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Tea Cups

Age 7 to 14
Challenge Level

Tea Cups

pic 2

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 sixteen cups and sixteen saucers altogether.

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.

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 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).


So here is the challenge to start off this investigation:
Place these sixteen different combinations of cup/saucer in this four by four 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 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.

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 with pupils in late primary and early secondary school. The problem lends itself for small group work, so that the learners have an opportunity to decide on approaches. Then, having completed the challenge, groups can discuss the different solutions, and each method's strengths and weaknesses.  If the introductory story is told to the full, it is also a good problem for having  pupils decide what information is useful and what is irrelevant.

Possible approach

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.

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.
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.   

The results of this investigation would make an engaging and attractive  classroom display.

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

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 more extension work

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.

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?

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.]


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.

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