This work brings for the first time in the history of magic squares a new kind of magic square of order 6. It is called cornered magic square of order 6. It contains at the corner a pandiagonal magic square of order 4. Before to this the author (click here) also introduced a new kind of magic square of order 6 based on magic rectangle of order 4x6. Using pandiagonal magic square of order 4, we have written four different kinds of cornered magic squares of order 6. Based on these four cornered magic squares of order 6, we have written some examples of magic squares of orders 8, 10, 12, 14 and 16.