I think I see what he means though. There should be one free variable in this system, assuming the bounding rectangle is fixed.

Let's say the piece of wood is a rectangle from (0,0) to (A,B). Inside that piece of wood, you place 4 nails, also arranged in an axially-aligned rectangle. Assuming you want the string to touch the center of the 4 sides of the outer rectangle, the placement of the nails and the length of the string should all depend on a single free variable (sort-of a roundness factor, correct me if I'm wrong though).

Obviously, if this is 0, the nails go in the corners, and the string fits snugly all around, but as this value increases, the nails move inward, but not in such a simple way.

Let's say the nail closest to the origin is at (C,D). When the string stretches to the bottom center, it'll form a pentagon with side lengths

. Similarly, when it's on the left side, it'd have sides

. Since the string is the same length in either case, these two cases must sum to the same length. Since A and B are fixed, this gives us one equation in two variables (or if you add that the length of the string is equal to either of these, two equations in three variables). Solving these equations should give you all the information you need, technically, but you may have to apply some substitutions to get a more meaningful free variable.

Well, for now I've got to go, but it's pretty likely someone else here can poke holes in my equations and help you figure the rest of this out.