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<td width="50%">The diagram illustrates the formula:<br></br>
<p>1 + 3 + 5 + ... + (2n - 1) = n²</p>
<p>Use the diagram to show that any odd number is the difference of
two squares.</p>
<p>Note that 15 = 8² - 7² as well
as 4² - 1².</p>
<p>Write the number 105 as the difference of two squares in as many
different ways as you can?</p>
<p>The number 1155 can be written as the difference of two squares
in eight different ways, can you find them?</p>
</td>
<td width="50%"><mdo:image height="200" width="200" src="odd_sq.gif" alt=""></mdo:image></td>
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<p><span class="editorial">Tom started us off with the following
idea. Thanks Tom.</span></p>
In the square of dots you can imagine the"L" of any odd number and
if you take the square for the odd number below it from the square
of the odd number you get the number you stated with. To find the
two squares you have to find half the odd number and round it up
and down. So half of $13$ is $6.5$ so the two squares have sides
$7$ and $6$. <br></br>
<br></br>
Here is a picture:<br></br>
<div style="text-align: center;"><mdo:image height="414" width="470" alt="" src="square1.gif"></mdo:image></div>
<p class="editorial">So, using the diagram you know
that $$105= \frac{105-1}{2} + \frac{105+1}{2}$$</p>
<p class="editorial">$$ 105 = 52 +53$$</p>
<p class="editorial">$$ 105 = 53^2 - 52^2$$</p>
<p class="editorial">You can always find one way using this method.
So</p>
<p class="editorial">$$1155= \frac{1155-1}{2} +
\frac{1155+1}{2}$$</p>
<p class="editorial">$$ 1155 = 577+578$$</p>
<p class="editorial">$$ 105 = 578^2 - 577^2$$</p>
<p class="editorial">Alex, from the Grammar School at Leeds extends
the algebra building on the second visualisation:</p>
<p style="text-align: center;" class="editorial"><mdo:image height="200" width="495" alt="" src="OddSquaresNotes.gif"></mdo:image></p>
<div>The task is to find eight pairs of numbers $(a,b)$, satisfying
the following equation: $$a^2 + b^2 =115$$</div>
<div>The LHS can be factorized to $(a+b)(a-b)$, so now we can begin
to find eight factor pairs of $1155$:</div>
<br></br>
<div>Using prime factorization,</div>
<div>$155 = 5 \times 231 = 5 \times7\times33 = 5 \times7 \times3
\times11$</div>
<div>Now we use different selections of these prime factors to
build the list of eight factor pairs:</div>
<div>\begin{eqnarray} 1155 &=& 5 \times231 \\ &=& 7
\times165 \\ &=& 3 \times385 \\ &=&35 \times33 \\
&=&21 \times55 \\ &=& 15 \times77 \\ &=&11
\times105 \\ &=&1\times 115\end{eqnarray}</div>
<br></br>
<div>Then it is simply a matter of substituting each pair into the
following procedure:</div>
<div>$1155 = (a+b)(a-b) = $ Large factor $\times $ small
factor</div>
<div>$a + b =$ large factor</div>
<div>$a - b =$ small factor</div>
<div>$2b =$ large factor$-$ small factor</div>
<div>$b =$ (large $-$small)/$2$</div>
<div>$a = b + $small factor</div>
<br></br>
<div>Applying this to each pair: \begin{eqnarray} 1155 &=&
46^2 - 31^2 \\ &=&58^2- 47^2 \\ &=& 578^2 -577^2 \\
&=&118^2 - 113^2 \\ &=&86^2 - 79^2 \\
&=&194^2 - 191^2 \\ &=&34^2 - 1^2 \\
&=&38^2- 17^2\end{eqnarray}</div>
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<h3>Why do this problem :</h3>
This connection between the set of odd numbers and squares is not often encountered in school and offers a very accessible example of visualisation.<br></br>
<p>The 'difference of two squares' is a key algebraic transformation and problems like this can lead students into a deeper appreciation of that form through a further visualisation.</p>
<h3>Possible approach :</h3>
<div>Start by giving learners time to make sense of the idea that any odd number can be written as the difference of two squares using the dotty grid. Encourage them to investigate with small numbers (for example, $9=5^2 - 4^2$ ) before generalising for larger numbers such as $1155$ and then devising a general rule for anyone to apply.</div>
<br></br>
Remind them of the identity $a^2 - b^2 = (a + b)(a -b)$ and give them time to consider and see the connection with what they have already done. For example: $$9=5^2 - 4^2 = (5+4)(5-4).$$<br></br>
<br></br>
At this point another visualisation might be useful in order to suggest how further differences might be found.<br></br>
<div style="text-align: center;"><mdo:image alt="" height="200" src="OddSquaresNotes.gif" width="500"></mdo:image></div>
<br></br>
<p>The problem of writing the number $105$ as the difference of two squares becomes a problem about factor pairs that make a product of $105$.</p>
<p>Draw out from the students how this transformation helps [we now seek factors of $105$ rather than guessing squares and calculating differences].</p>
<div>Moving to the last part, and even pushing beyond that to a general result, may require the group to spend time making sure that they are secure with an algorithm for producing prime factors.</div>
<h3>Key questions :</h3>
<ul>
<li>Explain how this image shows us the sum of the first n odd numbers - what is that sum ?</li>
<li>
<p>For $105$ (and then for $1155$) how many ways might there be, and why do you think that ?</p>
</li>
</ul>
<h3>Possible extension :</h3>
<div>Is it true that no number can be written as the difference of $2$ squares in exactly three ways, and if so why?</div>
<h3>Possible support :</h3>
<div>Perhaps try the problem <a href="http://nrich.maths.org/658&part=">Plus Minus</a> first.</div>
<div>For students not yet ready for this problem, time spent on finding factors will be valuable. When the group are ready, check that they can use an algorithm to find prime factors and invite them to suggest how they can use that to assist factor-finding.</div>
<br></br>
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<p>Here is a further challenge for you after you have seen this
solution: is it true that no number can be written as the
difference of 2 squares in exactly three ways, and if so why?</p>
<p>Counting the dots which are joined by lines in the diagram we
get the odd numbers 1, 3, 5 , 7 and 9. To make larger squares add
11 dots (i.e. twice five plus one), then 13 (twice six plus one)
and so on, each time adding an odd number of dots. In this way any
odd number (2n+1) can be represented by dots along two sides of an
(n+1) by (n+1) square, showing that:</p>
<p>2n+1 = (n+1)² - n² .</p>
<p>The rest of the question can be done algebraically or by using a
program or a spreadsheet.</p>
<p>Expressing a number as the difference of two squares is
equivalent to factorising the number.</p>
<p>N = a² - b² = (a - b)(a + b).</p>
<p>If we know pairs of factors we can find (a -b) and (a + b) and
then a and b.</p>
<p>To find every possible way of writing 105 as the difference of
two squares first find all the pairs of factors:</p>
<p>105 = 1 x 105 = 3 x 35 = 5 x 21 = 7 x 15.</p>
<p>The results are given in the following table.</p>
<table width="100%">
<tbody>
<tr>
<td align="center" colspan="4">PAIRS OF FACTOR OF 105</td>
<td align="center">DIFFERENCE OF 2 SQUARES</td>
</tr>
<tr>
<td align="center">a - b</td>
<td align="center">a + b</td>
<td align="center">a</td>
<td align="center">b</td>
<td align="center">a² - b²</td>
</tr>
<tr>
<td align="center">1</td>
<td align="center">105</td>
<td align="center">53</td>
<td align="center">52</td>
<td align="center">53² - 52²</td>
</tr>
<tr>
<td align="center">3</td>
<td align="center">35</td>
<td align="center">19</td>
<td align="center">16</td>
<td align="center">19² - 16²</td>
</tr>
<tr>
<td align="center">5</td>
<td align="center">21</td>
<td align="center">13</td>
<td align="center">8</td>
<td align="center">13² - 8²</td>
</tr>
<tr>
<td align="center">7</td>
<td align="center">15</td>
<td align="center">11</td>
<td align="center">4</td>
<td align="center">11² - 4²</td>
</tr>
</tbody>
</table>
<p>The same method gives the eight ways of representing 1155 as the
difference of two squares.</p>
<table width="100%">
<tbody>
<tr>
<td align="center" colspan="4">PAIRS OF FACTOR OF 1155</td>
<td align="center">DIFFERENCE OF 2 SQUARES</td>
</tr>
<tr>
<td align="center">a - b</td>
<td align="center">a + b</td>
<td align="center">a</td>
<td align="center">b</td>
<td align="center">a² - b²</td>
</tr>
<tr>
<td align="center">1</td>
<td align="center">1155</td>
<td align="center">578</td>
<td align="center">577</td>
<td align="center">578² - 577²</td>
</tr>
<tr>
<td align="center">3</td>
<td align="center">385</td>
<td align="center">194</td>
<td align="center">191</td>
<td align="center">194² - 191²</td>
</tr>
<tr>
<td align="center">5</td>
<td align="center">231</td>
<td align="center">118</td>
<td align="center">113</td>
<td align="center">118² - 113²</td>
</tr>
<tr>
<td align="center">7</td>
<td align="center">165</td>
<td align="center">86</td>
<td align="center">79</td>
<td align="center">86² - 79²</td>
</tr>
<tr>
<td align="center">11</td>
<td align="center">105</td>
<td align="center">58</td>
<td align="center">47</td>
<td align="center">58² - 47²</td>
</tr>
<tr>
<td align="center">15</td>
<td align="center">77</td>
<td align="center">46</td>
<td align="center">31</td>
<td align="center">46² - 31²</td>
</tr>
<tr>
<td align="center">21</td>
<td align="center">55</td>
<td align="center">38</td>
<td align="center">17</td>
<td align="center">38² - 17²</td>
</tr>
<tr>
<td align="center">33</td>
<td align="center">35</td>
<td align="center">34</td>
<td align="center">1</td>
<td align="center">34² - 1²</td>
</tr>
</tbody>
</table>
<p><em><strong>The best solution sent in came from Michael Pryor,
Kunal Patel and Matthew Loffman of the Simon Langton Boys School
Canterbury</strong></em> . They set up a spreadsheet with two
columns for values of a and b using the formula</p>
<p>b = sqrt (a² - 1155)</p>
<p>for the entries in the second column (giving a² -
b² =1155 in every row). They were then able to pick out
the required whole number solutions.</p>
<p>They also recognised (from the first part of the question) that
the largest squares would be 578 and 577 squared, and that the
lowest had to be over the square root of 1155, so they ran their
spreadsheet from a = 34 to a = 578 in the first column knowing that
this would give them all the required solutions.</p>
<hr></hr>
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Odd Differences
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.
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