Is it possible to divide a given square into n squares for any n≥6?

We first note that if you have a sub-division into n squares, you also have a sub-division into n+3 squares by choosing any one square and dividing it into 4 parts.

So if you find a division for n = 6,7,8 you would have proved for all n

n = 7 is easy because you can divide into n = 4 and so you can divide into 4+3 = 7

for n = 6, consider a 3X3 and then consider a 2X2 in it as one square. That gives you 9 -4+1 = 6 squares

for n = 8, consider a 4X4 and consider a 3X3 as one square. Now you have 16-9+1 = 8 squares and so we have proved for all n