Is there a mathematical way to determine a solution for puzzle games?

Question

Or more specifically, a mathematical way to determine HOW to solve a puzzle game.

Take a look at this screenshot of a puzzle game "The Talos Principle."

As you can see, the purpose of this mini-puzzle is to fill in the rectangle with Tetris blocks. I've struggled through some of them, but I managed to brute force my way through, while always having in mind "there's gotta be an easier way to solve these problems."

Is there any developed... stuff? to solve these puzzles?

Answer 1

Googling comes up with the paper

• Erik D. Demaine and Martin L. Demaine. Jigsaw Puzzles, Edge Matching, and Polyomino Packing: Connections and Complexity. Graph. Comb. 23, 1 (February 2007), 195-208.

where corollary 3 states that

It is NP-complete to decide whether $n$ given polyomino pieces, each fitting within an $\Theta(\log n) \times \Theta(\log n)$ rectangle, can be exactly packed into a specified square box whose area equals the total area of the pieces.

This doesn't quite say that there's not going to be any simple answer to your question, but it's fairly close to saying that.

The paper is available at

http://erikdemaine.org/papers/Jigsaw_GC/paper.pdf

if you're interested.