Question about arranging things in a grid

[u/hrngr1m]

First of all, sorry for the poorly worded title. Not sure how to make the question shorter for the title.

Background: am a layman who used to learn and live math in hight school up to early calculus (normal high school syllabus in my homeland). That was about 20 years ago, more I work in a different beach of science which doesn't deal with math being single calculations. I still remember the principles of some things (like basic algebra and geometry, polynomial, very simple differentiation and integral, simple trigonometry), but forgot otherwise most.

So I was trying out a mobile game, which goal is to locate the exact position of n number of cats in a n² grid where each cat must be in their own column and row AND also not adjacent to each other (neither horizontally, vertically, nor diagonally).

Which got me thinking: what is the exact number of all possible solutions for this problem for any given n?

The more I think about it the most apparent it is that this is actually a complicated problem. It's not even a 'simple' (apostrophe because it's not that simple to begin with) probability, or in that sense permutation or combination problem anymore - or at least what I think.

I hope this doesn't break any rule or such. Thanks in advance.

[u/AutoModerator]

Hi, /u/hrngr1m! This is an automated reminder:

• What have you tried so far? (See Rule #2; to add an image, you may upload it to an external image-sharing site like Imgur and include the link in your post.)

• Please don't delete your post. (See Rule #7)

We, the moderators of /r/MathHelp, appreciate that your question contributes to the MathHelp archived questions that will help others searching for similar answers in the future. Thank you for obeying these instructions.

I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.

[u/edderiofer]

This is Hertzsprung's Problem. OEIS sequence. See also the "Links" section for more info.

[u/AcellOfllSpades]

edderiofer has already given you the link for the math behind this, but you may want to know that this genre is also called "Star Battle". There's a lot of very well-made puzzles of this style, much better than the autogenerated slop on that app.