Mage Merlin's Unsolved Mathematical Mysteries

This slim volume appears to be aimed at middle and high school students who possess a taste for the design and style of fantasy media. The authors use the literary device of a storyteller, named for the late Fields Medalist Maryam Mirzakhani, who shares 16 puzzles purported to be recorded in Merlin’s journal. The connection to mathematics is that these are all unsolved problems—mostly from number theory, graph theory, and geometry—and the point is to help readers see that mathematics remains a living, growing discipline. Indeed, the authors use the metaphor of an ice cream cone to argue that the mathematics yet to be discovered is far vaster than the mathematics that is already known. They also note that they selected problems whose statements can be understood by anyone, even though the solutions are as yet known to no one.

 

Each chapter provides a “journal entry” to introduce each problem as it might have arisen in the imagined daily life of Camelot, then shifts into a nonfiction voice to state the problem and the current state of efforts to solve it. Thus, readers encounter the questions of whether: six square tiles are sufficient to cover a slightly larger square; there are infinitely many twin primes; every polyhedron can be unfolded into a single flat surface; all perfect numbers are even; a single connected tile tiles a plane periodically; every planar graph can be drawn with whole-number segments; a thrackle can have more edges than vertices; every even number can be written as the sum of two primes; all trees are graceful; the 5th and kth Schur numbers can be calculated; any room with flat mirrored walls can be fully illuminated by a single light source; any convex polygon can be fairly partitioned into n convex pieces; a perfect brick exists; and every whole number iterated by the Collatz algorithm arrives at 1. 

 

The 16 vignettes do extend concepts from high school mathematics, so I think the book will be enjoyed by those who find its medieval conceit appealing. If you gift it, please first correct the statement on p. 31 that “Leonhard Euler proved in the nineteenth century,” since his lifespan was entirely in the 18th century. (His chronology is stated correctly on p. 85.) I did not find any other historical or mathematical errors that were as egregious, although some of the explanations seemed terse to me and a bit more mathematical detail would have furthered my own understanding. I do now have a better idea of how much I need to learn about graph theory, in particular. If I had written the book, my own choice would have been to put further reading recommendations, including additional websites, at the end of each chapter to encourage readers to stop and explore each unsolved problem. Perhaps, though, the publisher mandated a single bibliography at the end of the book; it does contain an interesting mix of recent popular accounts, research publications, textbooks, and classic works.

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Amy Ackerberg-Hastings (aackerbe@verizon.net) co-edits MAA Convergence with Janet Heine Barnett. She is a historian of mathematics whose interests lie especially in the history of mathematics education in 18th-century Scotland and 19th-century American colleges; women in the history of science, technology, and mathematics; and the history of scientific instruments.