# Tiling a 5-by-5 bathroom with L-shaped triomino tiles

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.

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.

• Does this answer your question? puzzling.stackexchange.com/questions/90301/… Even though it isn't an exact duplicate, the answer is practically the same. Mar 8 at 18:20
• @mathlander I agree that puzzling.stackexchange.com/questions/90301/… 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. Mar 8 at 21:39

The missing square has to be one of these:

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:

and rotations

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.

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.

Here are the solutions in a GIF anim.