The diagram shows that it is possible to fit five T shapes in the
square. In order to fit six T shapes into the square, exactly one
of the 25 squares would be left uncovered; hence at least 3
corner squares must be covered.
We now label a corner square H or V
if it is covered by a T shape which has the top part of the T
horizontal or vertical respectively. If all four corner squares
are covered then there must be at least two cases of an
H corner with an adjacent V corner.
Each such combination produces a non-corner square which cannot be covered
e.g. the second square from the right on the top row of the diagram.
If only 3 corner squares are covered, there must again be at least one
H corner with an adjacent V corner and therefore a non-corner square
uncovered, as well as the uncovered fourth corner. In both cases,
at least 2 squares are uncovered, which means that it is impossible to fit six
T shapes into the square.