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**Looking for an interesting Mathematical problem "family"**

And what do I mean by that. All right, I think this was the most fitting category, I hope I'm not too off-topic.

So I want to write a puzzle game, but, as everyone who loses his time playing puzzle flash games (like me), knows that you cannot be original enough. What I'm looking for:

1. A mathematical problem that can be randomly generated. So that's why I want a "problem family".

2. It should not be directly related to math-I do not want equations, formulas, etc. I want something that can be given to someone with graphics- a table, a grid, anything you fancy.

3. Must be solvable without using math. They can, and since they are behind the whole thing, will help getting to the solution, but the "trial and error" or "think very hard about it" ways MUST be possible.

4. I must be able to predict if the generated problem can be solved. A simple way of doing that, something that can be transfered to a program easily, is preffered, but not a must.

5. Things that have been going around for decades, things from scientific papers, models, anything, suit me. Not being popular, or already used for the same purpose is also preffered.

I remember playing a game based on a "problem". There was a grid with boxes, and each colored box was supposed to be a person. Based on certain condition, after each "round", persons either held "offsprings" (new persons next to them), did nothing (...just stayed there), or "died" (dissapear). The game gave you a challenge, for example, have 20 at least persons after so many rounds, and you were allowed to place some (limited) boxes on your own, were you saw fit. After some point it got harder and harder (I cant remember beating it, lol). I read it was based on some experiment, paper, or something like that, but saddly I cant remember what it was exactly. BTW, if anyone knows what the theory behind it is , let me know, I've been looking for that game :D

I think those are good for now.

This is going to be a self project, made for fun. If the idea comes from a paper, I will give credit, if it comes from a member here, I will give credit. The goal is not to make a sellable product, just have some programming fun.

Come on people, show me what you got! :D