7821
/www/nrich/html/content/id/7821/
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0000-00-00T00:00:00
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><p style="text-align: center;"><mdo:image src="MagicV.png" style="float: right;"></mdo:image></p>
<p><br></br>
Charlie has made a Magic V with five consecutive numbers:</p>
<p>It is a Magic V because each 'arm' has the same total.</p>
<p> </p>
<p>Alison drew this magic V: <mdo:image src="AlisonMagicV.png" style="text-align: center; vertical-align: middle;"></mdo:image></p>
<p> </p>
<p>Charlie said "That's really just the same Magic V as mine!"</p>
<p>What do you think Charlie meant?</p>
<p>Click below to see the other Magic Vs that Charlie considers to be the same as his:</p>
<p> </p>
<div class="toggle"><mdo:image src="magicv2.png"></mdo:image> <mdo:image src="magicv1.png"></mdo:image> <mdo:image src="magicv3.png"></mdo:image> <mdo:image src="magicv4.png"></mdo:image> <mdo:image src="magicv5.png"></mdo:image> <mdo:image src="magicv6.png"></mdo:image></div>
<p> </p>
<p><strong>Can you find other Magic Vs using the numbers $1$ to $5$ that are <em>different</em> from Charlie's?</strong></p>
<p>How will you know when you have found all the different Magic Vs using the numbers $1$ to $5$?</p>
<p><br></br>
What happens if you use the numbers from $2$ to $6$?</p>
<p>From $3$ to $7$? $\dots$</p>
<p> </p>
<p>You can use <a class="spreadsheetlink" href="/content/id/7821/Magic%20V.xls">this spreadsheet</a> to investigate Magic Vs made from any five consecutive numbers.</p>
<p>Try to find a strategy to find efficiently all Magic Vs and their totals for any given set of numbers.</p>
<p><strong>Can you describe how to find the possible Magic Vs using the numbers $987, 988, 989, 990, 991$?</strong></p>
<p><strong>Can you describe an efficient strategy for finding a Magic V where each arm has a total of 1000?</strong></p>
<p> </p>
<p>Charlie and Alison drew some more magic letters.</p>
<p> </p>
<table border="0" cellpadding="1" cellspacing="1" style="width: 500px;">
<tbody>
<tr>
<td style="text-align: center;">A Magic L</td>
<td style="text-align: center;">A Magic N</td>
<td style="text-align: center;">A Magic W</td>
</tr>
<tr>
<td><mdo:image src="MagicL.png"></mdo:image></td>
<td><mdo:image src="Magic%20N.png"></mdo:image></td>
<td><mdo:image src="Magic%20W.png"></mdo:image></td>
</tr>
</tbody>
</table>
<p> </p>
<p>Investigate some of these Magic Letters in the same way that you explored Magic Vs. What general conclusions can you reach? You can use <a class="spreadsheetlink" href="/content/id/7821/Magic%20Letters.xls">this spreadsheet</a> to explore.<br></br>
<br></br>
<a href="http://nrich.maths.org/public/viewer.php?obj_id=6814&part=">Click here</a> for a poster version of the Magic Vs.<br></br>
</p></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><p><span class="editorial">This was an interesting problem that was open to a lot of interpretation, so many of you came up with new and inventive ways of answering the question, or answered questions we hadn't even asked - it's great to see that you all enjoyed the challenge!</span></p>
<p><span class="editorial">Many people gave ideas as to why Charlie might think that Alison's Magic V was the same as his. Shreya, from Claremont High School, wrote:</span></p>
<p>Charlie's V had 3, 4 and 1 down one side and 5, 2 and 1 down the other side, so each side added up to 8. Alison's V also had the numbers from 1 to 5 arranged in such positions so that one side of the V added up to 8. Therefore, she had the same magic V as Charlie had!</p>
<p><span class="editorial">Ben, from Wilson's Grammar School, had the same thoughts. Ayngharran and Rohan, also from Wilson's, thought something a little more specific. Ayngharran said:</span></p>
<p>Charlie used the same numbers as Alison, with the same 'middle' number 1, and the same numbers on each side but just in a different order. So the pairs 3, 4 and 5, 2 are the same in both cases.</p>
<p><span class="editorial">Zoya, from St. Hilda's C of E Primary School, said:</span></p>
<p>You will know if you have found all the Magic Vs adding up to a certain number, as each Magic V will always have eight ways of rearranging the arms to still add up to the same total.</p>
<p><span class="editorial">Elliott, from Solihull School, found a different Magic V with the same numbers:</span></p>
<table style="width: 143px; height: 96px" border="1">
<tbody>
<tr>
<td>2</td>
<td> </td>
<td> </td>
<td> </td>
<td>1</td>
</tr>
<tr>
<td> </td>
<td>3</td>
<td> </td>
<td>4</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td>5</td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>
<p><span class="editorial">Ben also found a similar Magic V. Some people made interesting comments about what sorts of Magic Vs might exist with these numbers. Abopakr, from Globe Academy, said:</span></p>
<p>The base can only be an odd number if you are using numbers 1 - 5, because if you put an even number at the base there will be only one even number not at the base. So it has to be an odd number at the base. Whichever is the most common kind of number - even or odd - is the kind that goes at the base.</p>
<p><span class="editorial">Michelle, also from Globe Academy, had the same idea, and so did Utkarsh, from Sancta Maria International School. Murat and Callum, from Globe Academy, said:</span></p>
<p>You need to put the lowest number on the same side as the highest number.</p>
<p><span class="editorial">Adam, from Collis Primary School, made the following Magic V and commented:</span></p>
<table style="width: 144px; height: 96px;border-spacing:1px;" border="1">
<tbody>
<tr>
<td>4</td>
<td> </td>
<td> </td>
<td> </td>
<td>5</td>
</tr>
<tr>
<td> </td>
<td>3</td>
<td> </td>
<td>2</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td>1</td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>
<p>All you do is start at the bottom, then zigzag up and right, then left, then up and left, then right.</p>
<p><span class="editorial">That's a nice systematic way of constructing a Magic V - well spotted! (I wonder whether we can make bigger Magic Vs in this way?)</span></p>
<p><span class="editorial">Kavi, from Wilson's, suggested:</span></p>
<p>To construct a Magic V with the numbers 2-6, just take a Magic V with the numbers 1-5 and raise each number by 1. Then each arm will sum to 3 more.</p>
<p><span class="editorial">Maciej, from Wilson's, gave us the following example:</span></p>
<table style="width: 146px; height: 96px;border-spacing:1px;" border="1">
<tbody>
<tr>
<td>4</td>
<td> </td>
<td> </td>
<td> </td>
<td>3</td>
</tr>
<tr>
<td> </td>
<td>5</td>
<td> </td>
<td>6</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td>2</td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>
<p><span class="editorial">An anonymous student in the UK gave the following example:</span></p>
<table style="width: 145px; height: 96px;border-spacing:1px;" border="1">
<tbody>
<tr>
<td>7</td>
<td> </td>
<td> </td>
<td> </td>
<td>9</td>
</tr>
<tr>
<td> </td>
<td>8</td>
<td> </td>
<td>6</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td>5</td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>
<p><span class="editorial">Ben solved our 987-991 puzzle in the same way:</span></p>
<table style="width: 139px; height: 96px;border-spacing:1px;" border="1">
<tbody>
<tr>
<td>988</td>
<td> </td>
<td> </td>
<td> </td>
<td>987</td>
</tr>
<tr>
<td> </td>
<td>989</td>
<td> </td>
<td>990</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td>991</td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>
<p><span class="editorial">Some people came up with different types of Magic Vs. For example, Daniel, from Wilson's, solved our '1000' puzzle in this way:</span></p>
<p>What I did was start with a magic V I already knew:</p>
<table style="width: 145px; height: 96px;border-spacing:1px;" border="1">
<tbody>
<tr>
<td>2</td>
<td> </td>
<td> </td>
<td> </td>
<td>1</td>
</tr>
<tr>
<td> </td>
<td>3</td>
<td> </td>
<td>5</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td>4</td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>
<p><br></br>
Each arm had a total of ten. I then multiplied each number in the V by 100.</p>
<table style="width: 107px; height: 96px;border-spacing:1px;" border="1">
<tbody>
<tr>
<td>200</td>
<td> </td>
<td> </td>
<td> </td>
<td>100</td>
</tr>
<tr>
<td> </td>
<td>300</td>
<td> </td>
<td>500</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td>400</td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>
<p><span class="editorial">Interesting - this doesn't use consecutive numbers, but it's still a Magic V. Well done! Ben gave lots of examples of non-consecutive Magic Vs like this:</span></p>
<table style="border-spacing: 1px; width: 650px; height: 135px;" border="1">
<tbody>
<tr>
<td>1</td>
<td> </td>
<td> </td>
<td> </td>
<td>1</td>
<td> </td>
<td>1.5</td>
<td> </td>
<td> </td>
<td> </td>
<td>2.5</td>
<td> </td>
<td>250</td>
<td> </td>
<td> </td>
<td> </td>
<td>250</td>
</tr>
<tr>
<td> </td>
<td>3</td>
<td> </td>
<td>3</td>
<td> </td>
<td> </td>
<td> </td>
<td>4.5</td>
<td> </td>
<td>3.5</td>
<td> </td>
<td> </td>
<td> </td>
<td>250</td>
<td> </td>
<td>250</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td>5</td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td>5</td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td>500</td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>
<p><span class="editorial">The last one also solves our '1000' puzzle.</span></p>
<p><span class="editorial">Shavindra, from Wilson's, gave us a formulaic way of finding a Magic V with each arm summing to N (as long as N is divisible by 3):</span></p>
<table style="width: 389px; height: 132px;border-spacing:1px;" border="1">
<tbody>
<tr>
<td>(N/3) + 1</td>
<td> </td>
<td> </td>
<td> </td>
<td>(N/3) + 2</td>
</tr>
<tr>
<td> </td>
<td>(N/3) - 1</td>
<td> </td>
<td>(N/3) - 2</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td>(N/3)</td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>
<p><span class="editorial">Ellis, from Westfield Middle School, gave us a Magic L, and a comment:</span></p>
<table style="width: 50px; height: 127px;border-spacing:1px;" border="1">
<tbody>
<tr>
<td>1</td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>3</td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>4</td>
<td> </td>
<td> </td>
</tr>
<tr>
<td>5</td>
<td>2</td>
<td>6</td>
</tr>
</tbody>
</table>
<p>The number in the corner must be odd. After you've done one, you can jumble the numbers on each arm.</p>
<p><span class="editorial">Finally, Rebecca and Angus came up with this observation:</span></p>
<p>If you take the middle number of the five consecutive numbers for a Magic V and multiply it by 3, it will give you the middle number of the range of totals you could get for that magic V.<br></br>
Also, there are only 3 consecutive totals for each V depending on which number is on the bottom of the V.<br></br>
For example, for numbers 12-16, the middle number is 14.<br></br>
14 times 3 is 42, so the range of totals is 41 to 43.</p>
<p><span class="editorial">Excellent! Thanks to everyone for all your contributions.</span></p></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div><a href="/7821">This problem</a> gives students opportunities to perceive structure in a simple case and make generalisations based on that structure.</div>
<div> </div>
<div>While offering students a chance to practise addition in an engaging context, there will also be opportunities to appreciate the effect of combining odd and even numbers.</div>
<div> </div>
<h3>Possible approach</h3>
<div>Draw 5 circles on the board in a V shape, and give students mini-whiteboards or paper.</div>
<div>"I want you to arrange the numbers from 1 to 5 in a V shape like mine; you can put the numbers in any order you like."</div>
<div> </div>
<div>Ask students to hold up their V shapes. Divide the board into two columns, headed 'Magic' and 'Not Magic'. Select some of the class's Vs and copy them on the board in the appropriate column, according to the definition in <a href="/7821">the problem</a>.</div>
<div>"While I'm writing up your Vs, be thinking about what criteria I might be using to determine whether a V is magic or not."</div>
<div> </div>
<div>Once students have worked out what makes a V magic, clarify what makes Vs different; for example, these Vs are equivalent:</div>
<div> </div>
<div><mdo:image src="magicv1.png"></mdo:image> <mdo:image src="magicv2.png"></mdo:image> <mdo:image src="magicv3.png"></mdo:image></div>
<div> </div>
<div>"Have we found all possible non-equivalent Magic Vs that can be made with the numbers from 1 to 5? Work with your partner to find any more, or convince yourselves that there are no more to find."</div>
<div> </div>
<div>Take some time to discuss their arguments for why there are only three possible Magic Vs. Possible suggestions might touch upon:</div>
<ul>
<li>As the numbers from 1 to 5 add up to an odd total, the repeated number at the bottom must be odd.</li>
<li>If we put a 2 or a 4 at the bottom, we'd have an odd total left over to split between the two arms, so it can't be done equally.</li>
<li>If there was an even number at the bottom, there'd be three odds and one even left, so they can't be split to make the same total on each side.</li>
</ul>
<div> </div>
<div>"Close your eyes and imagine you wanted to make a Magic V using the numbers from 2 to 6 instead. Can you picture what the Vs could look like, using the insights you have from making the Vs with 1 to 5?"</div>
<div> </div>
<div>"Now open your eyes and talk to your partner. Write down your Vs and convince each other that you found them all."</div>
<div> </div>
<div>Bring the class together and share their convincing explanations.</div>
<div> </div>
<div>"In a few minutes, I'm going to give you five consecutive numbers bigger than 100. I want you to be able to tell me straight away how to make the complete set of Magic Vs that use those five numbers. Talk to your partner and agree a strategy."</div>
<div> </div>
<div>Once students have had a chance to consider how they will tackle the challenge, bring the class together and challenge some pairs of students to produce Magic Vs from randomly selected sets of 5 consecutive numbers.</div>
<div> </div>
<div><strong>Next</strong>, move on to exploring the arm totals of Magic Vs.</div>
<div>"Here is the set of Magic Vs we made with the numbers 1 to 5. The arm totals were 8, 9 and 10. In a while, I'm going to ask you to produce a Magic V with a particular arm total. You need to devise a strategy for working this out quickly."</div>
<div> </div>
<div>As students are working on this, circulate and look out for students who are struggling to get started. You could prompt them:</div>
<div>"What were the arm totals when we used the numbers 2 to 6?"</div>
<div>"Why is the arm total for the V with a 2 at the bottom 3 more than the arm total for the V with a 1 at the bottom?"</div>
<div>"What is special about the arm total when the number at the bottom of the V is the middle number of the five consecutive numbers?"</div>
<div> </div>
<div>Notice too any students who are developing efficient strategies.</div>
<div>If appropriate, bring the class together so that those who have developed a strategy can demonstrate (without explanation!).</div>
<div> </div>
<div>Finally, bring the class together and challenge pairs to find Magic Vs with particular arm totals, sharing and explaining how they do it.</div>
<div> </div>
<h3>Possible extension<br></br>
</h3>
<div>Magic Ls, Ns and Ws can be explored in the same way.</div>
<h3> </h3>
<h3>Possible support</h3>
<div> </div>
<div>Giving students digit cards or inviting them to write the numbers on small pieces of paper that can be moved around easily might help them to experiment with different arrangements.</div>
<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><p><br></br>
What do you notice about the number at the base of the V in all your magic Vs?</p></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><h3>Magic Vs</h3>
<p>Always 3 solutions</p>
<p>If the 5 numbers contain more evens than odds, the number at the botttom will be even. Otherwise odd.</p>
<p>If the average of the five numbers is n, the totals of each arm will be 3n-1, 3n and 3n+1</p>
<p>Alison's method to find a magic V with a particular arm total:</p>
<p>Say we wanted a magic V with an arm total of 157. There are three possible 1-5 magic Vs with totals of 8, 9 and 10. 157 is one more than a multiple of 3, so I need to start from the magic V with a total of 10. To get from 10 to 157, I need to add 147, which is 3 times 49. So if I add 49 to each of the numbers in the magic V with a total of 10, I'll get a total of 157.</p>
<h3> </h3>
<table border="0" cellpadding="1" cellspacing="1" style="width: 500px; height: 96px">
<tbody>
<tr>
<td>1</td>
<td> </td>
<td> </td>
<td> </td>
<td>3</td>
<td> </td>
<td>50</td>
<td> </td>
<td> </td>
<td> </td>
<td>52</td>
</tr>
<tr>
<td> </td>
<td>4</td>
<td> </td>
<td>2</td>
<td> </td>
<td> </td>
<td> </td>
<td>53</td>
<td> </td>
<td>51</td>
<td> </td>
</tr>
<tr>
<td> </td>
<td> </td>
<td>5</td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td>54</td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>
<h3> </h3>
<h3>Magic Ls</h3>
<p>Sometimes 3 solutions (e.g. when using the numbers 2 to 7)</p>
<p>Sometimes 2 solutions (e.g. when using the numbers 4 to 9)</p>
<p>Sometimes 1 solutions (e.g. when using the numbers 6 to 11)</p>
<p>Sometimes no solutions (e.g. when using the numbers 11 to 16)</p>
<p>The number at the vertex will always be odd.</p>
<p> </p>
<h3>Magic Ns</h3>
<p>Always 7 solutions.</p>
<p>If the 7 consecutive numbers contain 4 evens and 3 odds, the number at the centre of the N will be odd. Otherwise even.</p>
<p> </p></mdoxml>
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Magic Letters
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
Numbers and the Number System
Properties of numbers
Using, Applying and Reasoning about Mathematics
Generalising
Numbers and the Number System
Odd and even numbers