# Wilhelm Ahrens' *Mathematical entertainment and games*

In 1910 Wilhelm Ahrens published

*Mathematische Unterhaltungen und Spiele*. We give below a translation of the Preface of this book into English. This translation was made by Tom Schnabel.In past decades, this field of study, dubbed by [French mathematician] Claude Bachet as the "problèmes plaisants et délectables qui se font par les nombres", has doubtlessly not seen a more successful devotee than Edouard Lucas. This is strongly evinced not only by his numerous published works in a variety of Journals and many sections of his "Théorie des nombres", but also by his works "Arithmétique amusante" (Paris, 1895) and above all "Récréations mathématiques" (v. I-IV). However, this last work, originally intended as a cohesive summary of the entire field of study, famously remained incomplete following the sudden death of its author. Indeed, the last 2 volumes were only published posthumously and as such many important topics intended for later volumes received only superficial treatment at best, or none at all. To name but one example, the important "Rösselsprungproblem", far from occupying half a volume by itself, as envisioned by Lucas, was covered only briefly. Given the broader nature of the compiled work, it regretfully had to give way to other topics that are likely to be of far less interest to both the mathematician and the layman reader. - German literature does not possess a work similar to that of Lucas. The only recent publications that might be considered for their independent value, "Mathematische Muβestunden" and "Zwölf Geduldspiele" by Prof. Dr. Schubert, are written primarily for the non-mathematical reader. As in the generally far more rigorous British work "Mathematical Recreations", by Rouse Ball, a variety of topics are thus covered that have been treated exhaustively in purely scientific literature, for instance the problem of squaring the circle etc.

In contrast, the author [of this book] strove primarily to fill those gaps left in Lucas's work, that should be of interest not only to the layman, but also the mathematician, and to provide a rigorous and critical treatment of the historic literature thereof. Topics that would seem trivial to the professional, or at least incapable of awakening scientific interest, were excluded. Also scattered throughout this book are numerous topics that cannot be found in any of the above mentioned literature. It will become quickly apparent to the experts of the field that the author has not limited himself to merely reproducing or adapting old theories, but also conducted extensive research of his own, as demonstrated by the meticulous referencing of the background literature throughout the book. Regarding this literary research, the author has spared no efforts and plowed through as much of the literature as was possible, given a permanent position at an official body of a city without even the most modest scientific library. If, despite his efforts, the author has failed to consider all relevant publications, he asks for forgiveness, as the literature in this field is not only very extensive, but is also not always exclusively scientific and thus distributed among many, difficult to access publication organs. As a result, many of the relevant original works lack awareness of prior publications and thus fail to reference or comment on these. In particular the acquisition of chess literature, generally very scarce in public libraries, proved difficult. So I feel obliged to express my gratitude to the "Berliner Schachgesellschaft" and the "Magdeburger Schachclub", both of which granted me full access to their respective libraries. Not being a regular chess player myself, I also owe thanks to senior teacher E Schollwer, a previous president of the "Magdeburger Schachclub", for his guidance on chess technique and literature.

I envisioned this to be a work that is not only accessible to the educated layman, but also covers, however briefly, the most important scientific aspects and links to pure mathematics. Given the fundamental character of most problems, this did not strike me as impossible. Understanding the book at hand should require only rudimentary mathematical knowledge, even though a certain familiarity with the mathematical way of thinking might be an indispensable prerequisite. In order to make the text more accessible, sections which address primarily the expert and the layman is thus advised to skip, are marked by a smaller font. The author freely admits that he might not always have managed to meet the demands of both parties, but asks the reader to appreciate the difficulty of providing a generally comprehensible, yet scientifically thorough treatment of the subject matter.

The extensive literary index attached to the book relies, with regard to the earlier centuries, mainly on the known bibliographies. It contains only the literature that is directly relevant to the topics discussed in the book. As such, publications, such as the extensive literature on the Analysis Situs, are not included if they were not directly used, even though they may be cited in the text. My original intention of including a brief summary of each article cited, later proved impossible given the scale of the work. Instead, included with every citation are the pages of this work, printed in bold, for which the article is relevant. As such, the book itself gives a quick overview of the content of the most significant recently published works. Also in the pursuit of brevity, smaller articles without a special name from the "Educational Times Reprints" or the "Intermédiaire des mathématiciens" etc. were excluded, despite their often great value, unless a suitable place in the text was found in which mention them. Similarly, smaller publications which were later included in larger publications of the same author were excluded. These include, for instance, several small essays of Edouard Lucas's, amongst others - The literary index is followed by a Subject and name index.

I would also like to thank those who have assisted me with the editing of my work.

Magdeburg, in October 1900

W Ahrens

Last Updated April 2016