Placing figures on a grid
What is the largest number of these tetrominoes which can fit on a
grid without any overlap?
The pieces can be rotated and reflected. However, they cannot overlap and go off the grid.
The answer is 9.
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It is easy to see that each figure will be on exactly one of the 9 gray-colored squares below, we only need to show that it'll be on at least one and at most one of them.
There are only 9 gray-colored squares, thus making it impossible for 10 figures to fit (else we'd get a contradiction -- 10 gray-colored squares would be with figures on them).
But 9 figures don't contradict anything yet -- there are exactly 9 gray-colored squares, the same amount.
If we prove we can fit 9 figures, this will be our maximum. And we can fit them, as shown in the diagram below.