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


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

  • 2
    $\begingroup$ 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
  • 1
    $\begingroup$ @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

it could be (any) one of the corners:
black one is the empty space.


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

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.