The Muffin Problem

TL;DR

This paper investigates the Muffin Problem, establishing the existence and computability of the optimal minimum piece size, and providing formulas and methods to approximate or determine it for various cases.

Contribution

It introduces a formal framework for the Muffin Problem, deriving bounds and formulas for the optimal minimum piece, and develops methods to handle exceptional cases.

Findings

f(m,s) exists, is rational, and computable

f(m,s) often equals FC(m,s), with exceptions

Developed formulas and methods for many cases

Abstract

You have $m$ muffins and $s$ students. You want to divide the muffins into pieces and give the shares to students such that every student has $\frac{m}{s}$ muffins. Find a divide-and-distribute protocol that maximizes the minimum piece. Let $f(m,s)$ be the minimum piece in the optimal protocol. We prove that $f(m,s)$ exists, is rational, and finding it is computable (though possibly difficult). We show that $f(m,s)$ can be derived from $f(s,m)$; hence we need only consider $m\ge s$. We give a function $FC(m,s)$ such that, for $m\ge s+1$, $f(m,s)\le FC(m,s)$. It is often the case that $f(m,s)=FC(m,s)$. More formally, for all $s$, for all but a finite number of $m$, $f(m,s)=FC(m,s)$. This leads to a nice formula for $f(m,s)$, though there are exceptions to it. We give a formula $INT(m,s)$, which has 6 parts, such that for many of the exceptional $m$, $f(m,s)=INT(m,s)<FC(m,s)$. This works for most of the exceptional $m$ where ceil${2m/s}\ge 4$. There are still some exceptional $m$ with ceil${2m/s}=3$ (if its $\le 2$ then the problem is trivial). For these cases we have a way to {\it generate theorems}. For $1\le d\le 7$ we have generated formulas for $f(s+d,s)$. We do not have a theorem here but we do have a methodology which leads to, for some of the $m$, a value $BM(m,s)$ such that often $f(m,s)=BM(m,s)<INT(m,s)<FC(m,s)$. So far it seems like, for $m\ge s$, $f(m,s) = \min\{FC(m,s), INT(m,s), BM(m,s) \}$, though we have not prove this. For $1\le s\le 50$ and $s\le m\le 60$ we have obtained $f(m,s)$ for all but 20 values.