  In the roleplaying game Tunnels & Trolls the characteristics of player
characters are determined by rolling dice in the following manner: First, one
rolls three dice and calculates their sum. If the three dice all give the same
result, another three dice are rolled and added to the total. This is continued
until the three dice no longer match. We calculate the average result of the
stochastic sum: 10 + 4/5.
  We also consider a generalized dice rolling scheme where we roll an arbitrary
number of dice with arbitrary number of sides. This generalization is motivated
by various exotic dice that are used in many roleplaying games. We calculate
the expectation, and how much it differs from the situation where we only roll
the set of dice once, with no rerolling and adding. As the number of dice
increases, or the number of sides the dice have increases, this difference
approaches zero, unless there are two dice (with the number of sides
increasing), in which case the difference approaches one.
