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.

Age

7 to 11

Challenge level

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Problem

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.

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.

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.

So, for example:

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;

b) there is a red cup on a red saucer but that's the only one like it.

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).

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:-

1) In any row there must only be one cup of each colour;

2) In any row there must only be one saucer of each colour;

3) In any column there must only be one cup of each colour;

4) In any column there must be only one saucer of each colour.

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.

There are a lot of different ways of approaching this challenge.

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.

This challenge is also found at nrich.maths.org.uk/7397

Student Solutions

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.

Pupils from Chesterbrook Academy sent in this,

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.

Our answers were:

First row

g r b w

b g w r

Second row

w b r g

g b r w

Third row

b w g r

r w g b

Fourth row

r g w b

w r b g

Then someone else from the same school added,

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.

Here's Emma's Solution that she sent in in a doc.form:

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!

The answer is

BB WR GW RG

RW GG WB BR

WG BW RR GB

GR RB BG WW

B stands for blue

G stands for green

R stands for red

W stands for white

The first letter of each double is the saucer

The second letter of each double is the cup

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.

Eg BB GG

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.

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?).

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.

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.

These four solutions could be viewed as:-

Teachers' Resources

Those Tea Cups

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.

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.

So, for example:

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;

b) there is a red cup on a red saucer but that's the only one like it.

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:-

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.

There are a lot of different ways of approaching this challenge.

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.

Why do this problem?

Doing this problem 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.

See also this article by a PGCE student about using this activity.

Possible approach

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.

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.

The results of this investigation would make a lovely classroom display.

If you need to check solutions you might like to use this interactive.

(When you have a complete solution it will say Well Done!!)

Key questions

Tell me about this.

What do you think you have to do?

What have you tried so far?

Have you checked that all your cup and saucer combinations are different?

Possible extension

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.

For the highest-attaining

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.

If the diagonals also have to be different can a system for getting all the possible answers be found?

Possible support

For youngsters who have difficulties with colours you might want to use this image:

(A bigger version can be seen here.)

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 Teddy Town problem is essentially the same investigation as this one, but starts at a simpler point.)

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.]

Extra

Suppose we are using playing cards. There are quite a few solutions, but I'll take just one particular kind.

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.

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.

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.

Then there are the suits to look at, you can ask the pupils to describe, draw, record etc. what they've noticed.

Here, from the first solution above, is one of the many ways that they may come up with.

This shows how the suits form shapes, we have Green for Hearts, Blue for Spades, Grey for Clubs and Red for Diamonds.

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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.

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These two show patterns also derived from looking firstly at the suits and then the values.

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