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It's rather easy to fill a $7 \times 7$ board with 16 long triominos, leaving the center square void: see the picture below. But if I want to move the void square in another position, where else could I place it?

enter image description here

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The trick to this puzzle is to:

tri-color the board.
enter image description here

Any tromino must cover one square of each color. There are 16 blue squares, 16 yellow squares, but 17 red squares, so a red must be the uncovered one. This is true for the reverse coloring as well, which gives our final result: the only possible squares are the corners, the center of the edges, and the center.

(And here are those tilings: the center was already given, and the rest are obtainable from these by rotation.)

enter image description here

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    $\begingroup$ That's not actually a full solution. You've shown that certain squares are impossible, but you haven't shown that the others are possible. $\endgroup$ – Peter Taylor Jul 10 '17 at 12:57
  • $\begingroup$ why not the square on second row, second column? $\endgroup$ – Marius Jul 10 '17 at 13:00
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    $\begingroup$ @Marius because you can use a mirror-image colouring, which has the same number of red squares but in different places; between them, these two colourings rule out all squares but the ones Deusovi says. $\endgroup$ – Gareth McCaughan Jul 10 '17 at 14:08
  • $\begingroup$ Ah. Makes sense now. $\endgroup$ – Marius Jul 10 '17 at 14:14
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it could be (any) one of the corners:
corner
black one is the empty space.

or

it could be one of the center squares on the first/last row/column.
enter image description here

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  • $\begingroup$ that's right (and if you think about it, there is another position you can have :-) ) $\endgroup$ – mau Jul 10 '17 at 12:27
  • $\begingroup$ @mau I edited the answer and added a new position, also explained the first finding a bit more. $\endgroup$ – Marius Jul 10 '17 at 12:30

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