Teddy Town

'Teddy Town' printed from https://nrich.maths.org/

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Teddy Town

In Teddy Town, teddies are either red or yellow and they live in red or yellow houses. There are 4 teddies - 2 red and 2 yellow, and 4 houses - 2 red and 2 yellow.

Can you match each teddy to a house so that the 4 pairs are all different from each other?

Imagine now that there are 3 different colours of teddies and houses - red, yellow and blue. In Teddy Town there are now 9 teddies and 9 houses. All 9 pairs of houses and teddies are different from each other.

Here is a map showing Teddy Town:

Map of Teddy Town.

The streets are very special. If you walk along a street from east to west, or west to east, all the houses are a different colour and the teddies living in the houses are a different colour too. The same is true if you walk along the streets in a north-south or south-north direction.

In other words, looking at the map grid, each row and column must have different coloured houses and different coloured teddies.

Can you arrange the houses and teddies on the map grid, making sure that all 9 pairs of houses and teddies are different from each other?

Teddy Town is expanding rapidly as green teddies move to the area and green houses are built. Now there are 16 teddies and 16 houses.

There are 16 different ways to combine the teddies and houses. How could these 16 households be organised on the map now? Remember that in each row and column there must be both different coloured houses and teddies.

Now... yes, you've guessed it. Another colour of teddy bear has moved to Teddy Town. As well as red, yellow, blue and green teddies there are now purple teddies. Of course, this means that purple houses will have to be built. So, now in Teddy Town there are 5 of each colour bear, making 25 teddies in all, and also 25 houses, again 5 of each colour.

Can you make the 25 different combinations of teddy and house now? Arrange these on the street map below in the same way as before:

Teddy Town is becoming very overcrowded! However, there is just enough room for some turquoise teddies to join. Living there now are 36 teddy bears: 6 red, 6 yellow, 6 blue, 6 green, 6 purple and 6 turquoise. There are 36 houses for them to live in: 6 red, 6 yellow, 6 blue, 6 green, 6 purple and 6 turquoise.

Make the 36 combinations of teddies and houses. Do you think it will be possible to put these 36 combinations in the street grid? Maybe it won't be. Have a go!

Now, look back at what you have done and ask yourself some of these questions:

Was it easier to arrange the combinations in some of the grid sizes compared with others?
Why do you think this is?
What was your strategy for solving the arrangement puzzle each time?
What would happen if the two diagonals on the map also had to have different coloured houses and different coloured teddies? Can you solve the problem for each street plan now?

Printable NRICH Roadshow resource.

Acknowledgements

This activity is based on a Bernard's Bag problem from December 1997 called Tea Cups. The idea for the teddies came from Andrew Massey who is an Advisor for Worcestershire County Council.

Why do this problem?

The original version of this problem uses only a 4x4 grid, but reducing the size makes this investigation accessible to younger children too. Doing this problem is an excellent way to work at problem solving with learners. The problem lends itself to small group work, and provides an engaging context for pupils to use the skills of trial and error, and working systematically.

Possible approach

No matter how old the children, it would be advisable to have objects to represent the teddies and houses as an introduction to the activity. These can be, for example, coloured counters and coloured squares if there are no plastic teddies available (or you could print these teddies). Coloured magnets would be ideal for use on a whiteboard as a demonstration. If you prefer, click on the following links to download word documents of the different coloured houses which you could print, laminate and cut out: yellow, red, blue, green, purple and turquoise.

It would be worth clarifying the very first instruction. Work out the four different combinations together with the children, using teddies and houses of two different colours.

Throughout all of this investigation, encourage the children to explain their thinking orally. This may be to each other, or to the class as a whole. Either way, it is vital in allowing them to clarify their own ideas, reflect critically on their work and so move themselves forward.

Key questions

How can we make sure they are all different?

Is there a way to go about making the combinations so we don't leave any out?

Talk about being methodical and systematic i.e. planning and checking.

Possible support

Some children might find it easier to work out all the possible combinations of houses and teddies before putting them onto the grid - others might prefer to put all of the houses onto the grid first and then put the teddies on afterwards. If pupils are finding it difficult to check their solutions, they might benefit from using a checking mechanism for the first grid:

Possible extension

Encourage pupils to solve each problem again, while also making sure that the diagonals follow the same rules as the rows and columns. Children might like to have a go at Tea Cups, which is a similar activity.