Puzzle 29: Stargazers

[zotmeister]

I have found janko.at to be an excellent resource for challenges, prizes, and even inspiration. The site is in German, but that doesn't stop me from trying to figure out the logic of the puzzles anyway. It was there that I discovered Magnetic Field. The puzzle I present today is another I was unfamiliar with until experiencing it there. I was actually emailed by Otto Janko with a request to make my puzzles available on es site; I accepted, but haven't seen them there nor heard back from em since. Oh well.



On the left is an unsolved Stargazers puzzle; its solution is on the right. Technically, only the five stars (rendered in web-safe "dark hard yellow") are the solution; the "pale turquoise" crossouts are just a solving aid.

Okay, so they're not stars. They're asterisks. I tried to draw five-pointed stars in a 5×5-pixel square - none of them looked good.

Anyway: for each arrow in the grid, there is exactly one star centered in an otherwise-empty cell; the object is to locate all the stars. If a ray were extended from each arrow in the direction it is pointing, each star would be incident upon exactly one such ray, and each ray would have exactly one star incident upon it. Numbers above columns and to the left of rows indicate exactly how many stars must be in their associated rows and columns.

For those who never took geometry and don't know what a "ray" is:

1) Some of the empty cells of the grid contain stars. There's never more than one star in a cell; you'll never find a star in the same cell as an arrow. The task is to find all the stars; there'll be the same number of stars as there are arrows.
2) The arrows represent stargazers. Each arrow is "looking at" a star: traveling straight out from each arrow, there is exactly one star between it and the edge of the puzzle. (Think of a chess queen on the arrow, moving in the direction the arrow points, all the way to the outer border of the grid.) Not zero stars, not two or more - exactly one star is aligned with each arrow. They're stargazers, not starsgazers♪ Note that arrows "see" right through each other; in actuality, they're all looking upward at an angle.
3) Each star has exactly one arrow looking at it. That is, no two stargazers are gazing at the same star!
4) You may find some numbers above the grid; each says exactly how many stars must be in the column they're positioned over. Similarly, any numbers to the left of the grid show how many stars must be in the row they're lined up with.

The sample is quite straightforward, but I would not be performing my civic duty [places fist over heart] if I didn't provide a walkthrough:


I start with the leftmost arrow on the top row. It points down and right. There are three cells aligned with it. The one right in front of the arrow can't be a star, because it's pointed at by a second arrow (bottom row, center column, pointing straight up). Crossing out that cell, I move on to the next one: the cell down and right from there can't be a star either for the same reason (look up two cells). Therefore, the cell just below center in the rightmost column has a star. When I find a star, I like to mark the arrow pointing at it so that I know it's been dealt with.

Now I look at the number on top of the rightmost column. It tells me there's one star in that column, and I just put one there... so I can cross out the other four cells in that column. That can work both ways, and in fact, that's exactly what happens next: the top row needs two stars, and there are now only two empty cells left up there! This is important: after I fill those stars in, I look for the arrows that are pointing at them, mark them, and cross out all the other cells they are looking at. (Remember that each star can only have one arrow looking at it, so figuring out which arrow goes with which star is simple.) This crosses out three cells I hadn't already eliminated.

The arrow in the bottom-left corner only has two cells left for its star, and one is right in front of another arrow. That last arrow in turn has two spots for the last star, but it's the number that decides it - there's no star on the bottom row yet, and there needs to be one there; since this is the last arrow and last star, they have to satisfy that '1' outside the grid, so that's where it goes. This solves the puzzle - five stars, five marked arrows - but being anal I crossout the remaining empty cells anyway.


This puzzle isn't necessarily hard, but it does require observation. No pun intended. Ahem. Actually, when I originally constructed it, there were twelve numbers outside it when I finished, but upon testsolving I found that fully a quarter of them were rendered unneeded for solving, presumably by arrows I'd added to the puzzle later than those numbers. I of course deleted these numbers, as can be seen below. I should have Puzzle 30 up tomorrow, but until then, enjoy this one, let me know what you think of it, and send in solutions. Maybe I'll send something back. - ZM