2661
/www/nrich/html/content/id/2661/
1
2011-02-01T00:00:01
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Start with the set of the twenty-one numbers $0$ - $20$.<br></br>
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<div style="text-align: center;"><mdo:image alt="numbers from 0 to 20" height="143" src="numbers.gif" width="390"></mdo:image></div>
<p style="font-style: italic;"> </p>
<p style="font-style: normal;"> </p>
<p style="font-style: normal;">Can you arrange these numbers into seven subsets each of three numbers so that when the numbers in each are added together, they make seven consecutive numbers?</p>
<p style="font-style: normal;">For example, one subset might be $\{2, 7, 16\}$</p>
<p style="text-align: center; font-style: normal;">$2 + 7 + 16 = 25$</p>
<p style="font-style: normal;">another might be $\{4, 5, 17\}$</p>
<p style="text-align: center; font-style: normal;">$4 + 5 + 17 = 26$</p>
<p style="font-style: normal;">As $25$ and $26$ are consecutive numbers these sets are the kind of thing that you need.</p>
<p style="font-style: normal;">[Remember that consecutive numbers are numbers which follow each other when you are counting, for example, $4$, $5$, $6$, $7$ or $19$, $20$, $21$, $22$, $23$.]</p>
<p style="font-style: normal;"> </p>
<p style="font-style: normal;"><a href="http://nrich.maths.org/public/viewer.php?obj_id=6011&part=">Click here for a poster of this problem</a>.</p>
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<p class="editorial">Many of you approached this problem in the
same way to start with. Maddie and Alex from The Mount School
wrote:</p>
We found out that 1+2+3+4+5+6...+20=210. Because we needed 7
different subsets that when added together made 7 consecutive
numbers, we divided 210 by 7. 210/7=30<br></br>
This told us that the seven numbers had to be around 30. The
numbers turned out to be 27, 28, 29, 30, 31, 32 & 33! <br></br>
We tried a variation of numbers but ended up using: <br></br>
0+7+20=27 <br></br>
<div style="clear: both;">6+10+12=28<br></br>
<div style="clear: both;">2+8+19=29<br></br>
<div style="clear: both;">5+9+16=30<br></br>
<div style="clear: both;">3+11+17=31<br></br>
<div style="clear: both;">1+13+18=32<br></br>
<div style="clear: both;">4+14+15=33</div>
<div style="clear: both;">This is one of many solutions!</div>
<p style="clear: both;" class="editorial">Well done, Maddie and
Alex. In fact, not many of you mentioned that there are lots of
solutions to this problem. Boris from Gresham's Preparatory School
found the seven consecutive numbers in a similar way and then
said:</p>
So now we check which numbers from 0 to 20 can make them in groups
of 3:
<div style="clear: both;">27 - 7,3,17<br></br>
<div style="clear: both;">28 - 8,5,15<br></br>
<div style="clear: both;">29 - 19,4,6<br></br>
<div style="clear: both;">30 - 9,10,11<br></br>
<div style="clear: both;">31 - 1,18,12<br></br>
<div style="clear: both;">32 - 2,16,14<br></br>
<div style="clear: both;">33 - 0,13,20<br></br>
<div style="clear: both;">To find these it's easier to make two of
the numbers end with a zero (e.g. 3+17=20, 5+15=20, 4+6=10,
9+11=20, 18+12=30, 16+14=30, 0+20=20) and then add the other number
to finish it.</div>
<p class="editorial" style="clear: both;">Thank you, Boris. Many of
you suggested ways to make pairs of numbers and then add a third to
total one of the consecutive numbers like Boris' method.</p>
<p class="editorial" style="clear: both;"> </p>
<p class="editorial" style="clear: both;">Ivo from Gresham's Prep
School used a different method to work out which consecutive
numbers to aim to make:</p>
<p class="editorial" style="clear: both;"> </p>
<div>We can see that 0+1+2+3+.....+17+18+19+20=210.</div>
<div>Each consecutive number is made of three numbers between 0 and
20. Let the first of the consecutive number be made of A,B and C.
That means that the next consecutive number will be A+B+C+1. This
led me to my solution.</div>
<div>(A+B+C) + (A+B+C+1) + (A+B+C+2)+ (A+B+C+3) + (A+B+C+4) +
(A+B+C+5) + (A +B+C+6) = 210</div>
<div>7A + 7B + 7C + 21 = 210</div>
<div>7(A+B+C) = 189</div>
<div>A+B+C=27</div>
<div>This means that the first of the consecutive numbers is 27.
The next is 28 and so on. So the consecutive seven are 27, 28, 29,
30, 31, 32, 33.</div>
<p class="editorial" style="clear: both;">Thank you, Ivo! Zoe,
Andrew, Nikita and Ben from Aqueduct Primary School went about the
problem in a slightly different way:</p>
<div style="clear: both;">We looked at the numbers from 0 - 20. We
used trial and error to solve this problem. The first thing we did
was decide what our first set of three numbers was going to be. We
looked at the "Nrich" example:</div>
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2+7+16 = 25<br></br>
<div style="clear: both;">
<div style="clear: both;">4+5+17= 26</div>
<div>These sets when added together obviously gave consecutive
answers. We decided that for there to be consecutive answers there
must be some relationship between each number in each set.</div>
<div>
<div style="clear: both;">We separated the numbers into 3 columns:
a + b + c = answer:</div>
</div>
<table border="1">
<tbody>
<tr>
<td style="clear: both;">A</td>
<td>B</td>
<td>C</td>
<td>Answer</td>
</tr>
<tr>
<td>2</td>
<td>7</td>
<td>16</td>
<td>25</td>
</tr>
<tr>
<td>4</td>
<td>5</td>
<td>17</td>
<td>26</td>
</tr>
</tbody>
</table>
<br></br>
We noticed that the pattern in A = adding 2 each time<br></br>
<div style="clear: both;">We noticed that the pattern in B =
subtract 2 each time<br></br>
<div style="clear: both;">We noticed that the pattern in C = adding
1 each time<br></br>
<div style="clear: both;">We followed the pattern on to see if we
could come up with 7 sets however, we only managed to achieve
5.<br></br>
<div style="clear: both;">Here are our results:<br></br>
<div style="clear: both;">0 + 9 + 15 = 24<br></br>
<div style="clear: both;">2 + 7+ 16 = 25<br></br>
<div style="clear: both;">4 + 5 + 17 = 26<br></br>
<div style="clear: both;">6 + 3 + 18 = 27<br></br>
<div style="clear: both;">8 + 1 + 19 = 28</div>
</div>
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<div>
<div style="clear: both;">We decided that we needed to take a more
systematic approach to solve this problem. The most logical number
to start with was 0 because you can build on 0. We noticed that the
numbers 0-20 were already organised into 3 rows of 7 numbers:</div>
</div>
<table border="0">
<tbody>
<tr>
<td>0</td>
<td>1</td>
<td>2</td>
<td>3</td>
<td>4</td>
<td>5</td>
<td>6</td>
</tr>
<tr>
<td>7</td>
<td>8</td>
<td>9</td>
<td>10</td>
<td>11</td>
<td>12</td>
<td>13</td>
</tr>
<tr>
<td>14</td>
<td>15</td>
<td>16</td>
<td>17</td>
<td>18</td>
<td>19</td>
<td>20</td>
</tr>
</tbody>
</table>
<div><br></br>
<div style="clear: both;">We listed the numbers consecutively. We
made three columns. This gave us 7 sets of 3 numbers.</div>
</div>
</div>
<table border="1">
<tbody>
<tr>
<td>A</td>
<td>B</td>
<td>C</td>
<td>Answer</td>
</tr>
<tr>
<td>0</td>
<td>7</td>
<td>14</td>
<td>21</td>
</tr>
<tr>
<td>1</td>
<td>8</td>
<td>15</td>
<td>24</td>
</tr>
<tr>
<td>2</td>
<td>9</td>
<td>16</td>
<td>27</td>
</tr>
<tr>
<td>3</td>
<td>10</td>
<td>17</td>
<td>30</td>
</tr>
<tr>
<td>4</td>
<td>11</td>
<td>18</td>
<td>33</td>
</tr>
<tr>
<td>5</td>
<td>12</td>
<td>19</td>
<td>36</td>
</tr>
<tr>
<td>6</td>
<td>13</td>
<td>20</td>
<td>39</td>
</tr>
</tbody>
</table>
<br></br>
This did not give us consecutive answers. The answers increased by
3 each time. We looked more closely at our lists of numbers and
realised that although the numbers in each list were consecutive,
the lists were not consecutive to each other. We rearranged the
lists:<br></br>
<br></br>
<table border="1">
<tbody>
<tr>
<td>A</td>
<td>B</td>
<td>C</td>
<td>Answer</td>
</tr>
<tr>
<td>0</td>
<td>13</td>
<td>14</td>
<td>27</td>
</tr>
<tr>
<td>1</td>
<td>12</td>
<td>15</td>
<td>28</td>
</tr>
<tr>
<td>2</td>
<td>11</td>
<td>16</td>
<td>29</td>
</tr>
<tr>
<td>3</td>
<td>10</td>
<td>17</td>
<td>30</td>
</tr>
<tr>
<td>4</td>
<td>9</td>
<td>18</td>
<td>31</td>
</tr>
<tr>
<td>5</td>
<td>8</td>
<td>19</td>
<td>32</td>
</tr>
<tr>
<td>6</td>
<td>7</td>
<td>20</td>
<td>33</td>
</tr>
</tbody>
</table>
<br></br>
This did give us consecutive answers.<br></br>
We noticed that the numbers in columns A and B when added together
all give the same answer, for example:<br></br>
<div style="clear: both;">0 + 13 = 13<br></br>
<div style="clear: both;">1 + 12 = 13<br></br>
<div style="clear: both;">2 + 11 = 13<br></br>
<div style="clear: both;">3 + 10 = 13<br></br>
<div style="clear: both;">4 + 9 = 13<br></br>
<div style="clear: both;">5 + 8 = 13<br></br>
<div style="clear: both;">6 + 7 = 13<br></br>
<div style="clear: both;">These are number bonds for 13.</div>
<div style="clear: both;">We also noticed that the lists of numbers
had to follow on consecutively from each other. For example: 6 and
7 had to be in the same set, 13 and 14 had to be in the same set.
(see the chart of results)</div>
<div style="clear: both;">We noticed that the second and third
numbers in each set when added together also gave the same
total.</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<p class="editorial">Thank you Zoe, Andrew, Nikita and Ben. It is
always good to receive solutions which take us all the way through
the process that you followed to solve the problem. Your solution
shows us that "playing" with a problem can be a very good way to
start and will often lead to us finding something out that helps us
go about a solution more systematically (in other words more
logically).</p>
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<h3>Why do this problem?</h3>
<a href="http://nrich.maths.org/public/viewer.php?obj_id=2661&part=">This
problem</a> has several different solutions. The problem can be
solved using an experimental / trial and error approach but some
consideration of the structure can lead to more efficient solution
techniques. There will be no need for students to feel 'stuck' on
this problem: they will always be able to experiment with new
combinations.<br></br>
<h3>Possible approach</h3>
<div>Students could all write down the numbers $0$ to $20$. One
student could be asked to select a first triple and everyone writes
that down. All students search for a second triple, whose sum is
one more than the first's sum. One such triple is chosen, and
everyone writes it down and starts to search for the next - until
the task of finding triples whose sums are consecutive is fully
understood by the group, at which point, they can work alone or in
pairs to find a solution.</div>
<br></br>
With the whole group, ask students to describe what problems occur
and how they are dealing with them. Ask them to share any
observations, or inspirations they have had. Check that the points
in the key questions have been covered in the students comments.
<br></br>
<br></br>
<div>Students who wish to continue to work experimentally could be
encouraged to devise a clear recording system for the combinations
they are trying. For example, starting with the 20 and 19, what are
the possibilities for the other two cards. Students who want to
work analytically may choose to use algebra to determine the
smallest consecutive number.</div>
<br></br>
<h3>Key questions</h3>
<ul>
<li>Why have you run out of possibilities? Can you change anything
to avoid that problem?</li>
<li>What are the biggest and smallest sums we could get from a
triple?</li>
<li>Can you work out what the sevenconsecutive numbers will have to
add up to?</li>
<li>Can you select your triples in a logical/symmetrical way?</li>
</ul>
<h3>Possible extension</h3>
<ul>
<li>How many different solutions do you think that there might be?
Can you work out how many might be possible?</li>
<li>If the numbers $0$ to $20$ were changed to different sets of
$20$ numbers, would solutions still be possible? (specifically
designed sets e.g. $\{10, 11, .., 30\}$ or $\{1000, 1001,
...1020\}$ or $\{0, n, 2n, 3n, 4n, ..., 20n\}$, or totally random
sets)</li>
</ul>
<h3>Possible support</h3>
<div>It might be helpful to provide students with cards labelled
$0$ to $20$ to allow them to make their arrangements. You could
also provide calculators so that students can focus on the
structure of the problem, rather than getting stuck with the
additions.</div>
<br></br>
Alternative questions include:
<div>Can you find seven <span style="font-weight: bold;">pairs</span> of numbers which add up to
consecutive numbers?</div>
<div>Can you find seven sets of three cards which add up to the
SAME number?</div>
<div>Students could write such questions for each other, working in
pairs to ensure that these are phrased as accurately as
possible.</div>
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What is the total of all the numbers from $0$ to $20$? <br></br>
You could try dividing this by $3$ and so make seven equal numbers
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<mdoxml version="1.0">Many possible, for example:<br></br>
<br></br>
3 + 8 + 16 = 27 <br></br>
2 + 7 + 19 = 28 <br></br>
0 + 9 + 20 = 29 <br></br>
5 + 10 + 15 = 30
<br></br>
1 + 12 + 18 = 31 <br></br>
4 + 11 + 17 = 32 <br></br>
6 + 13 + 14 = 33<br></br><br></br>0 + 20 + 7 = 27<br></br>19 + 8 + 1 = 28<br></br>18 + 2 + 9 = 29<br></br>17 + 3 + 10 = 30<br></br>16 + 4 + 11 = 31<br></br>15 + 5 + 12 = 32<br></br>14 + 6 + 13 = 33<br></br></mdoxml>
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Consecutive Seven
Can you arrange these numbers into 7 subsets, each of three
numbers, so that when the numbers in each are added together, they
make seven consecutive numbers?
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