Problem Idea 1D tape simulating 2D tape

When I was taking a course on Computational Models, we had the following question appear on a Homework.

Consider a Turing Machine with a 2-dimensional tape. The head now has 4 possible moves, up, down, left, or right. (Everything else is to a standard Turing Machine.) Prove that a Turing Machine with a 2-dimensional tape can be simulated by a standard Turing Machine

A normal turing machine is only 1-dimensional and extends infinitely in either direction. A 2-dimensional turing tape is supposed to extend infinitely up,down,left,right.

Anyways, the key idea for showing how this could be done is basically summarized by this diagram

The problem statement I am imagining is this: We are given a Turing Machine that is only capable of increasing and decreasing it's index along a tape. We want to have this 1-dimensional Turing Machine simulate a 2-dimensional Turing Machine, so when it receives instructions to move up,down,left,or right, we have to tell the 1-dimensional turing machine how much to increase or decrease it's index. We can also assume that either the Turing machine will start at cell 0, or the starting cell could also be part of the input.

Assuming the machine starts at cell 0, and supposing we had the following test data:

U R U L L U U R U R D D L

The machine would visit these cells after executing each movement.

3 2 13 14 15 34 61 60 95 94 59 32 33

So the output of the program should be

3 -1 11 1 1 19 27 -1 35 -1 -35 -27 1

Also, another idea for a problem, that does something related to the above but adds a more complicated step. Based on this video: https://www.youtube.com/watch?v=3s7h2MHQtxc

The problem statement would be, given a list of floating point numbers, figure out which point each should map to when mapping it through the Hilbert curve function.

Also... the Hilbert curve appears to have a variety of applications in Computer Science: https://en.wikipedia.org/wiki/Hilbert_curve so an implementation of this can also be practically useful :P