Arithmagons

[From the notes of Emily Moore. Modified by Sam Rebelsky.]

A secret positive integer is assigned to each vertex of a triangle. On each side of the triangle is written the sum of the secret numbers at its ends.

For example, we might have a triangle with sides labeled 11, 18, and 27. What are the secret numbers from which these numbers are derived?

Find a simple rule for revealing the secret numbers. What triples of numbers are possible for labels for the sides? What are impossible?

Does it make a difference that the integers are positive? Why or why not?

Generalize this question to other polygons. Try to find other questions you might ask.