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The following puzzle appeared in the Cambridge, UK February 2024 MathsJam Shout:

Tiling with Triominoes

Imagine tiling a 5-by-5 bathroom with L-shaped triomino tiles.

5 x 5 grid AND L-shaped triomino

Clearly, not every square can be covered, since 3 doesn't divide exactly into 25. Can you find a way to cover 24 of the 25 squares? What are the possible positions for the un-tiled square?

Clarification: The triomino tiles can be rotated.

Will Octagon Gibson
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  • Does this answer your question? https://puzzling.stackexchange.com/questions/90301/can-you-put-l-trominos-to-fill-the-figure Even though it isn't an exact duplicate, the answer is practically the same. – mathlander Mar 08 '24 at 18:20
  • @mathlander I agree that https://puzzling.stackexchange.com/questions/90301/can-you-put-l-trominos-to-fill-the-figure is a very similar question but the answers to that question only partially answer this question. For example, those other answers don’t show if a corner square in this question can be the un-tiled square. – Will Octagon Gibson Mar 08 '24 at 21:39

3 Answers3

20

The missing square has to be one of these:

enter image description here
Since each triomino can only cover one of them, we can only cover 8 of the 9 with our 8 triominos.

Demonstration that any of these is possible:

enter image description here
and rotations

Carmeister
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4

Here is one way

A A H H G
A B H G G
B B . F F
C D D F E
C C D E E

Which is made up of four 3x2 sets.

Weather Vane
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2

There are

128

solutions. Of course, most of those solutions are related to each other via rotation and reflection.

Here are all the solutions, found using Knuth's Algorithm X. The solutions are displayed as an animated SVG image.

5x5 Tromino tilings, SVG

Here are the solutions in a GIF anim.

5x5 Tromino tilings, GIF

PM 2Ring
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