32
/www/nrich/html/content/97/12/bbprob1/
1
2011-02-01T00:00:01
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<p style="text-align: center;"><mdo:image width="356" height="79" alt="pic 2" src="Picture%202.jpg"></mdo:image></p>
<p>Aunt Jane had been to a jumble sale and bought a whole lot of
cups and saucers - she's having many visitors these days and felt
that she needed some more. You are staying with her and when she
arrives home you help her to unpack the cups and saucers.</p>
<p>There are four sets: a set of white, a set of red, a set of blue
and a set of green. In each set there are four cups and four
saucers. So there are sixteen cups and sixteen saucers
altogether.</p>
<p>Just for the fun of it, you decide to mix them around a bit so
that there are sixteen different-looking cup/saucer combinations
laid out on the table in a very long line.</p>
<p>So, for example:</p>
<p>a) there is a red cup on a green saucer but not another the
same, although there is a green cup on a red saucer;<br></br>
b) there is a red cup on a red saucer but that's the only one like
it.</p>
<p>There are these sixteen different cup/saucer combinations
on the table and you think about arranging them in a big square.
Because there are sixteen, you realise that there are going to
be four rows with four in each row (or if you like,
four rows and four columns).</p>
<a href="" class="control" onclick="tablealter(1)"></a>
<table width="50%">
<tbody>
<tr>
<td width="75%" style="text-align: center;"> </td>
<td width="25%">
<div style="text-align: center;"><mdo:image width="191" height="191" alt="b-rc" src="b_rc.gif"></mdo:image></div>
<br></br>
<br></br>
<br></br>
</td>
</tr>
</tbody>
</table>
<div>So here is the challenge to start off this
investigation:</div>
<div>Place these sixteen different combinations of cup/saucer in
this four by four arrangement with the following rules:-</div>
<p>1) In any row there must only be one cup of each colour;<br></br>
2) In any row there must only be one saucer of each colour;<br></br>
3) In any column there must only be one cup of each colour;<br></br>
4) In any column there must be only one saucer of each colour.</p>
<p style="text-decoration: underline; font-weight: bold;">Remember
that these sixteen cup/saucers are all different so, for
example, you CANNOT have a red cup on a green saucer somewhere and
another red cup on a green saucer somewhere else.</p>
<p style="text-decoration: none; font-weight: 400;">There are a lot
of different ways of approaching this challenge.</p>
When you think you have completed it, check it through very
carefully. It's even a good idea to get a friend who has seen
the rules to check it also.<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<p><span class="editorial">Sophie from Bridge Learning Campus,
Bristol sent in the following, which pleased me in the way that she
describes her work.</span></p>
<br></br>
1. I did the teacups first, white teacups first going down
diagonally, top left to bottom right, green tea cups next going
diagonally down from the top right to bottom left. It didn't matter
which one I did first.<br></br>
2. I did the red teacups in the places I could put them and
added the blue teacups to the empty spaces.<br></br>
3. Next I did the saucers on the top row to match the cup and
saucer, I did the column going downwards on the left and repeated
it on the other columns until I found the pattern.<br></br>
4. I looked at the shapes with my teacher and by joining the
blue cups, I started to see a parallelogram in the blue cups and
one the same in the red teacups.<br></br>
5. I changed teacups and saucers for numbers and saw that there
were patterns in the numbers, when I drew shapes over the same
numbers for the saucers, I saw four triangle like shapes with the
points going into the four corners. I enjoyed the
investigation!<br></br>
<p class="editorial">More than half of the solutions had a common
slip-up in that there were several combinations of cups and saucers
that were repeated. The challenge is to have them mixed up so that
no two cup/saucers werre the same. I was very pleased to see that
various computer programs were being used to send in the solutions,
like Samuel, Minc and Julian from Manilla who sent this in:</p>
<p class="editorial" style="text-align: center;"> <mdo:image height="142" width="142" src="Samuel%20M%20%26%20J%20Tea%20Cups.jpg" alt="Samuel M J"></mdo:image></p>
<p class="editorial">However I've put all the solutions into the
same format for easy comparison. Well done! All of you listed
here:</p>
<br></br>
<br></br>
<div style="text-align: center;"> <mdo:image height="810" width="540" src="New%20All%20C%20%26%20S.jpg" alt="New All"></mdo:image></div>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><div class="embed">
<h2>Tea Cups</h2>
<br></br>
<p style="text-align: center;"><mdo:image alt="pic 2" height="79" src="Picture%202.jpg" width="356"></mdo:image></p>
<p>Aunt Jane had been to a jumble sale and bought a whole lot of cups and saucers - she's having many visitors these days and felt that she needed some more. You are staying with her and when she arrives home you help her to unpack the cups and saucers.</p>
<p>There are four sets: a set of white, a set of red, a set of blue and a set of green. In each set there are four cups and four saucers. So there are sixteen cups and sixteen saucers altogether.</p>
<p>Just for the fun of it, you decide to mix them around a bit so that there are sixteen different-looking cup/saucer combinations laid out on the table in a very long line.</p>
<p>So, for example:</p>
<p>a) there is a red cup on a green saucer but not another the same, although there is a green cup on a red saucer;<br></br>
b) there is a red cup on a red saucer but that's the only one like it.</p>
<p>There are these sixteen different cup/saucer combinations on the table and you think about arranging them in a big square. Because there are sixteen, you realise that there are going to be four rows with four in each row (or if you like, four rows and four columns).</p>
<table style="" border="">
<tbody>
<tr>
<td style="text-align: center;" width="75%"> </td>
<td width="25%" style="">
<div style="text-align: center;"><mdo:image alt="b-rc" height="191" src="b_rc.gif" width="191"></mdo:image></div>
<br></br>
<br></br>
</td>
</tr>
</tbody>
</table>
<div>So here is the challenge to start off this investigation:</div>
<div>Place these sixteen different combinations of cup/saucer in this four by four arrangement with the following rules:-</div>
<p>1) In any row there must only be one cup of each colour;<br></br>
2) In any row there must only be one saucer of each colour;<br></br>
3) In any column there must only be one cup of each colour;<br></br>
4) In any column there must be only one saucer of each colour.</p>
<p style="font-weight: bold; text-decoration: underline;">Remember that these sixteen cup/saucers are all different so, for example, you CANNOT have a red cup on a green saucer somewhere and another red cup on a green saucer somewhere else.</p>
<p style="font-weight: 400; text-decoration: none;">There are a lot of different ways of approaching this challenge.</p>
When you think you have completed it, check it through very carefully. It's even a good idea to get a friend who has seen the rules to check it also.<br></br>
</div>
<br></br>
<h3>Why do this problem?</h3>
<div>Doing <a href="http://nrich.maths.org/public/viewer.php?obj_id=32&part=index">this problem</a> is an excellent way to work at problem solving for pupils in late primary and early secondary school. It lends itself to being worked on by small groups of learners so that they have an opportunity to decide on approaches. Then, having completed the challenge, groups can discuss the different
ways of solving it including each method's strengths and weaknesses. If the introductory story is told to the full it also is a good problem in which pupils have to decide what information is useful and what is irrelevant.</div>
<br></br>
<h3>Possible approach</h3>
<div>Telling the story with little or no warning about why you're telling it is a good way to get learners engaged with this problem. Having set the scene, you could begin by giving the class a chance to work in pairs to find the sixteen different combinations of cup/saucer.</div>
<br></br>
<div>It's a good idea to encourage children to work in pairs or small groups (perhaps up to four) when solving this challenge. Give them time to work on it and then gather the whole group together to have a discussion about where they have got to so far. This sharing of ideas will help move everyone forward - you will find that different groups have approached the task in different ways (for
example some might set out the saucers first, others might keep the cup/saucer pairs together). Some might come up with relevant points which they think are important.</div>
<div> </div>
<div>Once they have reached solutions, invite groups to share their approaches again. Is it possible to decide on a particuarly 'good' way of solving this challenge? (Learners themselves will be able to articulate what they mean by 'good' in this context.) It would then be interesting to encourage each group to have another go at the problem using an alternative method of their
choice. </div>
<br></br>
<div>The results of this investigation would make an engaging and attractive classroom display.</div>
<br></br>
<h3>Key questions</h3>
<div>Tell me about this.</div>
<div>What do you think you have to do?</div>
<div>What have you tried so far?</div>
<div>Have you checked that all your cup and saucer combinations are different?</div>
<br></br>
<h3>Possible extension</h3>
<div>The problem of course enters into its own when questions are asked like, " I wonder what would happen if we ...?" For example, children might consider using three sets of cups and saucers; using plates to go with the cups and saucers when you have three colours.</div>
<br></br>
<h3>For the exceptionally mathematically able</h3>
<h4 style="font-weight: 400;">You can expect these pupils to look carefully at different solutions (as opposed to different ways of solving the challenge) and then compare them to sort out similarities and differences as well as equivalences. Then the problem can be extended to include a third attribute, for example a plate, so that the cup, saucer, plate combination would use three different
colours. Deciding how to record solutions in this case is quite a challenge.</h4>
<h4 style="font-weight: 400;">Is it possible to arrange the cups and saucers if the diagonals also have to be different? Is there a system for getting all the possible answers?</h4>
<h3>Possible support</h3>
<div>Children have done this activity using a variety of different materials to help them - it can be made part of the challenge for them to decide on the materials they will use. You could start with just three differently coloured cups/saucers to be arranged in a 3 by 3 grid so that the aim of the investigation is understood. (The <a href="http://nrich.maths.org/public/viewer.php?obj_id=108&part=index">Teddy Town</a> problem is essentially the same investigation as this one, but starts at a simpler point.)</div>
<br></br>
<div>With children near the end of primary school, the activity can be approached in a different way although the challenges are essentially the same - it's all about using playing cards. The saucers would be replaced by the suits and the cups by the value of the cards. So we have four suits and four different values of the cards. [I've used the Ace, King, Queen and Jack, although of course any
four will do.]</div>
<br></br>
<h3 style="font-weight: bold;">Extra</h3>
<div style="font-weight: 400;">Suppose we are using playing cards. There are quite a few solutions, but I'll take just one particular kind.</div>
<div style="font-weight: 400;">The first solution here puts the Jack, Queen, King and Ace of differing suits in the central 2 by 2 square. Then, the outside ones in each row and column are gradually puzzled out - often through a kind of logic. So here is a typical result.</div>
<div style="text-align: center;"><mdo:image alt="" height="327" src="1stCards.jpg" width="259"></mdo:image></div>
<div style="text-align: left;">The pupils can then be asked to tell you what they notice about this arrangement. Some will recall the rules and say that is what they notice. Others notice patterns. So, one way of opening it out further is to concentrate on the patterns of the solutions found. Here is another solution based upon the same starting places in the centre square.</div>
<div style="text-align: center;"><mdo:image alt="" height="328" src="2ndCard.jpg" width="259"></mdo:image></div>
<div style="text-align: left;">You can get some discussion going concerning the patterns they see. Some will see the Aces forming a pattern - since they stand out more, so you may ask, " Do you see any other patterns in the way that the cards are placed.</div>
<div style="text-align: left;">Then there are the suits to look at, you can ask the pupils to describe, draw, record etc. what they've noticed.</div>
<div style="text-align: left;">Here, from the first solution above, is one of the many ways that they may come up with.</div>
<div style="text-align: center;"><mdo:image alt="" height="147" src="Rel4-1%20Suit.jpg" width="135"></mdo:image></div>
<div style="text-align: left;">This shows how the suits form shapes, we have Green for Hearts, Blue for Spades, Grey for Clubs and Red for Diamonds.</div>
<div style="text-align: center;">------------------------------</div>
<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt="" height="147" src="Rel4-1Numb.jpg" width="138"></mdo:image></div>
<div style="text-align: left;">This shows how the values of the cards link to form shapes, we have Green for Kings, Blue for Aces, Grey for Queens and Red for Jacks.</div>
<div style="text-align: center;">-------------------------------</div>
<div style="text-align: center;"><mdo:image alt="" height="146" src="Rel4-2%20Suit.jpg" width="135"></mdo:image><mdo:image alt="" height="150" src="Rel4-2Numb.jpg" width="130"></mdo:image></div>
<div style="text-align: left;">These two show patterns also derived from looking firstly at the suits and then the values.</div>
<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">Have you checked that all your cup and saucer combinations are
different?<br></br>
What could you try on the diagonals?<br></br>
Is there any pattern to the way you have arranged the cups and
saucers so far? Could you continue the pattern?<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
When I want to create a solution and I may have forgotten one then
I work on it in the following way;<br></br>
<mdo:image width="415" height="500" alt="" src="Ans.jpg"></mdo:image><br></br>
<br></br>
I place the four "doubles" in a square to br the middle [diag 1].
East and West of the top two must be red/blue and blue/red so Ihave
a 50% chance of being correct [diag 2] East and West of the lower
pair must be white/green and green/white [diag 3] I then do the
North and South of the left two of the original square, these must
be blue/green and green/blue [diag 4] Similarly for the right hand
side [diag 5] This just leaves the four corners and that's very
easy [diag 6].<br></br>
<p class="editorial">The very quick solution that Sarah found
satisfied her a lot. She used some bits of coloured papers, and
used the four doubles across the top to start off.</p>
<br></br>
<mdo:image alt="Sarah, Diagram A" src="sarah_a.gif"></mdo:image><br></br>
<p class="editorial">She noticed how each of the four corner
arrangements of four contained one of each colour for both the cups
and the saucers. She also had managed to achieve differences
throughout the diagonals as well, although that was not asked
for.</p>
<p class="editorial">This led me to look more closely:-</p>
<br></br>
<mdo:image alt="Modified Sarah, Diagram A" src="4A.gif"></mdo:image><br></br>
<p class="editorial">Here I copied Sarah's top left hand corner
four, but I slightly changed the top right hand four saucers so
that the clockwise order of saucers [ white, blue, red & green]
should remain the same, and the arrangement was just a rotation of
180'. The bottom left hand four then turned out to be a flip [along
the y axis] from the top left hand four. The bottom right hand then
came from either rotating the bottom left hand four OR flipping the
top right hand four . These were the saucers sorted out and the
cups worked in a similar way but with the transformations swopped
over, as shown in the diagramme.</p>
<p class="editorial">Many youngsters have produced a result like
this :-</p>
<br></br>
<br></br>
<mdo:image alt="Teacup diagram" src="4B.gif"></mdo:image><br></br>
<p class="editorial">Here the diagonals are of one saucer colour
each. It is a good solution to look at as it has some interesting
patterns in it.</p>
<br></br>
<p class="editorial">Many youngsters have gone on to have a look at
different numbers of cups and saucers.</p>
<br></br>
<mdo:image alt="Tecup diagram" src="3and5.gif"></mdo:image><br></br>
<p class="editorial" style="text-align: center;">Some new patterns
can be seen when you look at these arrangements that have an odd
number of sets of coloured cups and saucers.</p>
<div>When I want to create a solution and I may have forgotten one
then I work on it in the following way;</div>
<div style="text-align: center;"><mdo:image width="415" height="500" src="Ans.jpg" alt="Ans"></mdo:image></div>
<div style="text-align: left;">SOLUTIONS PRIOR TO FEB 2011</div>
<div style="text-align: left;">Well done all of you who sent in
these solutions. I am very pleased with the ways that you went
about doing this challenge and the different ways you showed your
results.</div>
<div style="text-align: left;">Pupils from Chesterbrook Academy
sent in this:</div>
<div style="text-align: left;">We first started off by doing column
by column and row by row. After we found out it would take us
forever, we tried putting the doubles (green green, blue blue,
etc.) in the middle four squares. This idea was given to us by our
teacher, Mrs. Johnson. After that, we built off the centre. Our
answers were:</div>
<div style="text-align: left;">First row</div>
<div style="text-align: left;">g r b w</div>
<div style="text-align: left;">b g w r</div>
<div style="text-align: left;">Second row</div>
<div style="text-align: left;">w b r g</div>
<div style="text-align: left;">g b r w</div>
<div style="text-align: left;">Third row</div>
<div style="text-align: left;">b w g r</div>
<div style="text-align: left;">r w g b</div>
<div style="text-align: left;">Fourth row</div>
<div style="text-align: left;">r g w b</div>
<div style="text-align: left;">w r b g </div>
<div style="text-align: left;"> </div>
<div style="text-align: left;">Then someone else from the same
school added:</div>
<div style="text-align: left;">We figured it out by putting the
doubles (white, white, green, green etc.) in the middle. Then we
put the ones that were opposites together and took each column too
see if the patterns could go there. If it didn't work we switched
the middle until we finally got it.</div>
<div style="text-align: left;">Here's Emma's Solution that she sent
in in a doc.form: </div>
<div style="text-align: center;"> <mdo:image width="539" height="360" src="TeaCupSolB.jpg" alt="TEACUPSOLB"></mdo:image></div>
<div style="text-align: left;"> </div>
<div style="text-align: left;">Andy sent in the following. I liked
his method for finding a solution by starting with the special
diagonal and then filling in. I've not come across this method
before. This solution has the added attraction (perhaps difficulty)
that the diagonals also have to obey the rule!</div>
<div style="text-align: left;">The answer is</div>
<div style="text-align: left;">BB WR GW RG</div>
<div style="text-align: left;">RW GG WB BR</div>
<div style="text-align: left;">WG BW RR GB</div>
<div style="text-align: left;">GR RB BG WW</div>
<div style="text-align: left;">B stands for blue</div>
<div style="text-align: left;">G stands for green</div>
<div style="text-align: left;">R stands for red</div>
<div style="text-align: left;">W stands for white</div>
<div style="text-align: left;">The first letter of each double is
the saucer</div>
<div style="text-align: left;">The second letter of each double is
the cup</div>
<div style="text-align: left;">First we have to put the cups and
saucers that are the same colour in a diagonal line. Then we work
out the rest of the space by using the diagonal.</div>
<div style="text-align: left;">Eg BB GG</div>
<div style="text-align: left;">The space between BB and GG will be
filled with RW and the opposite space would be filled with WR. In
the end all the space would be filled in the the shape of a square
and that is the answer.</div>
<div style="text-align: left;"> </div>
<div style="text-align: left;">Susie sent in these comments
accompanied by the Spreadsheet. This is the first time, out of
hundreds of times, that I've been presented with this method of
finding a solution. Well done Susie (from Cathedral School - I
wonder which one?).</div>
<div style="text-align: left;">I made a 4 by 4 square and wrote in
each square pairs of letters AA, AB, AC to mean the colours of cups
and saucers. I used Excel.</div>
<div style="text-align: left;">To select a cup and saucer I
coloured the letters black and crossed the same thing off in all
the other squares. If the thing I chose was AB then I would also
cross off the A-somethings from same line and row, and I would also
cross off the something-Bs from same line and row. I started off by
putting AA, BB, CC and DD on the top row. I wanted a diagonal going
down right of A-somethings and a diagonal going down left of
D-somethings. In one square I had the choice of DA or DB so I chose
one and carried on. I noticed a pattern of 2 short diagonals of
C-somethings and B-somethings appearing. My dad helpd me with the
spredsheet and with this email. </div>
<div style="text-align: center;"> <mdo:image width="343" height="265" src="Susie%20C%26S.jpg" alt="SISUEC&S"></mdo:image></div>
<div style="text-align: left;"> </div>
<div>The four solutions could be viewed as:-</div>
<div> <mdo:image width="550" height="129" src="4Sols.jpg" alt="4 sols"></mdo:image></div>
<div> </div>
<br></br>
<br></br></mdoxml>
2
5
0
1
0
0
0
Tea Cups
Place the 16 different combinations of cup/saucer in this 4 by 4
arrangement so that no row or column contains more than one cup or
saucer of the same colour.
Decision Mathematics and Combinatorics
Combinations
Using, Applying and Reasoning about Mathematics
Working systematically
Using, Applying and Reasoning about Mathematics
Investigations
Admin
Upper primary mapping document