4871 /www/nrich/html/content/id/4871/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <div style="text-align: center;"><mdo:image alt="seven hexagons" height="61" src="hexagons.gif" width="500"></mdo:image></div> <br></br> You have seven hexagonal-shaped mats in a line.<br></br> These mats all have to be turned over, but you can only turn over exactly three at a time.<br></br> You can choose the three from anywhere in the line.<br></br> A mat may be turned over on one move and turned back over again on another.<br></br> <br></br> What is the smallest number of moves you can do this in?<br></br> Try with other numbers of mats.<br></br> Do you notice any patterns in your findings?<br></br> Can you explain why these patterns occur?<br></br> <br></br> You might like to use the interactivity to have a go. Click on the three hexagons which you would like to turn over. The mats are red on one side (this side is face up to start with) and purple on the other. <p style="font-style: normal;"><a href="/content/id/4871/discs.swf">Full Screen Version</a></p> <mdo:flash height="400" id="/content/id/4871/discs.swf" width="400"><param name="allowfullscreen" value="true"></param><param name="movie" value="/content/id/4871/discs.swf"></param> <param name="flashplayerversion" value="7"></param> <param name="height" value="400"></param> <param name="width" value="400"></param> </mdo:flash> <p style="font-style: italic;"> </p> <p style="font-style: italic;"> </p> <p style="font-style: normal;"><a href="http://nrich.maths.org/public/viewer.php?obj_id=5587&amp;part=">Click here for a poster of this problem</a>.</p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p class="editorial">Joe from Bishop Ramsey School looked at the seven mat problem. He said:</p> In this problem it doesn't matter where you start on the diagram. My solution is written in a number of stages:<br></br> <div style="clear: both;">1) pick any 3 hexagons (they will turn blue)<br></br> <div style="clear: both;">2) pick 2 more hexagons and then pick one that you chose in the first go. You should now have 4 hexagons blue.<br></br> <div style="clear: both;">3) finally pick the three remaining red hexagons.<br></br> <div style="clear: both;">It would only have taken you 3 goes.</div> </div> </div> </div> <br></br> <p class="editorial">Many of you answered this part well, including Alistair of Cottenham Primary School.</p> <p class="editorial">Kahlia and Amy from Ardingly College Junior School then looked a bit further and tried with other numbers of mats:</p> If the number of tiles is a multiple of 3 it will divide equally into the number of tiles<br></br> <div style="clear: both;">eg: for 60 tiles, remove three each go: 60 divided by 3 = 20 turns</div> <br></br> <br></br> <p class="editorial">Amelia and Kathryn, also from Ardingly College Junior School, investigated many different numbers of mats in a very systematic way:</p> <br></br> 6= 2 moves <br></br> <div style="clear: both;">7= 3 moves<br></br> <div style="clear: both;">8= 4 moves<br></br> <div style="clear: both;">9= 3 moves<br></br> <div style="clear: both;">10= 4 moves<br></br> <div style="clear: both;">11= 5 moves<br></br> <div style="clear: both;">12= 4 moves<br></br> <div style="clear: both;">13= 5 moves<br></br> <div style="clear: both;">14= 6 moves<br></br> <div style="clear: both;">15= 5 moves<br></br> <div style="clear: both;">16= 6 moves<br></br> <div style="clear: both;">17= 7 moves<br></br> <div style="clear: both;">18= 6 moves<br></br> <div style="clear: both;">19= 7 moves<br></br> <div style="clear: both;">20= 8 moves</div> <div style="clear: both;"></div> <p style="clear: both;" class="editorial">Kahlia and Amy identified a pattern:</p> <div style="clear: both;"> <div style="clear: both;">If there is a number of tiles 1 more than a multiple of 3 you add 1 to the answer of the multiple below it eg: 18 tiles = 6 turns; 21 tiles = 7 turns<br></br> <div style="clear: both;">17 tiles = 7 turns; 20 tiles = 8 turns</div> <div style="clear: both;"></div> <p style="clear: both;" class="editorial">Jeff and Raphael from Zion Heights Junior High School relate this back to the strategy for flipping the mats:</p> <div style="clear: both;"></div> <div style="clear: both;">For numbers with one remainder after dividing by three, you follow the strategy for 7 mats stated above.<br></br> <div style="clear: both;">For numbers with 2 remainders, you flip over two mats instead of one on the second move, thus taking one more move.</div> </div> <div style="clear: both;"></div> <p style="clear: both;" class="editorial">So, thinking about this like Kahlia and Amy did, we could say that if the number of tiles is 2 more than a multiple of 3, you add 2 to the answer of the multiple below it.</p> <p style="clear: both;" class="editorial">Well done to everyone who tackled this problem - it wasn't easy at all.</p> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> </div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> This challenge is quite tricky but children will find it a motivating context in which to develop a logical, systematic approach. <br></br> <br></br> It would be useful to introduce the problem on an interactive whiteboard so that the whole group can be involved with deciding which mats to turn. Alternatively, cardboard mats coloured differently on each side could be used and pinned to a board. This initial whole group work will familiarise the children with the &quot;rules&quot; of the problem so that they will be confident to find the smallest number of moves in pairs. <br></br> <br></br> It will be important for them to devise a recording system that they are happy with, and this is something that can be addressed in the introduction, for example by asking whether they would be able to repeat the moves they made. Encourage them to think about odd and even numbers of flips, and when they come to investigate other numbers of tiles, you might expect them to generalise for multiples of three at least.<br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <div>Does the position of the mats you turn over in your first go matter?</div> <div>What will the mats look like just before you can make the last move?</div> <div>Perhaps you could try with smaller numbers of mats before you go on to more than seven mats.</div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Each hexagon must be flipped over an odd number of times.<br></br> <br></br> For seven hexagons - three moves. If hexagons are numbered from 1 to 7 starting at the left, then moves are, for example:<br></br> <br></br> 1. Flip 1, 2, 3<br></br> 2. Flip 3, 4, 5<br></br> 3. Flip 3, 6, 7<br></br> <br></br> For three hexagons - one move (easy!).<br></br> <br></br> For four hexagons - four moves. If hexagons numbered as above, then:<br></br> <br></br> 1. Flip 1, 2, 3<br></br> 2. Flip 1, 2, 4<br></br> 3. Flip 1, 3, 4<br></br> 4. Flip 2, 3, 4 <br></br> <br></br> For five hexagons - three moves:<br></br> <br></br> 1. Flip 1, 2, 3<br></br> 2. Flip 1, 2, 4<br></br> 3. Flip 1, 2, 5.<br></br> <br></br> For six hexagons - two moves:<br></br> <br></br> 1. Flip 1, 2, 3<br></br> 2. Flip 4, 5, 6.<br></br> <br></br> For eight hexagons - four moves: turn three in one move, then left with five hexagons which is three moves. <br></br> For nine - three moves.<br></br> For ten - one more than seven moves, so four.<br></br> <br></br> Generally:<br></br> If no. hexagons a multiple of 3, then smallest number of moves is number of hexagons divided by 3. <br></br> <br></br></mdoxml> 2 5 0 1 0 0 0 Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time. Information and Communications Technology Interactivities Numbers and the Number System Factors and multiples Numbers and the Number System Odd and even numbers Using, Applying and Reasoning about Mathematics Working systematically Using, Applying and Reasoning about Mathematics Practical Activity