4835
/www/nrich/html/content/id/4835/
1
2011-02-01T00:00:01
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<mdoxml version="1.0"><br></br>
<p>A $2$ by $3$ rectangle contains $8$ squares. Can you see how?<br></br>
<br></br>
<mdo:image alt="2 by 3 rectangle" height="86" src="squares3.gif" width="127"></mdo:image><br></br>
<br></br>
A $3$ by $4$ rectangle contains $20$ squares. Can you see how?<br></br>
<br></br>
<mdo:image alt="3 by 4 rectangle" height="124" src="squares1.gif" width="165"></mdo:image><br></br>
<br></br>
A $4$ by $6$ rectangle contains $50$ squares. Can you see how?<br></br>
<br></br>
<mdo:image alt="4 by 6 rectangle" height="165" src="squares2.gif" width="243"></mdo:image><br></br>
<br></br>
<br></br>
What size rectangle contains exactly $100$ squares?<br></br>
<br></br>
Is there more than one?<br></br>
Can you find them all?<br></br>
Can you prove that there are no more?<br></br>
<br></br>
<a href="http://nrich.maths.org/public/viewer.php?obj_id=5586&part=">Click here for a poster of this problem</a>.<br></br>
</p></mdoxml>
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<p> </p>
<p><span class="editorial">Many people suggested the answer should be $6$ by $8$, because this is "double" the $6$ by $4$ example given in the problem. However, this doesn't take into account squares contained in the</span> <span class="editorial">$6$ by $8$ rectangle which are not contained in the original</span> <span class="editorial">$6$ by $4$ rectangle,</span> <span class="editorial">like
the red ones below:</span><br></br>
<br></br>
<mdo:image alt="5 by 5 square" height="310" src="5by5cropped.gif" width="235"></mdo:image> <mdo:image alt="Squares" height="310" src="squaresanswercropped.gif" width="235"></mdo:image></p>
<br></br>
<p class="editorial">In the first picture, we have a $5 \times 5$ square. You cannot fit any of these in a $6 \times 4$ rectangle, but you can fit them in a $6 \times 8$ rectangle.</p>
<p><span class="editorial">Imagine the $6 \times 8$ rectangle is made up from joining two $6 \times 4$ rectangles. In the second picture, you can see a $2 \times 2$ square. Where the $6 \times 4$ rectangles meet there will be an extra row of these $2 \times 2$ squares that would not be in either of the $6 \times 4$ rectangles.</span></p>
<p> </p>
<div><span class="editorial">One way to approach this problem is to try some examples and set your working out in a table, and then look for patterns and try to explain them. Sam from OPGS sent us his table:</span></div>
<br></br>
<table border="1">
<tbody>
<tr>
<td style="font-weight: bold;">
<div style="text-align: center;">columns:</div>
<div style="text-align: center;">rows:</div>
</td>
<td style="font-weight: bold;">
<div style="text-align: center;">1</div>
</td>
<td style="font-weight: bold;">
<div style="text-align: center;">2</div>
</td>
<td style="font-weight: bold;">
<div style="text-align: center;">3</div>
</td>
<td style="font-weight: bold;">
<div style="text-align: center;">4</div>
</td>
<td style="font-weight: bold;">
<div style="text-align: center;">5</div>
</td>
<td style="font-weight: bold;">
<div style="text-align: center;">6</div>
</td>
<td style="font-weight: bold;">
<div style="text-align: center;">7</div>
</td>
</tr>
<tr>
<td style="font-weight: bold; text-align: center;">1</td>
<td style="font-style: italic; text-align: center;">1</td>
<td style="font-style: italic; text-align: center;">2</td>
<td style="font-style: italic; text-align: center;">3</td>
<td style="font-style: italic; text-align: center;">4</td>
<td style="font-style: italic; text-align: center;">5</td>
<td style="font-style: italic; text-align: center;">6</td>
<td style="font-style: italic; text-align: center;">7</td>
</tr>
<tr>
<td style="font-weight: bold; font-style: normal; text-align: center;">2</td>
<td style="font-style: normal; text-align: center;">2</td>
<td style="font-style: italic; text-align: center;">5</td>
<td style="font-style: italic; text-align: center;">8</td>
<td style="font-style: italic; text-align: center;">11</td>
<td style="font-style: italic; text-align: center;">14</td>
<td style="font-style: italic; text-align: center;">17</td>
<td style="font-style: italic; text-align: center;">20</td>
</tr>
<tr>
<td style="font-weight: bold; font-style: normal; text-align: center;">3</td>
<td style="font-style: normal; text-align: center;">3</td>
<td style="font-style: normal; text-align: center;">8</td>
<td style="font-style: italic; text-align: center;">14</td>
<td style="font-style: italic; text-align: center;">20</td>
<td style="font-style: italic; text-align: center;">26</td>
<td style="font-style: italic; text-align: center;">32</td>
<td style="font-style: italic; text-align: center;">38</td>
</tr>
<tr>
<td style="font-weight: bold; font-style: normal; text-align: center;">4</td>
<td style="font-style: normal; text-align: center;">4</td>
<td style="font-style: normal; text-align: center;">11</td>
<td style="font-style: normal; text-align: center;">20</td>
<td style="font-style: italic; text-align: center;">30</td>
<td style="font-style: italic; text-align: center;">40</td>
<td style="font-style: italic; text-align: center;">50</td>
<td style="font-style: italic; text-align: center;">60</td>
</tr>
<tr>
<td style="font-weight: bold; font-style: normal; text-align: center;">5</td>
<td style="font-style: normal; text-align: center;">5</td>
<td style="font-style: normal; text-align: center;">14</td>
<td style="font-style: normal; text-align: center;">26</td>
<td style="font-style: normal; text-align: center;">40</td>
<td style="font-style: italic; text-align: center;">55</td>
<td style="font-style: italic; text-align: center;">70</td>
<td style="font-style: italic; text-align: center;">85</td>
</tr>
<tr>
<td style="font-weight: bold; font-style: normal; text-align: center;">6</td>
<td style="font-style: normal; text-align: center;">6</td>
<td style="font-style: normal; text-align: center;">17</td>
<td style="font-style: normal; text-align: center;">32</td>
<td style="font-style: normal; text-align: center;">50</td>
<td style="font-style: normal; text-align: center;">70</td>
<td style="font-style: italic; text-align: center;">91</td>
<td style="font-style: italic; text-align: center;">112</td>
</tr>
<tr>
<td style="font-weight: bold; text-align: center;">7</td>
<td style="text-align: center;">7</td>
<td style="text-align: center;">20</td>
<td style="text-align: center;">38</td>
<td style="text-align: center;">60</td>
<td style="text-align: center;">85</td>
<td style="text-align: center;">112</td>
<td style="font-style: italic; text-align: center;">140</td>
</tr>
</tbody>
</table>
<p> </p>
<p>Since the number of squares in an $n \times m$ rectangle is the same as the number of squares in an $m \times n$ rectangle, we will look only at rectangles with at least as many columns as rows. These are italicised in the above table.<br></br>
<br></br>
First we notice that, if there is one row, increasing the number of columns by one increases the number of squares by $1$. This is because the only size of square that we can make is $1 \times 1$, so adding a column adds just one square.<br></br>
<br></br>
If there are two rows, adding an extra column increases the number of squares by $3$. Why is this? Well, if we add an extra column we can make $2$ additional $1 \times 1$ squares and $1$ additional $2 \times 2$ square. This is a total of three extra squares. If we continue the pattern $2, 5, 8, \ldots$ in the $2$ row, we eventually get to $\ldots, 98, 101, \ldots$ - missing out $100$. This tells
us that it isn't possible to make a rectangle with $100$ squares which has $2$ rows.<br></br>
<br></br>
If there are three rows, adding an extra column allows us to make 3 more $1 \times 1$ squares, $2$ more $2 \times 2$ squares and $1$ more $3 \times 3$ square. This is a total of $6 = 3 + 2 + 1$ squares. This gives us the sequence $14, 20, 26, 32, \ldots, 98, 104, \ldots$ which tells us that we can't make a rectangle with exactly $100$ squares which has $3$ rows.<br></br>
<br></br>
Using the same reasoning for four rows, we see adding a column increases the number of squares by $10$, and that we can make rectangles with $20, 30, \ldots, 90, 100, 110, \ldots$ squares. $100$ is in this list! In fact, a $4 \times 11$ rectangle contains exactly $100$ squares.<br></br>
<br></br>
We can repeat this for the other rows in the table to find all the rectangles with exactly $100$ squares. These are $1 \times 100$, $4 \times 11$ and $5 \times 8$.</p>
<p> </p>
<p><span class="editorial">Mr Lunn's Year 7 class from The Wensleydale School reasoned in a very similar way.</span> <u><a href="/content/id/4835/Squares%20within%20rectangles.xls">Here</a></u> <span class="editorial">is a summary of their findings, showing a link between the results and Triangle Numbers.</span><br></br>
<br></br>
<span class="editorial">Sandeep from Nrayana Junior College and Terence from Brumby Engineering College used a slightly different approach to count the number of squares in rectangles of different sizes, starting with the examples from the problem page. Here is Sandeep's method:</span><br></br>
<br></br>
In a $2 \times 3$ rectangle, squares of side $1$ unit and squares of side $2$ units can be formed.<br></br>
<br></br>
Number of squares of side $1$ unit is $2 \times 3 = 6$.<br></br>
Number of squares of side $2$ units is $1 \times 2 = 2$.<br></br>
Therefore the total number of squares in a $2 \times 3$ rectangle is $6 + 2 = 8$.<br></br>
<br></br>
Similarly if we consider a $3 \times 4$ rectangle there are a total of $20$ squares.<br></br>
<br></br>
Number of squares of side $1$ unit: $3 \times 4 = 12$.<br></br>
Number of squares of side $2$ units: $2 \times 3 = (3-1) \times (4-1) = 6$.<br></br>
Number of squares of side $3$ units: $1 \times 2 = (3-2) \times (4-2) = 2$.<br></br>
Total number of squares $= 12 + 6 + 2 = 20$.<br></br>
<br></br>
We can generalise this fact by taking a rectangle of $x$ rows and $y$ columns: $$\text{Total number of squares} = x \times y + (x-1) \times (y-1) + (x-2) \times (y-2) + \ldots$$ The addition series stops only when either the rows or the columns reduces to 1.<br></br>
<br></br>
By trial and error, 5 $\times$ 8 and 4 $\times$ 11 rectangles contain exactly 100 squares. There can be no other rectangles containing 100 squares. This can be verified by increasing or decreasing the number of rows and columns.<br></br>
</p>
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<mdoxml version="1.0"><h3>Why do this problem?</h3>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=4835&part=">This
problem</a> is a context for systematic number work, geometrical
thinking and problem solving.</div>
<div>It is an excellent example of a situation where the thinking
involved in analysing one rectangle can be applied directly to
other rectangles. These transferable insights about the structure
of the problem can then be expressed as algebraic statements about
all rectangles.</div>
<h3>Possible approach</h3>
<div>As students enter, display the $20$ and $50$ diagram, asking
how many squares there are. It may be appropriate to give the
answers then ask pairs to come and explain - one to talk, and one
to write/draw/record on the board. Well laid out number work will
help with the algebra later, so the students' boardwork should
prompt more suggestions about how to record working for this
problem.</div>
<br></br>
<div>Present the problem, give students an opportunity to share
first ideas. Several approaches (working backwards, trial and
error, building up from smaller ones, systematic searching) might
be suggested, and advantages/disadvantages discussed.</div>
<br></br>
<div>Encourage students to compare results with peers, and to
resolve discrepancies by mathematical argument, rather than relying
on the teacher's spreadsheet (see below). It might be useful to
gather the results of the students as they work, to help them to
see patterns and encourage them to conjecture the results for other
rectangles.</div>
<br></br>
<div>With a group who have not moved towards algebra, a final
plenary could ask for observations about the rectangles, and
discuss how each can be expressed algebraically.</div>
<h3>Key questions</h3>
Is there an obvious rectangle which contains $100$ squares?
<div>How might you organise a search for rectangles with exactly
$100$ squares?</div>
<div>Is what you're describing specific to this rectangle? how does
it generalise?</div>
<h3>Possible extension</h3>
<div>Prove that you have all the rectangles.</div>
<div>Can you find an algebraic rule for the number of squares
contained in an '$m \times m$' square? an '$m \times n$'
rectangle?</div>
<div>For a given area, which rectangle gives the largest total
number of squares? Can you show this in general?</div>
<div>If the original question didn't say $100$, what other numbers
(under $100$) would give a problem with non-trivial solutions? Is
there a pattern to these?</div>
<div>Set up a spreadsheet to calculate numerical solutions to these
problems.</div>
<h3>Possible support</h3>
<div>Struggling students could shade squares on <a href="/content/id/4835/3by4%20and%203by5.doc">worksheets</a> (<a href="/content/id/4835/4%20by%206.doc">2nd sheet</a> ) with lots of
small copies of the rectangles. Encourage them to work
systematically, in order to observe the structure, and then make
conjectures about the numbers of the next size of square, or in the
next rectangle.</div>
<br></br>
<div>Once they, either individually or as a group, have worked out
the full counting they could be asked to do this activity: In a
pair, each guess a rectangle which might have $40$ squares. Swap
squares and each work out the total. Whose guess was closest to
$40$? Between you, can you come up with a better guess?</div>
<br></br>
<br></br>
<br></br>
<br></br>
<div>This <a href="/content/id/4835/squares%20in%20rectangles%204835.xls">spreadsheet</a>
contains a calculator to work out the totals and also lists all of
the possibilities; you might find this helpful for checking
students' work quickly.</div>
<br></br>
<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
A $2$ by $3$ rectangle contains $8$ squares:<br></br>
<br></br>
six small $1\times 1$ squares and two larger $2 \times 2$
squares<br></br>
<mdo:image height="86" width="127" alt="A 2 by 3 rectangle" src="squares3.gif"></mdo:image><br></br>
<br></br>
<br></br>
A $3$ by $4$ rectangle contains $20$ squares: <br></br>
<br></br>
twelve $1\times1$ squares, six $2 \times 2$ squares and two $3
\times 3$ squares<br></br>
<br></br>
<mdo:image height="124" width="165" src="squares1.gif" alt="A 3 by 4 rectangle"></mdo:image><br></br>
<br></br>
<p>Consider rectangles with a height of $2$ units.<br></br>
Increase their width by $1$ unit at a time.<br></br>
What effect does this have on the total number of squares?</p>
What about rectangles with a height of $3, 4, 5, \ldots$?<br></br>
<br></br>
Make a note of the number of squares in rectangles with a height of
$2$ units.<br></br>
<br></br>
Do you notice anything special?<br></br>
<br></br>
Use your results to decide whether a rectangle with a height of $2$
units can contain exactly $100$ squares?<br></br>
<br></br>
What about rectangles with a height of $3, 4, 5, \ldots$?<br></br>
<br></br>
Draw up a table of results.<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
$1\times100$<br></br>
$4 \times11$<br></br>
$5 \times8$<br></br>
<br></br>
possibly provide the whole completed table of results showing the
few occasions when $100$ appears<br></br></mdoxml>
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Squares in rectangles
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle
contains 20 squares. What size rectangle(s) contain(s) exactly 100
squares? Can you find them all?
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