# Humans, vampires and mirrors

Here is a room composed of $$10 \times 10$$ squared cells. Some cells contain a double-faced mirror ( / or \ ) that reflects the light with an angle of $$90°$$ on both sides. The other cells can be empty ($$X$$) or can contain exactly one human ($$H$$) or exactly one vampire ($$V$$).

The numbers on the borders indicate how many creatures can be seen if someone steps on the number and looks inside the room. The visible creatures are: humans and vampires directly and humans reflected by the mirrors. Of course the vampires are not reflected by the mirrors.

Complete the grid by filling each cell with one of $$X$$,$$H$$ or $$V$$ using the rules above and knowing that:

• the four central cells are empty.
• if looking in some direction the same cell can be seen two times: the creature on it (if any) must be counted only once.
• There are $$29$$ vampires in the room.

Credits: I have found this kind of puzzle on a magazine but I have create this particular grid by myself. Enjoy!

EDIT: find the part 2 (with triangular mirrors) here.

• In reflections, can humans be seen through vampires? Mar 29 '20 at 17:46
• Yes, also the heights of the creatures don't matter: you can always see all the creatures on the "path" Mar 29 '20 at 17:53
• This is similar to Undead in Simon Tatham's puzzle collection. Mar 29 '20 at 18:21
• This was a fun puzzle! I've seen this puzzle type before but never done it, so this was a nice introduction.
– Deusovi
Mar 29 '20 at 18:27
• @DanielMathias Undead is based off my puzzle type “Haunted Mirror Maze”, which itself is a riff on mirror puzzles by Erich Friedman. Apr 6 '20 at 1:12

My color scheme is: yellow for "empty", dark gray for "human", red for "vampire", light red for "no human, maybe vampire", and light gray for "no vampire, maybe human".

Next,

look at the 2 and 1 near the bottom right.
In any path with clues on both sides, each side has access to the same things except for any vampires in the initial segments. That is, all differences in the numbers must arise from vampires in the first segments seen. (This is an important general principle.)

Here, there is a difference of 1, so there must be a vampire that the 2 can see that the 1 cannot. Combined with the "3" clue, this lets us resolve several cells in the lower-right corner.

A similar strategy can be used

with the "7" clue on the lower right, and the "4" clue on the end of its path looking the other way. Because the difference is 3, there must be at least 3 vampires in the 7's initial segment.

The 4 in the third-from-bottom row and its counterpart 3 can be similarly attacked, as can the other 3 in the middle right (that connects to the 1 in the middle of the top side).

We can also attack the top left in a similar way.

There are two "1...2" pairs that can be fully resolved.

And another pair can be attacked:

The 9 near the top left must have at least three more initial-segment vampires than the 6 it connects to. The 6 already has one initial-segment vampire, so the 9 must have at least 4. And this allows those clues to be fully resolved.

Unrelatedly, the 4 and 3 near the middle right can be resolved too. And both of these together allow us to resolve the 4 and 7 pair as well.

Some miscellaneous easy logic...

...and some more...

And more, specifically:

note that the 2 near the top middle requires the cell in R1C8 to be a human, and this finishes off the 1 clue starting next to it.

And more...

...and that lets us finish the rest of the clues off!

Finally,

the condition that there are 29 vampires means that all the undetermined cells must be filled with vampires. So the puzzle is solved!

• Ah well ... discards partly written answer. (I was using a different path of logic - there's some stuff in three of the corners that can be done immediately - but dunno how long it will take to finish everything completely.) Mar 29 '20 at 18:29
• Well done as usual! Do you sleep sometimes? Mar 29 '20 at 19:01
• @melfnt Occasionally!
– Deusovi
Mar 29 '20 at 19:06
• @Randal'Thor Yeah, I went back later and caught some of the easy corner stuff. I think there are a number of different paths. (Also, might be a good idea to post a comment when I start on a long logic puzzle answer...)
– Deusovi
Mar 29 '20 at 19:06
• I'll treat this as practice and hope another one of these gets posted soon :-) Mar 29 '20 at 20:19