The title almost completely describes the problem. So you have a $5 \times5$ square. You have to fit the minimum number of $T$ shaped tetrominoes such that no more $T$ shaped tetrominoes can be fitted into the square. By skimming through all the possible configurations using a computer program I can safely say that the answer is $3$.
What I cannot figure out is a correct purely mathematical proof. I've tried colouring the square and other things. My attempts prove that the number has to be greater than or equal to $2$ but I cannot really get to the $3$.