I came up with the following thought experiment in my research in order to better understand the way Turing machines can transfer information through their tapes (the motivation is detailed below, isn't necessarily needed to answer the question). I was wondering what sort of area of study this might fall under, or if the general problem itself already has a name.
You are in a very peculiar museum. There are two movable walls with lines of paintings on them. Each painting is one meter by one meter. You also have a canvas that is k meters by 1 meter for some constant k. You can only look at the paintings directly to your left and right at any given time (due to your poor eyesight), but you can always choose to have either wall move in order to show an adjacent painting.
Before moving a wall, you can choose to paint as much of a given painting on your canvas as you want, but you can't resize it. You can paint over other parts that you have already painted, too. You can also transfer any painted parts of your canvas to one of the two canvases containing the paintings you are currently able to see (overwriting the contents forever, unless you have them painted somewhere else).
Your goal is to, in the shortest number of wall movements, have each canvas on the wall contain an equal proportion (or close to an equal proportion) of area with contents from each of the original paintings. How well can you do this?
Of course, the question is fairly open-ended. I can give more details about specific parameters if that would help in finding related problems.
Motivation
A few complexity theoretic problems for one tape turing machines can be solved using information theoretic-esque techniques, such as those used to prove quadratic time lower bounds on palindrome recognition. I haven't really seen these techniques generalized to multi-tape machines, and spent quite a lot of time considering how we might even think about this added complexity. So I came up with the above thought experiment that, in a sense, models the information sharing between tape squares of a Turing machine.
