# Heteromino: Introduction

Heteromino is an area-dividing puzzle with very simple rules. I encountered this puzzle type on puzz.link, and seeing there's none posted here, I decided to give it a try.

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:

• Ooh this puzzle type is kinda memorable for me.. In my previous uni, I tried to prove that this is NP-hard :) Commented Oct 20, 2020 at 1:12
• @athin Interesting. Did you complete the proof? Commented Oct 20, 2020 at 1:16
• Fortunately yes, we were planning to rewrite the proof in English (because it was written in Indonesian haha), but until now we haven't continued it.. XD Commented Oct 20, 2020 at 1:25

First:

Take a look at the cell in row 2, column 3. It must be part of a region extending left -- where is the third cell in that region? If it goes left, it blocks off 2 cells in the upper left. If it goes down, then there will be two Γ-trominoes adjacent. So it must go up.

Similarly, the group with cell R3C2 now cannot go down or right, so it must go left.

Similar logic, but with a bit of lookahead:

If the group with cell R4C3 goes right, then the bottom will either be two — shapes, or a ⅂ shape that touches the one we already have. So it must go down.

Another bit of lookahead gives us this: If we have this pink-highlighted box, there's a problem in the upper right. (Either two ⅃ shapes touch, or we block off a cell in the right column.)

Extending the top region rightwards is also an issue...

so we have this:

Finishing off the puzzle with some more attacks on carefully-chosen borders:

This is impossible (because it will either make two |-shaped regions, or a 4-long region in the bottom.)

And then the two remaining cells in row 4 must be connected (to not have two |-shaped regions touching), and that finishes off the puzzle!

• Super fast and nice job, as always :) My logic for the top right region was that rot13(vs E1P4 vf n ubevmbagny V, nyy guerr pubvprf sbe E3P4 yrnq gb pbagenqvpgvbaf). Commented Oct 20, 2020 at 1:10
• @Bubbler Makes sense! I figured that out early on (not completely rigorously), before actually starting with the top left. But since it didn't immediately lead anywhere, I wanted to see if I could find a simpler logical path through the puzzle.
– Deusovi
Commented Oct 20, 2020 at 1:18
• Thank, this really helps! Commented Oct 20, 2020 at 2:11