Okay, as promised, here's the one about eigenvectors:
Problem 14: Community Events
The "World of Warcraft" tabletop roleplaying game is based in part on the computer game of the same name, but also includes other features, including rules for in-game towns and communities. In the game, communities have a "community behavior map" consisting of a list of attributes, including "wealth", "greed", "happiness", "disaster", and so on, each with a numerical value. Different events can affect different attributes. Also, there are "links" going from one attribute to another, and each link has a different "intensity," which can be positive or negative. If a link goes from A to B with intensity I, then whenever an event affects attribute A, attribute B also changes by an amount equal to the amount attribute A changed, times the intensity of the link. For example maybe "wealth" and "happiness" have a link with an intensity of 0.5. Then if a group of adventurers brings back lots of treasure, increasing the "wealth" by 10, the "happiness" will increase by 10 x 0.5 = 5. Of course the intensity can be negative - an increase in the "disaster" attribute will probably reduce other attributes.
The rules specifically state that impact is only felt one link away from a factor. For example, suppose attribute A is linked to B and attribute B is linked to C. If an event affects A, then B will be affected by the link, but that change does not then continue on to affect C through its link. Suppose, however, that this rule were removed; you calculate the changes based on the initial event; then the changes based on the links; then the changes caused by the links from those links, and so on.
What conditions on the links and their intensities are necesary to ensure that this process always converges?
The answer is here.