Tiling

Chetan sent in this solution.

The total area of each one of the square shaped patio (3 X 3, 4 X 4, 5 X 5) is 9, 16 and 25 respectively.
Hence, with the available size of 1 X 1, 2 X 2, 3 X 3 and 4 X 4, we just need to find the tile values that total the area of the square.
example: for 4 X 4 = 16. Hence, the sum of the tiles areas should add up to 16.
no. of 4 X 4 = 1 (area 16)
no. of 1 X 1 = 16 (area 16)
no. of 2 X 2 = 4 (area 16)
no. of 3 X 3 = 1; 1 X 1 = 7 (area 16)
Hence, with respect to the total area of the square we will need to calculate the factors that sum up to the total number.

However, Paul from Brentside High School said:

So with a 5x5 square you could assume that: 1 tile of 3x3 = 9 and 4 tiles of 2x2 =16.
As 9+16 = 25 therefore you could use 5 tiles.
But in fact you can't as there is no configuration that will fit (without cutting the tiles which is not allowed).
So to say that we just need to find tile values that total the area of the square is not correct.

You are absolutely right, Paul. Paul goes on to wonder whether there is a purely numerical solution to the problem. I am not sure that there is, as you would have to take into account the spatial aspect too. What do you think?