Using LINQ to solve puzzles

The Puzzle

Here's a puzzle similar to the one in the puzzle hunt. The diagram below is a bunch of weights (A-M) hanging from a system of bars. Each weight has an integer value between 1 and 13, and the goal is to figure out what each weight must be for the the diagram below to balance correctly as shown:

                          |
                        |
            +--+--+--+--+--+--+--+
            |                    |
            |                    |
         +--+--+--+--+--+        |
         |     L        M        |
         |                       |
+--+--+--+--+--+--+     +--+--+--+--+--+
H              |  I     |  J        K  |
               |        |              |
      +--+--+--+--+--+  |     +--+--+--+--+--+
      E              F  |     G              |
                        |                    |
            +--+--+--+--+--+  +--+--+--+--+--+--+

            A              B  C                 D

The rules for this kind of puzzle are: (1) The weights on either side of a given pivot point must be equal, when weighted by the distance from the pivot, and (2) a bar hanging beneath another contributes it's total weight as through it were a single weight. For instance, the bar on the bottom right must have 5*C=D, and the one above it must have 3*G=2*(C+D).

I think there approach is way too brute force, this it seems like a little set theory mixed in with some linear algebra and recursive decent would be able to solve this problem. More on it later, if I come up with my own solution.