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 four pairs are all different from each other?

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Imagine now that there are three different colours of teddies and houses. red, yellow and blue. In Teddy Town now there are 9 teddies and 9 houses:

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What are the nine different combinations of teddies and houses?

Here is a map showing 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 nine different combinations you've found on the map grid?

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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:

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Find the sixteen different ways to combine the teddies and houses now.

How could these sixteen households be organised on the map now? Remember that in each row and column there must be both different coloured houses and teddies.

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Now ... yes, you've guessed it. Another colour 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?

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Arrange these on the street map below in the same way as before:

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

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Do you think it will be possible to put these 36 combinations in the street grid? May be it's not. Have a go!

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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. Thank you! Many thanks also to learningresources.com for the use of the bear images.