# Expected characteristic in Tunnels & Trolls character creation, with generalizations

## TL;DR

This paper analyzes the expected outcomes of a specific dice-rolling process in Tunnels & Trolls and generalizes it to arbitrary dice, providing formulas for expectations and differences from simple rolls.

## Contribution

It introduces a generalized model for dice rolling with rerolls, deriving expectations and asymptotic behavior for various dice configurations.

## Key findings

- Expected value of the reroll process is calculated as 10 + 4/5.
- As the number of dice or sides increases, the difference from a single roll diminishes.
- For two dice with increasing sides, the difference approaches one.

## Abstract

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.