7397
/www/nrich/html/content/id/7397/
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2011-02-01T00:00:01
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<p style="text-align: center;"><mdo:image width="356" height="79" src="Picture%202.jpg" alt="pic 2"></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 16 cups and 16 saucers altogether.</p>
<p>Just for the fun of it, you decide to mix them around a bit so
that there are 16 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 16 different cup/saucer combinations on the
table and you think about arranging them in a big square. Because
there are 16 you realise that there are going to be 4 rows with 4
in each row (or if you like, 4 rows and 4 columns).</p>
<div style="text-align: center;"> <mdo:image width="191" height="191" alt="a halp" src="b_rc.gif"></mdo:image> <a href="" class="control" onclick="tablealter(1)"></a></div>
<div> </div>
<div>So here is the challenge to start off this investigation.
Place these 16 different combinations of cup/saucer in this 4 by 4
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 16 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>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>
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<p class="editorial">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.</p>
<p class="editorial">Pupils from Chesterbrook Academy sent in
this,</p>
<br></br>
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 center. <br></br>
Our answers were: <br></br>
First row <br></br>
g r b w <br></br>
b g w r <br></br>
<br></br>
Second row <br></br>
w b r g <br></br>
g b r w <br></br>
<br></br>
Third row <br></br>
b w g r <br></br>
r w g b <br></br>
<br></br>
Fourth row <br></br>
r g w b <br></br>
w r b g<br></br>
<br></br>
<p><span class="editorial">Then someone else from the same school
added,</span></p>
<br></br>
<br></br>
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.<br></br>
<p class="editorial">Here's Emma's Solution that she sent in in a
doc.form:</p>
<br></br>
<div style="text-align: center;"><mdo:image height="360" width="539" alt="sol" src="TeaCupSolB.jpg"></mdo:image></div>
<br></br>
<p class="editorial">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!</p>
The answer is<br></br>
BB WR GW RG <br></br>
RW GG WB BR <br></br>
WG BW RR GB <br></br>
GR RB BG WW <br></br>
<br></br>
B stands for blue <br></br>
G stands for green<br></br>
R stands for red <br></br>
W stands for white<br></br>
<br></br>
The first letter of each double is the saucer <br></br>
The second letter of each double is the cup<br></br>
<br></br>
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. <br></br>
Eg BB GG <br></br>
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.<br></br>
<p class="editorial">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?).</p>
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.<br></br>
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. <br></br>
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. <br></br>
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.<br></br>
<div style="text-align: center;"><mdo:image height="265" width="343" src="Susie%20C%26S.jpg" alt="Susie"></mdo:image></div>
<div style="text-align: center;"></div>
<br></br>
<p><span class="editorial">These four solutions could be viewed
as:-</span></p>
<br></br>
<div style="text-align: center;"><mdo:image height="129" width="550" src="4Sols.jpg" alt="4 Sols"></mdo:image></div>
<div style="text-align: center;"></div>
<br></br>
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<h2>Those 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 16 cups and 16 saucers altogether.</p>
<p>Just for the fun of it, you decide to mix them around a bit so that there are 16 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 16 different cup/saucer combinations on the table and you think about arranging them in a big square. Because there are 16 you realise that there are going to be 4 rows with 4 in each row (or if you like, 4 rows and 4 columns).</p>
<div style="text-align: center;"> <mdo:image alt="a halp" height="191" src="b_rc.gif" width="191"></mdo:image></div>
<div> </div>
<div>So here is the challenge to start off this investigation. Place these 16 different combinations of cup/saucer in this 4 by 4 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 16 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>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>
<h3> </h3>
<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 is also a valuable challenge for pupils to work in a small group deciding on approaches. If the introductory story is told to the full it also is a good problem for pupils to have to sift
out what information they've been given is useful and what is irrelevant.</div>
<br></br>
<h3>Possible approach</h3>
<div>Telling the story with little or no warning as to the reason for them listening to 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 4) 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>
<br></br>
<div>The results of this investigation would make a lovely 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 highest-attaining</h3>
<h4 style="font-weight: 400;">These pupils can look carefully at the different solutions and then compare them to sort out similarities and differences as well as equivalences. Then the challenge 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;">If the diagonals also have to be different can 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>
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<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>
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<mdoxml version="1.0">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" src="Ans.jpg" alt=""></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 src="sarah_a.gif" alt="Sarah, Diagram A"></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 src="4A.gif" alt="Modified Sarah, Diagram A"></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 src="4B.gif" alt="Teacup diagram"></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 src="3and5.gif" alt="Tecup diagram"></mdo:image>
<br></br>
<p style="text-align: center;" class="editorial">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" alt="Ans" src="Ans.jpg"></mdo:image></div>
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Those 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