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Let's imagine we are tiling a floor or patio area.
Suppose the area is square and is 3 by 3.

Tiles come in three sizes: 1 by 1, 2 by 2 and 3 by 3.
You are allowed to use any arrangement of these tiles to cover the space completely.
However, none of the tiles can be cut.

For this challenge, we are interested in the total number of tiles you use for any arrangement.
For example, for the 3 by 3 square you could do it by using:

nine 1 by 1 tiles,


one 2 by 2 and five 1 by 1 tiles,


one 3 by 3 tile.

Therefore the smallest number of tiles is one and the largest is nine and the only other value you can get between one and nine is six (using one 2 by 2, and five 1 by 1).

Now imagine you also have tiles which are 4 by 4 in size.
What total numbers of tiles can you use for a square patio that is 4 by 4?
If you could have tiles which are 5 by 5 as well, what total numbers of tiles can you use for a square patio that is 5 by 5?

The final part of this activity is to examine carefully the answers you get for each of the three sizes of floors - 3 by 3, 4 by 4 and 5 by 5.

Now talk about and record what you notice.
Make some statements about what you therefore think will happen if someone takes the long time needed to explore a 6 by 6 and 7 by 7 square.