2790
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2011-02-01T00:00:01
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You might like to try <a href="http://nrich.maths.org/public/viewer.php?obj_id=2782&part=index">A Ring of Numbers</a> and <a href="http://nrich.maths.org/public/viewer.php?obj_id=2783&part=index">More Rings of Numbers</a> before this problem.<br></br>
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Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. (You must use each of the numbers once.)<br></br>
Can you find some other ways to do this? Explain how you do this.<br></br>
<br></br>
Can you put the numbers in the squares so that the difference between joined squares is even?<br></br>
Explain your answer.<br></br>
<br></br>
What general statements can you make about odd and even numbers?<br></br>
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<a href="/content/id/2790/Differences.swf" onclick="mediaSave()">Full size version</a><br></br>
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<h5>This problem is based on an idea taken from "Apex Maths Pupils' Book 2" by Ann Montague-Smith and Paul Harrison, published in 2003 by Cambridge University Press. To order a copy of this book, see their <a href="http://titles.cambridge.org">o</a><a href="http://education.cambridge.org/subject/mathematics/apex-maths">nline catalogue .</a></h5>
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<mdoxml version="1.0"><p><span class="editorial">The Maths Group from Devonshire Primary
sent in some solutions that they found so that the differences
between joined squares is odd:</span></p>
<br></br>
9 2 7<br></br>
<div style="clear: both;">8 5 4<br></br>
<div style="clear: both;">3 6 1</div>
<div style="clear: both;"></div>
<div style="clear: both;"><br></br>
9 6 7<br></br>
<div style="clear: both;">8 5 4<br></br>
<div style="clear: both;">3 2 1</div>
<div style="clear: both;"></div>
<div style="clear: both;"><br></br>
3 6 1<br></br>
<div style="clear: both;">8 5 4<br></br>
<div style="clear: both;">9 2 7</div>
<div style="clear: both;"></div>
<div style="clear: both;"><br></br>
7 6 9<br></br>
<div style="clear: both;">8 5 4<br></br>
<div style="clear: both;">1 2 3</div>
<div style="clear: both;"></div>
<div style="clear: both;"><br></br>
5 6 7<br></br>
<div style="clear: both;">8 9 4<br></br>
<div style="clear: both;">3 2 1</div>
<div style="clear: both;"></div>
<div style="clear: both;"><br></br>
9 6 7<br></br>
<div style="clear: both;">8 1 4</div>
<div style="clear: both;">3 2 5</div>
<div style="clear: both;"></div>
<div style="clear: both;"><br></br>
9 6 7<br></br>
<div style="clear: both;">8 3 4<br></br>
<div style="clear: both;">5 2 1</div>
<div style="clear: both;"></div>
<div style="clear: both;"><br></br>
9 6 5<br></br>
<div style="clear: both;">8 7 4<br></br>
<div style="clear: both;">3 2 1</div>
<div style="clear: both;"></div>
<div style="clear: both;"><br></br>
9 6 7<br></br>
<div style="clear: both;">2 5 4<br></br>
<div style="clear: both;">3 8 1</div>
<div style="clear: both;"></div>
<div style="clear: both;"><br></br>
9 6 7<br></br>
<div style="clear: both;">8 5 2<br></br>
<div style="clear: both;">3 4 1</div>
<div style="clear: both;"></div>
<div style="clear: both;"><br></br>
9 4 7<br></br>
<div style="clear: both;">8 5 6<br></br>
<div style="clear: both;">3 2 1</div>
<div style="clear: both;"></div>
<div style="clear: both;"><br></br>
3 6 7<br></br>
<div style="clear: both;">8 5 4<br></br>
<div style="clear: both;">9 2 1</div>
<div style="clear: both;"></div>
<div style="clear: both;"><br></br>
1 2 3<br></br>
<div style="clear: both;">4 5 8<br></br>
<div style="clear: both;">7 6 9</div>
<div style="clear: both;"></div>
<div style="clear: both;"><br></br>
3 2 1<br></br>
<div style="clear: both;">8 5 4<br></br>
<div style="clear: both;">9 6 7</div>
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<br></br>
<p><span class="editorial">Children from Merton Park
said:</span></p>
You have to have an odd number next to an even number or it (the
difference) won't be odd.<br></br>
<div style="clear: both;">You can have any numbers in any place as
long as the odd numbers are in the corners and middle.</div>
<br></br>
<p class="editorial">Well done - you're spot on and in fact that
means there are many solutions - thousands! Taranjot and Amrita
at Alexandra Junior School point out that:</p>
It is not possible to make even differences using each number once.
Even numbers cannot be next to even numbers and odd numbers cannot
be next to odd, because their differences would be even.<br></br>
<br></br>
<p class="editorial">Harmohan, Akash and Ayman also from Alexandra
School came up with some general rules about odd and even
numbers:</p>
Even is e and odd is o<br></br>
e-o=o <br></br>
<div style="clear: both;">o-e=o<br></br>
<div style="clear: both;">e+o=o<br></br>
<div style="clear: both;">o+e=o</div>
<div style="clear: both;">e-e=e<br></br>
<div style="clear: both;">o-o=e<br></br>
<div style="clear: both;">o+o=e<br></br>
<div style="clear: both;">e+e=e</div>
</div>
</div>
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<p><span class="editorial">Thank you to all those of you who sent
in solutions.</span></p>
<br></br>
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<mdoxml version="1.0"><div class="embed">
<h2>Number Differences</h2>
<br></br>
You might like to try <a href="http://nrich.maths.org/public/viewer.php?obj_id=2782&part=index">A Ring of Numbers</a> and <a href="http://nrich.maths.org/public/viewer.php?obj_id=2783&part=index">More Rings of Numbers</a> before this problem.<br></br>
<br></br>
Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. (You must use each of the numbers once.)<br></br>
Can you find some other ways to do this?<br></br>
Is it possible to put the numbers in the squares so that the difference between joined squares is even?<br></br>
Explain your answer.<br></br>
What would you need to change for it to be possible?<br></br>
What general statements can you make about odd and even numbers?<br></br>
<br></br>
<a href="/content/id/2790/Differences.swf" onclick="mediaSave()">Full size version</a><br></br>
<br></br>
<mdo:flash height="400" id="/content/id/2790/Differences.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param>
<param name="movie" value="/content/id/2790/Differences.swf"></param>
<param name="flashplayerversion" value="7"></param>
<param name="height" value="400"></param>
<param name="width" value="550"></param>
</mdo:flash><br></br>
<br></br>
<h5>This problem is based on an idea taken from "Apex Maths Pupils' Book 2" by Ann Montague-Smith and Paul Harrison, published in 2003 by Cambridge University Press. To order a copy of this book, or others published by CUP, see their <a href="http://titles.cambridge.org">online catalogue</a> .</h5>
<br></br>
</div>
<br></br>
Building on <a href="http://nrich.maths.org/public/viewer.php?obj_id=2782&part=index">Ring a Ring of Numbers</a> and <a href="http://nrich.maths.org/public/viewer.php?obj_id=2783&part=index">More Numbers in the Ring</a> , this problem encourages pupils to form early stages of proof by using their knowledge of odd and even numbers to construct mathematical arguments, leading to
generalisation. If not using the interactivity, it may be useful for children to have a <a href="/content/id/2790/Number%20Differences.pdf">sheet of blank grids</a> in order to try out their ideas.<br></br>
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Try putting any number in one square to start with.<br></br>
What numbers could go in the squares which join this first square? What do all these options have in common? Can this help you find other arrangements of the numbers? <br></br>You might like to print out this <a href="/content/id/2790/numdiff.doc">sheet of blank grids</a> if you are not using the interactivity.<br></br></mdoxml>
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Number Differences
Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?
Information and Communications Technology
Interactivities
Numbers and the Number System
Odd and even numbers
Calculations and Numerical Methods
Addition & subtraction
Using, Applying and Reasoning about Mathematics
Generalising
Admin
Upper primary mapping document