This thesis explores the problem of the distribution of the sum of $m$-sided dice, and how close this distribution can be to a uniform distribution. First, we show using generating functions that a uniform distribution of the random variable representing the sum of two six-sided dice is not possible. We then generalize the result to the case of $n$ dice with $m$ sides. Since the proof using generating functions fails in this case, we take a different approach via a theorem. In the second part of the thesis, we explore how to approximate the uniform distribution of the sum of two $m$-sided dice as closely as possible using optimal dice. We find that the optimal dice are symmetric, but for $m > 2$ they are not identical. We then prove that in a theoretical scenario where negative probabilities are allowed, a uniform distribution of the sum is possible. Finally, using Python code, we verify the validity of the theorem about two optimal dice, and then use the code to test general cases for arbitrary $n$ and $m$.
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