Let no. of each type of tile be k
2*2 covers 4 unit squares
3*1 covers 3
Adding,
4k + 3k = n2
So, it is easy to see that 7 must divide n2 and so n.
As 7 is prime it follows that 49 divides n2.
We need 2nd 7 so 7 must also divide k.
So, such a config is possible only when 7 divides n
There are many ways to cover a 7*7 area with such tiles, eg
1 1 2 2 2 1 1
1 1 2 2 2 1 1
2 1 1 1 1 1 1
2 1 1 1 1 1 1
2 2 2 2 2 2 2
1 1 1 1 2 2 2
1 1 1 1 2 2 2
Where 11 is a 2*2 tile and 222 is a 3*1 tile, which can
11
Be horizontal or vertical. Using these 7*7 areas as blocks, one can make such a config for any n=7m, m belonging to N
So 7 divides n is a necessary and sufficient condition.