7547
/www/nrich/html/content/id/7547/
1
2011-06-01T17:04:33
<mdoxml version="1.0"><br></br>
Charlie has been thinking about which numbers can be written
as a sum of two square numbers. He took a $10\times10$
grid, and shaded the square numbers in blue and the sums of two
squares in yellow.<br></br>
<br></br>
<div style="text-align: center;"> <mdo:image height="293" width="496" alt="Sums of two squares in ten columns" src="Sums_of_two_squares_10_columns.jpg"></mdo:image></div>
<br></br>
He hoped to find a pattern, but couldn't see anything
obvious. <br></br>
Vicky suggested changing the number of columns in the grid, so they
reduced it by one:<br></br>
<br></br>
<div style="text-align: center;"> <mdo:image height="321" width="438" alt="Sums of two squares in nine columns" src="Sums_of_two_squares_9_columns.jpg"></mdo:image></div>
<div>"There seems to be a diagonal pattern."</div>
<div>"If the rows were one shorter, then those diagonals would line
up into vertical columns, wouldn't they?"</div>
<div>"Let's try it..."</div>
<div> </div>
<div style="text-align: center;"> <mdo:image height="346" width="390" alt="Sums of two squares in eight columns" src="Sums_of_two_squares_8_columns.jpg"></mdo:image></div>
<br></br>
<span style="font-weight: bold;">What do you notice about the
positions of the square numbers?</span><br></br>
<span style="font-weight: bold;">What do you notice about the
positions of the sums of two square numbers?</span><br></br>
<span style="font-weight: bold;"> </span> <br></br>
<span style="font-weight: bold;">Can you make any conjectures about
the columns in which squares, and sums of two squares, would
appear if the grid continued beyond 96?</span><br></br>
<span style="font-weight: bold;"> </span> <br></br>
<span style="font-weight: bold;">Can you prove any of your
conjectures?</span><br></br>
<br></br>
You might like to look back at the nine-column grid and ask
yourself the same questions.<br></br>
<br></br>
Charlie couldn't write every number as a sum of two squares.
He wondered what would happen if he allowed himself three squares.
<br></br>
<br></br>
<div style="text-align: center;"> <mdo:image height="347" width="389" src="Sums_of_three_squares.jpg" alt="Sums of three squares"></mdo:image></div>
<div><span style="font-weight: bold;">Will any of the numbers in
the seventh column be a sum of three squares?</span></div>
<div><span style="font-weight: bold;">Can you prove
it?</span></div>
<div style="font-weight: bold;"> </div>
<div>"We <span style="font-style: italic;">must</span> be able to
write every number if we are allowed to include sums of four
squares!"</div>
<div>"Yes, but it's not easy to prove. Several great
mathematicians worked on it over a long period before <a href="http://www-groups.dcs.st-and.ac.uk/%7Ehistory/Biographies/Lagrange.html">
Lagrange</a> gave the first proof in 1770."</div>
<div> </div>
<div><span style="font-style: italic;">With thanks to Vicky Neale
who created this task in collaboration with
NRICH.</span> </div>
<br></br></mdoxml>
<mdoxml version="1.0">
<p><span class="editorial">Well done to Christopher from Sale Grammar School who sent us</span> <a href="/content/id/7547/Christopher.pdf">this solution</a> <span class="editorial">using modular arithmetic.</span></p>
</mdoxml>
<mdoxml version="1.0"><br></br>
<h3>Why do this problem?</h3>
This problem provides an interesting context in which students can
apply algebraic techniques and ideas about modular
arithmetic. It also gives them a taste of an area of number
theory that they might study if they go beyond the school
curriculum.<br></br>
<br></br>
<h3>Possible approaches</h3>
<div><span style="font-style: italic;">This problem follows on
nicely from</span> <a href="http://nrich.maths.org/7405&part=" style="font-style: italic;">What Numbers Can We Make?</a></div>
<div> </div>
<div>"Mathematicians have been interested in which numbers can be
written as a sum of square numbers. Here are the numbers that
we can make as a sum of two squares."</div>
<div><span style="font-style: italic;">Show</span> <a style="font-style: italic;" href="/content/id/7547/Sums_of_two_squares_10_columns.jpg">grid</a><span style="font-style: italic;">.</span> </div>
<div> </div>
<div>"There don't seem to be any obvious patterns here. But the
numbers are only in ten columns because we're used to grids like
this. Perhaps we should try a different number of columns
instead, like nine."</div>
<div><span style="font-style: italic;">Show</span> <a style="font-style: italic;" href="/content/id/7547/Sums_of_two_squares_9_columns.jpg">grid</a><span style="font-style: italic;">.</span> </div>
<div> </div>
<div>"Can we see any patterns this time?"</div>
<div>[Students might notice the two empty vertical columns.
They might also notice a diagonal pattern (top right to bottom
left). Suggest that this diagonal pattern could become
vertical if we make each row one shorter.]</div>
<div><span style="font-style: italic;">Show</span> <a style="font-style: italic;" href="/content/id/7547/Sums_of_two_squares_8_columns.jpg">grid</a><span style="font-style: italic;">.</span> </div>
<div> </div>
<div>Hand out <a href="/content/id/7547/FillingGaps.pdf">this copy
of the grid</a> and give students some time, working in pairs, to
look for patterns, make predictions, and explain those
predictions. You might want to encourage students to start by
looking at the three completely empty columns.</div>
<div> </div>
<div>Possible prompts if students are having difficulties providing
convincing/rigorous explanations:</div>
<div>In which columns do the square numbers appear?</div>
<div>In which columns do the squares of even numbers appear?
Can you explain why? </div>
<div>And the squares of odd numbers? Can you explain
why?</div>
<div>How can we describe the numbers in a particular
column?</div>
<div> </div>
<div>Bring the class together to pool ideas, and then offer <a href="/content/id/7547/FillingGaps2.pdf">this grid</a> with sums of
three squares for further investigation. Some students might
also like to consider what will happen when we add four
squares.</div>
<br></br>
<br></br>
<h3>Possible extension</h3>
<div>Suggest that students look for patterns that they can explain
in the nine-column grid.</div>
<div> </div>
<div>Students could also experiment with grids with different
numbers of columns.</div>
<br></br>
<br></br>
<h3>Possible support</h3>
Ensure that students have worked on <a href="http://nrich.maths.org/7405&part=">What Numbers Can We
Make?</a><br></br>
<br></br></mdoxml>
<mdoxml version="1.0">It may be helpful to look at the problem <a href="http://nrich.maths.org/7405&part=">What Numbers Can We
Make?</a><br></br>
<br></br>
In which columns do the square numbers appear?<br></br>
In which columns do the squares of even numbers appear? Can
you explain why?<br></br>
And the squares of odd numbers? Can you explain why?<br></br>
How can we describe the numbers in a particular column?<br></br>
<br></br>
You might want to start by looking at the completely empty
columns.<br></br></mdoxml>
2
4
0
0
0
1
0
Filling the gaps
Which numbers can we write as a sum of square numbers?
Numbers and the Number System
Square numbers
Numbers and the Number System
Modulus arithmetic
Numbers and the Number System
Properties of numbers