To understand the problem clearly, we first have to label the boxes. First, let's assume that we are solving a 1 x 2 version of the Connecting Averages problem. Let the boxes be A and B. Now, we can use a rule that A + B = (The sum of all the 6 numbers surrounding A and B) divided by 3.For example, if the numbers surrounding A and B have a sum of 48, then A + B = 16.
For problems that are 1x3 or larger, we have to use different methods. Let's assume that we are solving a 1 x 3 version of the Connecting Averages problem, with 3 boxes. Let the boxes be A, B and C. After labeling them, we can derive formulae such as A= B + the numbers surrounding A, B = A + C + the 2 numbers surrounding B, etc. A demonstration of both methods is shown in the file attached.