# Mythematics: Solving the Twelve Labors of Hercules

A combination of Greek mythology and mathematics is certainly unusual. Nevertheless, probably inspired by the popularization of Hercules by the Disney animation film, Huber has connected to each of the 12 tasks for Hercules some relatively elementary mathematical challenges for the reader.

The text by Apollodorus for the 12 works is taken as a basis to describe the labors. Next, Huber rephrases the texts and formulates some subtasks in the form of mathematical problems. For each of these, more details and data are provided to make them more concrete. That should be enough for the reader to solve the problems and he or she can start off and solve the problem on his or her own. For those who do not want to solve the problem for themselves, solutions are provided. These solution sections are essentially independent of the preceding introduction. The relevant section of the Apollodorus text is repeated, the mathematical problem with all the data is repeated and the solution is given. As a consequence of this format, there is a lot of repetition in the book.

The mathematical problems vary very much in their physical setting as well as in the mathematical skills needed to solve them. The degree of difficulty is on average increasing towards the end of the book. For example, a simple problem is associated to Hercules' cleaning of the Augean stables. It is required to compute the number of herds that Augeas has when you know that *half of them are at the stream of Alpheus, the eighth part pasture around the hill of Cronos, the twelfth part far away by the precinct of Taraxipus, the twentieth part feed in Elis, and the thirtieth part is left in Arcadia, and what you see is the remaining fifty herds.* A problem can be as simple as that, but it can also become more complicated to compute the pressure on the walls of the stables when they are filled with water. Or to compute the time it takes to drain the stables when Hercules makes a hole of a prescribed size in the wall. These examples as well as the other problems, illustrate the nice interplay of the laws of physics and the mathematics to model the problem and to eventually solve it.

In a similar way many different areas of mathematics are used like algebra, difference and differential equations, geometry, probability and statistics, and of course calculus in one and more variables. An appendix shows which skills are used in each of the 12 labors. One of the more advanced topics is the use of the Laplace transform in the eleventh labor (The apples of the Hesperides). Here Hercules has to wrestle with the giant Antaeus. The giant looses strength when he is lifted off the ground but he is ``reloaded'' when he is in contact with the ground. The model is that Hercules keeps him 5 minutes in the air, and then throws Antaeus on the ground for 1 minute. That adds a kind of pulse train to the differential equation describing the decay in force of the giant. Solving this is done by using the Laplace transform. The Laplace transform might be just outside the reach of the average reader of this book. Therefore some extra information on the Laplace transform is given in an appendix.

These are just a couple of examples of the 34 problems that are formulated, modeled and solved in the book. Other examples are the trajectory of an arrow (shooting the Nemean lion), harmonic oscillation and resonance (chasing the Stymphalian birds from the wood), population dynamics (the city of Abdera), an optimal illumination problem (between the pillars of Hercules), a packing problem (blocking the river Strymon), a steepest descent problem (descending to the underworld), etc.

As an entertaining intermezzo, there are three sudokus (with a twist) inserted after every fourth labor. The sudokus grow in difficulty just as is the tendency of the problems to be solved as one progresses through Hercules's labors. The bulk of the book is about the math problems and their solution but for the readers who got interested in the mythology, a bit of extra information about Hercules and the authors of his legend can be found in an appendix.

The book is unique in its mixture of ancient Greek mythology and applied mathematics. The problems to be solved are not really different from what could be found in other textbooks. The mixture and diversity of the problems is however rather unusual. The fact that they are connected to Hercules and his labors, is in retrospect not really essential. Nevertheless, placing them in that context may arise the interest of some youngsters who are mainly interested in the Greek legends and less in technological applications. It will certainly be a valuable source of inspiration for math teachers who have to teach these students

**Submitted by Adhemar Bultheel |

**21 / Sep / 2014