MATHEMATICAL MUFFIN MORSELS: NOBODY WANTS A SMALL PIECE

Suppose you have five muffins that you want to divide and give to Alice, Bob, and Carol. You want each of them to get 5/3. You could cut each muffin into 1/3-1/3-1/3 and give each student five 1/3-sized pieces. But Alice objects! She has large hands! She wants everyone to have pieces larger than 1/3. Is there a way to divide five muffins for three students so that everyone gets 5/3, and all pieces are larger than 1/3? Spoiler alert: Yes! In fact, there is a division where the smallest piece is 5/12. Is there a better division? Spoiler alert: No. In this book we consider THE MUFFIN PROBLEM: what is the best way to divide up m muffins for s students so that everyone gets m/s muffins, with the smallest pieces maximized. We look at both procedures for the problem and proofs that these procedures are optimal. This problem takes us through much mathematics of interest, for example, combinatorics and optimization theory. However, the math is elementary enough for an advanced high school student. Sample Chapter(s) Preface Chapter 1: Five Muffins, Three Students; Three Muffins, Five Students Chapter 2: One Student! Two Students! Some Basic Theorems! Contents: Preface About the Authors Acknowledgments Five Muffins, Three Students; Three Muffins, Five Students One Student! Two Students! Some Basic Theorems! Our Plan Three Students! Four Students! The Floor–Ceiling Theorem! Finding Procedures The Half Method A Formula for ƒ(m,5) The Interval Method The Midpoint Method The Easy Buddy–Match Method The Hard Buddy–Match Method The Gap and Train Methods Scott Huddleston's Method Appendices: Math Notation Fair Division ƒ(m,s) Exists! ƒ(m,s) is Rational! ƒ(m,s) is Computable! References Index Readership: High school and undergraduate students, computer scientists, mathematicians, and anyone interested in recreational mathematics.