2401
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2011-02-01T00:00:01
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<p>Here are four tiles:</p>
<p align="center"><mdo:image align="top" alt="4 similar squares with 2 blue and 2 green edges" bgcolor="" height="73" src="greenbluesquares.gif" width="314"></mdo:image></p>
<p>They can be arranged in a $2$ by $2$ square so that this large square has a green edge:</p>
<p align="center"><mdo:image align="top" alt="4 squares arranged to make a bigger square with green edges" bgcolor="" height="124" src="bluesqu.gif" width="124"></mdo:image></p>
<p>If the tiles are moved around, we can make a $2$ by $2$ square with a blue edge:</p>
<p align="center"><mdo:image align="top" alt="4 squares arranged to make a bigger square with blue edges" bgcolor="" height="124" src="greensqu.gif" width="124"></mdo:image></p>
<p>If I had nine tiles it would be quite easy to paint them so that, when they were arranged in a $3$ by $3$ square, the edge of this large square is green. I would need four tiles for the corners of the square, four tiles for the edges and one tile would go in the middle of the square so wouldn't need painting at all.</p>
<p>This is how the green-edged square would be made:</p>
<p align="center"><mdo:image align="top" alt="3 by 3 square with green edges" bgcolor="" height="184" src="squ.gif" width="190"></mdo:image></p>
<p>But I also want to be able to make a square with a blue edge and another square with a yellow edge.</p>
<p>How can the other sides of these tiles be painted so that all nine tiles can be rearranged to make two more $3$ by $3$ squares - one with a blue edge and one with a yellow edge?</p>
<p>Now try to colour sixteen tiles so that four $4$ by $4$ squares can be made - one with a green edge, one with a blue edge, one with a yellow edge and one with a red edge.</p>
<p>Find a way to colour $25$ tiles so that five $5$ by $5$ squares can be made, each with a differently coloured edge.</p>
<p>Do you think this is possible for $36$ tiles and six coloured edges?</p>
<p>Will it always be possible to add an extra colour as the squares get larger?</p>
<p>For a 3D version of this problem why not try " <a href="http://nrich.maths.org/public/viewer.php?obj_id=2424&part=index">Inside Out</a> "?</p>
<p> </p>
<p><a href="http://nrich.maths.org/7195&part=">Click here for a poster of this problem</a>. </p>
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<span class="editorial">Adam of Moorside School sent us in his
description of a possible painting for the three by three
square:</span> <br></br>
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The outside of the square should be green. The rest of one of the
corner squares should be yellow. The rest of another corner square
should be blue. The other two corner squares should have one side
blue and the other side yellow. Then two of the squares in between
the corner squares should have two touching sides yellow and one
side blue. The other two squares in between the corner squares
should have two touching sides blue and one side yellow. The middle
square should be half yellow, half blue. <br></br>
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<span class="editorial">Adam's square looks like this:</span>
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<div style="text-align: center;"><mdo:image height="182" width="182" alt="Adam's square" src="On%20the%20edge.gif"></mdo:image></div>
<div style="text-align: center;"></div>
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<div><span class="editorial">Arthur sent us these great pictures
for four by four and five by five squares. Thank you
Arthur!</span></div>
<div style="text-align: center;"><mdo:image height="215" width="215" alt="Four by four solution" src="squares4small.gif"></mdo:image></div>
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<div style="text-align: center;"><mdo:image height="273" width="273" src="squares5small.gif" alt="five by five solution"></mdo:image></div>
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<span class="editorial">Jenny made a prediction:</span> <br></br>
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``I noticed that in an $n \times n$ square there are a total on
$n^2$ small squares, and therefore a total of $4 n^2$ edges to be
coloured. Each colour takes up $4 n$ of these edges, so it should
be possible to colour an $n \times n$ square with $n$ different
colours. To do this, we need to arrange for all the colours to have
four cornerpieces and $4 (n-2)$ edge pieces." <br></br></mdoxml>
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<h3><span style="font-weight: bold;">Why do</span> <span style="font-weight: bold;">this problem</span> <span style="font-weight: bold;">?</span></h3>
<a href="http://nrich.maths.org/public/viewer.php?obj_id=2401">This
problem</a> is a familar context involving perimeters and squares
that requires careful analysis. There are opportunities to
experiment but a need to think ahead.
<h3>Possible approach</h3>
<div>Arrange four large squares into a $2 \times 2$ array. Colour
the edges - counting from $1$ to $8$ as you go along.</div>
<div>Rearrange the four squares to enable you to colour the edges
in a different colour (without repeat and counting $1$ to $8$
again).</div>
<br></br>
<div>Arrange nine large squares into a $3 \times 3$ array and
colour the edges - counting from $1$ to $12$ this time.</div>
<br></br>
<div><span style="font-weight: bold;">Rearrange</span> the nine
squares in a haphazard way to enable you to colour the edges in a
different colour (counting $1$ to $12$).</div>
<br></br>
<div style="margin-left: 40px;">"I wonder whether we might be able
to colour the edge in a third colour?"</div>
<div style="margin-left: 40px;"></div>
<div style="margin-left: 40px;"></div>
<div><span style="font-weight: bold;">Rearrange</span> the nine
squares again and "discover" that you are not going to be able to
do it - at least not on this occasion.</div>
<br></br>
<div style="margin-left: 40px;">"Does this mean we cannot do
it?"</div>
<div style="margin-left: 40px;">or</div>
<div style="margin-left: 40px;">"What went wrong?"</div>
<div style="margin-left: 40px;">or</div>
<div style="margin-left: 40px;">"Does this mean it is
impossible?"</div>
<br></br>
<div>Look out for responses that develop the idea that you need to
be able to draw $12$ lines and there are $12$ "blank" edges.
However, not all of these are on the edge of the new array.</div>
<br></br>
<div style="margin-left: 40px;">"So it feels like it should be
possible with a bit more thought".</div>
<br></br>
<div>Hand out sets of paper squares to pairs of students and
challenge them to complete the task successfully.</div>
<br></br>
<div>When any pair is successful challenge them to repeat the
task:</div>
<br></br>
<div style="margin-left: 40px;">"Well done. Would you be able to do
this again without making any "false moves"?"</div>
<br></br>
<div>After the first colouring, can students position the squares
for the second colouring knowing it will work for the third?</div>
<br></br>
<div>Move on to larger square arrays.</div>
<br></br>
<div style="margin-left: 40px;">"Can we use more colours when we
have $16$ squares?"</div>
<div style="margin-left: 40px;">"How about with $25$
squares...?"</div>
<br></br>
<h3>Key questions</h3>
<div>Can you anticipate where certain squares will have to go in
future rounds?</div>
<br></br>
<h3>Possible extension</h3>
<div>Can students do all three colourings without rearranging the
squares?</div>
<br></br>
<div>Will it always be possible to add an extra colour as the
squares get larger?</div>
<br></br>
<div>For a 3D version of this problem students could try <a href="http://nrich.maths.org/public/viewer.php?obj_id=2424&part=">Inside
Out</a></div>
<br></br>
<h3>Possible support</h3>
<div>Lots of paper squares so students do not have to worry about
making mistakes.</div>
<br></br>
<div>A $4 \times 4$ array is probably easier.</div>
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<mdoxml version="1.0">Take one colour at a time and see how many edges are being used up
on the edge of the big square.
<p>Once the edges have been used they can be 'hidden' in the
middle</p>
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<mdoxml version="1.0"> If you have a square of side n then its perimeter is 4n and the number of tiles you have is n x n. This gives you 4n x n edgesto the individual tiles. These can be arranged so that they can create n different squares with n diffent colours on the edges.<br></br><br></br></mdoxml>
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On the Edge
Here are four tiles. They can be arranged in a 2 by 2 square so
that this large square has a green edge. If the tiles are moved
around, we can make a 2 by 2 square with a blue edge... Now try to
colour sixteen tiles so that four 4 by 4 squares can be made - one
with a green edge, one with a blue edge, one with a yellow edge and
one with a red edge.
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Perimeters
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Squares
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smartphone
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