6571
/www/nrich/html/content/id/6571/
1
2011-02-01T00:00:01
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Look at the following row of numbers:<br></br>
<h1 style="text-align: center; font-weight: bold;">\[10\quad
15\quad 21\quad 4\quad 5\]</h1>
<br></br>
<div>They are arranged so that each pair of adjacent numbers adds
up to a square number:</div>
<br></br>
<div style="text-align: center;">\[<br></br>
10 + 15 = 25<br></br>
\]<br></br>
\[<br></br>
15 + 21 = 36<br></br>
\]<br></br>
\[<br></br>
21 + 4 = 25<br></br>
\]<br></br>
\[<br></br>
4 + 5 = 9<br></br>
\]<br></br>
</div>
<br></br>
<div>
<h4>Can you arrange the numbers 1 to 17 in a row in the same way,
so that each adjacent pair adds up to a square number?</h4>
<br></br>
This printable set of <a href="/content/id/6571/cards.doc">cards</a>, or the number cards in
the interactivity below, might help you to test different
options.</div>
<br></br>
<h4>Can you arrange them in more than one way? If not, can you
justify that your solution is unique? </h4>
<br></br>
<br></br>
<mdo:flash height="400" width="600"><param value="true" name="allowFullScreen" ></param><param value="/content/id/6571/StickyNumbers.swf" name="movie" ></param><param value="9" name="flashplayerversion" ></param></mdo:flash><br></br></mdoxml>
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<p><span class="editorial">We had an amazing response to this problem, with everyone who submitted a solution getting the 17 numbers in the right order, that is:</span></p>
<p>\[16\quad 9\quad 7\quad 2\quad 14\quad 11\quad 5\quad 4\quad 12\quad 13\quad 3\quad 6\quad 10\quad 15\quad 1\quad 8\quad 17\]</p>
<p><span class="editorial">We also received fantastic explanations, Ashleigh from the Sirius Academy used a systematic approach:</span></p>
<p>I started by listing all the square numbers up to 36, I then worked out all the addition pairs using a systematic approach to make sure I didnt miss any. I used this to make a separate list of all the pairs that made a square number when added together. I used a trial and improvement method to work out the order of the line.</p>
<p><span class="editorial">Sam from Acland writes</span>:</p>
<p>I worked out that 17 + 16 made 33, this means the highest square number I could make was 25. This means 17 and 16 only had one way to make a square number, so they would be at either end. After 17 it must be 8 and, after 8, 1. At this point there are two options, 3 or 15. But if 15 didn't go with 1 then it would have to go at the end which would have been impossible to complete. After this the
problem became simple as from this point it becomes linear with only one option available after every point.</p>
<p><span class="editorial">Christian and Pippa worked along similar lines</span>:</p>
<p>We have started to solve this problem by finding out what numbers go with only one other number. Those are 16 and 17 and they had to go at the end. Then we found all the numbers that go with only 2 other numbers. Once we found that, we matched the remaining numbers. This can be the only solution because if we swapped one number, it would make a cut in the line and it wouldn't match. For
example: 3 can go with more than 2 numbers. It can go with 13, 6 and 1 but if we put 1 next to 3, it wouldn`t work because 15 can only go with 1 and 10.</p>
<p><span class="editorial">Robin from Oxford noticed this too</span>:</p>
<p>I started by listing all pairs of numbers that added together to make a square number. Then I took the numbers that only added up with one other number to make a square number (16 and 17)and put them at the ends of the sequence. The two numbers that added up with 16 and 17 to make a square number (9 and 8) went next to 16 and 17 and so on.</p>
<p class="editorial">Emma, Joe and Lucy Attwood, James Nash, Nathan Wilson, Pavan Murali, Sara Jafar, Rachel Musgrave, Philip Knott, Tom Short, Matthew Clark, Alasdair Haines, Arjun Gill and Henry McEntyre and gave fantastic explanations, well done!</p>
<p class="editorial">Rob from Thurston explains why there is only one solution,</p>
<p>17 can only be paired with 8 to make 25 so has to be at one end but it doesn't matter which end it is at as all the numbers add up the same if you just swap them round.</p>
<p><span class="editorial">Mrs Dillon's year 7 class also noticed a pattern</span>:</p>
<p>We have noticed that the square numbers produced form a pattern: 25, 16, 9, 16, 25, 16, 9 etc.</p></mdoxml>
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<h3>Why do this problem?</h3>
<a href="http://nrich.maths.org/public/viewer.php?obj_id=6571&part=">
This problem</a> requires some knowledge of square numbers, but
more importantly requires students to think strategically and
provide convincing arguments and justifications for their
findings.<br></br>
<h3>Possible approach</h3>
<div>Introduce the problem using the example $10, 15, 21, 4, 5$,
explaining that this list of numbers is special because each pair
of adjacent numbers add up to make a square number.</div>
<br></br>
<div>Set them the challenge to produce a similar list using all the
numbers from $1$ to $17$. Hand out sets of <a href="/content/id/6571/cards.doc">these cards</a> and ask students
to work in pairs on this challenge.</div>
<br></br>
After the class have had a few minutes to work on this, add that
you would like to know <span style="font-weight: bold;">all</span>
the possible different ways of making such a list. Challenge them
to produce a convincing argument that they have found all the
possibilities.<br></br>
<br></br>
Finally, ask students to present their solutions and justifications
to the rest of the class.<br></br>
<br></br>
An alternative starting point for this lesson could be to ask for
17 volunteers to stand at the front holding one of <a href="/content/id/6571/1-17%20NumberCards.doc">these cards</a>.
With help from the rest of the class, ask them to arrange
themselves to satisfy the criteria above. Once they have found a
solution, ask them to work in pairs as outlined above to find all
the possible arrangements and justify that they have the complete
set.
<h3>Key questions</h3>
<div>Are there any numbers which cannot be linked to another
number?</div>
<div>Are there any numbers which can only link to one other
number?</div>
<div>Are there any numbers which can only link to two
numbers?</div>
<div>Are there any numbers which can link to more than two
numbers?</div>
<br></br>
How do you know you've found every possible solution?<br></br>
<h3>Possible extension</h3>
<div>This graph has the numbers from $1$ to $31$ at its vertices.
Each edge connects two numbers which add together to make a square
number.</div>
<br></br>
Can you use the graph to produce a list of all the numbers from $1$
to $31$ so that pairs of adjacent numbers add up to a square
number? Is there more than one way to do it?<br></br>
<br></br>
<mdo:image width="598" height="374" src="31graph.png" alt=""></mdo:image><br></br>
<br></br>
Can you find any other numbers $n$ less than $31$ so that all the
numbers from $1$ to $n$ can be written in a list in this way? How
does the graph help you? <a href="/content/id/6571/31graph.png">Here</a> is a printable version
of the graph. <br></br>
<h3>Possible support</h3>
Suggest that students could make a table listing the numbers which
can be paired with each of the numbers from $1$ to $17$ to make a
square total.<br></br>
<br></br>
<br></br></mdoxml>
<mdoxml version="1.0">Are there any numbers which can only link to one other
number?<br></br>
Where will you need to place these in your list?<br></br></mdoxml>
<mdoxml version="1.0"><br></br>
1-29:<br></br>
18,7,29,20,16,9,27,22,14,2,23,13,12,4,5,11,25,24,1,3,6,19,17,8,28,21,15,10,26<br></br>
<br></br>
1-30:<br></br>
18,7,29,20,16,9,27,22,14,2,23,13,12,4,5,11,25,24,1,3,6,30,19,17,8,28,21,15,10,26<br></br>
<br></br>
1-31:<br></br>
31,18,7,29,20,16,9,27,22,14,2,23,13,12,4,5,11,25,24,1,3,6,30,19,17,8,28,21,15,10,26<br></br>
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<a href="/content/id/6571/icon.jpg">icon</a><br></br>
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Sticky Numbers
Can you arrange the numbers 1 to 17 in a row so that each adjacent
pair adds up to a square number?
Numbers and the Number System
Square numbers
Using, Applying and Reasoning about Mathematics
Working systematically
Using, Applying and Reasoning about Mathematics
Mathematical reasoning & proof