# Those Tea Cups

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).

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

What could you try on the diagonals?

Is there any pattern to the way you have arranged the cups and saucers so far? Could you continue the pattern?

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 isBB 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:-

### Why do this problem?

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

### Possible approach

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

### Possible extension

### 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

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.)