Before asking my questions, allow me to begin with a separate example to help clarify what I'm driving at. For terms that are not defined formally, please interpret them as you feel would be most appropriate, and I welcome any attempts to modify this question so as to make it more interesting and/or understandable.

**Background Example:** Suppose you have Cartesian coordinates for $n$ different points $p_1, \ldots, p_n$ in the plane, and you want to find a point $p$ that minimizes $\sum_{i}^{n}|p-p_i|.$ (In other words, you want to find the "geometric median" for these $n$ points.)

Finding such a $p$ is trivial for $n = 1$ and $n = 2$, and well-known for $n = 3.$

Note that even for $n = 4,$ finding a general approach that leads to an exact answer is tough. For larger values, say, $n = 10,$ we would (in most real life situations) estimate $p$ using a computer.

But here's a way to estimate $p$ using a physical model: take a piece of plexi-glass, draw a grid on it, drill tiny holes corresponding to the $n$ points, thread equally-weighted wires through each of the holes (tie, say, a $10g$ weight to the ends below the plexi-glass), and fuse the tops of all the wires together. Holding the plexi-glass level, a minimal energy argument suggests that the final resting point for this "fused top" will be (approximately, because we're talking about the real world here) at $p.$

**Question 1:** Is there some way to build a physical model where you start with a $9$x$9$ Sudoku grid and its clues (i.e. the numbers filled in at the start) and then separately have all the numbers yet to be placed, and by "doing something" (dropping balls, something with weighted-strings, I'm not sure -- hence the question) release the non-placed numbers so that they "quickly" fall into place (e.g. as a "minimal energy" or "path of least resistance" consequence)?

**Remark:** On the one hand, Sudoku is "discrete" in a way that the aforementioned background example is not, which suggests to me that it might be possible to create such a physical model. On the other hand, the $NP$-completeness of Sudoku has me nearly convinced that no such model could be built. However, I haven't the faintest idea as to how one *proves* the non-existence of this sort of physical model.

**Question 2:** What would the implications of the existence (or non-existence) of a physical model to "quickly" solve Sudoku be for the question of $P = NP$?