# Heteromino: Solitude

Heteromino is an area-dividing puzzle with very simple rules.

Here is an example puzzle with solution:

The rules of Heteromino are as follows:

1. Divide the white area into L- or I-trominoes.
2. No two trominoes of identical shape and orientation may share an edge.

Now, solve the following puzzle. This time, I tried to come up with a puzzle that has all walls isolated (in terms of four-way neighborhood).

Some thought will show that this line is impossible:

The gray line at the bottom is impossible, because it will break the bottom left corner.

And then this one is as well, because it causes a contradiction in the region just above it. (We must have that ⅃ region because otherwise it would cut off the wrong number of cells in the lower left corner.)

That gets us this far:

We must have this line marked in pink. Otherwise, we'd have the wrong number of cells in the lower right corner, and we'd be forced to make a region of 2 or 4.
This resolves the bottom right corner:

and near the top, the same argument applies two more times to get the two dashes marked above.

We can now start to finish off some more regions:

This gray-marked region is bad, because it causes a problem immediately below it.

This lets us finish off the middle region...

The top-left corner must go down at least once. If it goes down twice, we have two L shapes touching along the left wall.

So this lets us finish off that corner...

And finally, in the top right corner, we can't have this wall or it breaks the upper left of its region:

and placing a region like this also causes a problem:

...so we get the final answer:

• Excellent. I used small case splits with backtracking, and your logic looks much more elegant. Btw, I thought "is there something wrong with my puzzle?" when I read the first line... Commented Oct 21, 2020 at 1:17
• @Bubbler Yep, Heteromino is a surprisingly difficult genre! I haven't (successfully) done too many of them before, but I know that the logic can be trickier than you might expect. This was a nice puzzle though!
– Deusovi
Commented Oct 21, 2020 at 1:20