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 $V$
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
This problem is taken from the UKMT Mathematical Challenges.