision Mathematics and Combinatorics
Combinations
Using, Applying and Reasoning about Mathematics
Working systematically
Calculations and Numerical Methods
Addition & subtraction
Mathematics Tools
Digit cards