2783
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2011-02-01T00:00:01
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Before doing this problem, it would be a good idea to look at <a href="http://nrich.maths.org/public/viewer.php?obj_id=2782&part=index">Ring a Ring of Numbers</a>
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Change the ring so that there are only 3 squares. <br></br>
Can you place three different numbers in them so that their differences are odd? <br></br>
Can you make the differences even? <br></br>
What do you notice about the sum of each pair in each case?<br></br>
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Try with different numbers of squares around the ring. <br></br>
What happens with 5 squares? 6 squares?<br></br>
What do you notice?<br></br>
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<a href="/content/id/2783/Multiring.swf">Full size version</a>
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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>
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<p class="editorial">Vidhya from Kensri School in India sent in a very well reasoned solution:</p>
When we subtract an even number from an odd number, or vice versa, the
difference is always odd. So if we fill up odd numbers and even numbers
alternately, if there are an even number of squares, the differences will all
be odd. But there is no solution <span class="editorial">(in other words the differences cannot all be odd)</span>
if there is an odd number of squares.<br></br>
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<h2>More Numbers in the Ring</h2>
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Before doing this problem, it would be a good idea to look at <a href="http://nrich.maths.org/public/viewer.php?obj_id=2782&part=index">Ring a Ring of Numbers</a> .<br></br>
<br></br>
Change the ring so that there are only 3 squares.<br></br>
Can you place three different numbers in them so that their differences are odd?<br></br>
Can you make the differences even?<br></br>
What do you notice about the sum of each pair in each case?<br></br>
<br></br>
Try with different numbers of squares around the ring.<br></br>
What happens with 5 squares? 6 squares?<br></br>
What do you notice?<br></br>
<br></br>
<a href="/content/id/2783/Multiring.swf">Full size version</a><br></br>
<br></br>
<mdo:flash height="480" id="/content/id/2783/Multiring.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="movie" value="/content/id/2783/Multiring.swf"></param>
<param name="flashplayerversion" value="7"></param>
<param name="height" value="480"></param>
<param name="width" value="550"></param>
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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, or others published by CUP, see their <a href="http://titles.cambridge.org">online catalogue</a> .</h5>
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<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<a href="http://nrich.maths.org/2783&part=">This problem</a> builds on <a href="http://nrich.maths.org/2782&part=">Ring a Ring of Numbers</a>. It encourages children to start from different examples and then begin to draw some more general conclusions based on their understanding of odd and even numbers.<br></br>
<h3>Key questions</h3>
<div>What happens when you put one more number in the ring?</div>
<div>What happens when you put two more numbers in the ring?</div>
<div>What happens when there is an odd number of numbers in the ring?</div>
<div>What happens when there is an even number of numbers in the ring? </div>
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<h3>Possible extension</h3>
The problem <a href="http://nrich.maths.org/2790&part=">Number Differences</a> makes a good follow-up challenge.<br></br>
<h3>Possible support</h3>
<div>Some children will benefit from spending more time on the <a href="http://nrich.maths.org/2782&part=">Ring a Ring of Numbers</a> problem. Having digit cards to move around on a large piece of paper will also help if they are not using the interactivity. Pupils might also find it useful to have sheets of blank rings so that they can try different combinations of numbers:</div>
<div><a href="/content/id/2783/3sheet.doc">Sheet of 3 rings</a></div>
<div><a href="/content/id/2783/ringaringsheet.doc">Sheet of 4 rings</a></div>
<div><a href="/content/id/2783/5sheet.doc">Sheet of 5 rings</a></div>
<div><a href="/content/id/2783/6sheet.doc">Sheet of 6 rings</a></div>
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Try putting one of the numbers in any square to start with. What numbers could go on each side of it?
When you add the numbers in two joined squares, what kind of number do you get?
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You might like to print off these sheets of blank rings to help you try out some different numbers:<br></br>
<br></br><a href="/content/id/2783/3sheet.doc">
Sheet of 3 rings</a><br></br><a href="/content/id/2783/ringaringsheet.doc">
Sheet of 4 rings</a><br></br><a href="/content/id/2783/5sheet.doc">
Sheet of 5 rings</a><br></br><a href="/content/id/2783/6sheet.doc">Sheet of 6 rings</a><br></br></mdoxml>
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With 3 squares, not possible to make odd difference as need
alternate odd/even around circle.<br></br>
Can make even difference by all odd numbers or all even
numbers.<br></br>
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When there are an odd number of squares in the ring, making an odd
difference is impossible.<br></br>
When there are an even number of squares in the ring, making an
odd difference is possible by having alternate odd/even numbers
round ring.<br></br>
The sum of adjacent squares is always odd.<br></br>
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It is always possible to make an even difference as long as the
numbers are all odd or all even.<br></br>
The sum of adjacent pairs is always even.<br></br>
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More Numbers in the Ring
If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?
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