The works of KŐNIG Dénes (1884–1944) in the domain of mathematical recreations and his treatment of recreational problems in his works of graph theory. Mitsuko Wate Mizuno To cite this version: Mitsuko Wate Mizuno. The works of KŐNIG Dénes (1884–1944) in the domain of mathematical recreations and his treatment of recreational problems in his works of graph theory.. Mathematics [math]. Université Paris-Diderot - Paris VII, 2010. English. tel-00591307 HAL Id: tel-00591307 https://tel.archives-ouvertes.fr/tel-00591307 Submitted on 9 May 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. UNIVERSITE PARIS. DIDEROT (Paris 7) ECOLE DOCTORALE : numéro 400 : SAVOIRS SCIENTIFIQUES : EPISTEMOLOGIE, HISTOIRE DES SCIENCES, DIDACTIQUE DES DISCIPLINES DOCTORAT Histoire des mathématiques AUTEUR : WATE MIZUNO Mitsuko TITRE : The works of KŐNIG Dénes (1884–1944) in the domain of mathematical recreations and his treatment of recreational problems in his works of graph theory. TITRE EN FRANÇAIS : Les oeuvres de KŐNIG Dénes (1884–1944) dans le domaine des récréations mathématiques et son traitement de problèmes récréatifs dans ses travaux de théorie des graphes. Thèse dirigée par Karine CHEMLA Soutenue le 3 décembre 2010 JURY Mme. Agathe KELLER, President Mme. Evelyne BARBIN Mme. Karine CHEMLA Mme. Anne-Marie DÉCAILLOT Mme. Hélène GISPERT Mme. Sylviane SCHWER Acknowledgments I would like to thank my advisor Karine Chemla for her helpful and perse- vering advice. I would like to thank the judges of this dissertation Evelyne Barbin, Anne-Marie Décaillot, Hélène Gispert, Agathe Keller and Sylviane Schwer. For my reading of Hungarian texts, Imre Toth, Jean-Luc Chevillard, Péter Gábor Szabó and Katalin Gosztonyi helped me. On mathemat- ical recreations, Eric Vandendriessche gave me useful information and suggestion. I would like to thank them all. Finally, I would like to thank REHSEIS team who received me as a doctor course student. 3 4 Contents 1 Introduction 9 1.1 Motivation for this research . 9 1.2 Historical background of this research . 10 1.2.1 Graph theory . 10 1.2.2 Kőnig’s works . 10 1.2.3 Mathematical recreations . 12 1.3 Significance of this research . 12 2 Kőnig’s works 15 2.1 Introduction . 15 2.2 Kőnig’s works . 18 3 Mathematical recreations before Kőnig 25 3.1 Mathematical recreations before Bachet’s compilations . 25 3.2 Collections as mathematical recreations . 27 3.3 Bloom of mathematical recreations in the second half of 19th century . 28 3.4 Mathematical approach to recreational problems . 29 3.4.1 Seven bridges of Königsberg . 30 3.4.2 Labyrinths . 48 3.5 Citation by Kőnig ........................ 54 4 Mathematikai Mulatsagok 1: Mathematical recreations 1 55 4.1 Convention of translation . 55 4.0 Előszó: Preface . 56 4.1 Nagy számok: Large numbers . 61 4.2 Érdekes számok és eredmények: Interesting numbers and results 69 4.3 Számok kitalálása: Guessing numbers . 86 4.4 Bűvös négyzetek: Magic squares . 102 4.5 Mathematikai hamisságok: Mathematical errors . 117 5 6 CONTENTS 4.6 Síkidomok szétszedése és összeállítása: Decomposition and re- composition of plane figures . 139 4.7 Felhasznált munkák: List of works used . 159 5 Mathematikai Mulatsagok 2: Mathematical recreations 2 163 5.1 A mathematikai valószínűségről: About the mathematical prob- ability . 163 5.2 A kettes számrendszerről: About the binary numeral system . 175 5.3 A négyszínű térkép: The four colour map . 189 5.4 A königsbergi hídak: The bridges of Königsberg . 199 5.5 Az iskoláslányok sétái: Daily walk of schoolgirls . 207 5.6 Tait problémái és hasonló feladatok: Tait’s problem and sim- ilar problems . 221 5.7 Elhelyezkedések körben: Positions on a ring . 230 5.8 Átkelési, átöntési és vasúti feladatok: Problems of traversing, pouring and railway . 243 5.9 Apróságok (Örök naptár. Versenyszámolás. Meglepő ered- mények): Trivial matters (Perpetual calendar, race-calculation, surprising results) . 254 5.10 Az első és második sorozat problemáinak eredetéről és iro- dalmáról: About sources and bibliography of the problems of the first and second series . 266 5.10.1 ELSŐ SOROZAT: FIRST SERIES . 266 5.10.2 MÁSODIK SOROZAT: SECOND SERIES . 272 6 Kőnig’s works on Mathematical recreations 283 6.1 Introduction . 283 6.2 Recreational problems in the book of 1902 . 285 6.3 Recreational problems in the book of 1905 . 286 6.4 Recreational problems in the book of 1936 . 290 6.4.1 Recreational problems not treated in the book of 1905 but in the book of 1936 . 296 6.4.2 Summary regarding the book of 1936 . 298 6.5 Difference in treatment of problems between the books of 1905 and 1936 . 299 6.5.1 Bridges in the book of 1905 . 299 6.5.2 Bridges in the book of 1936 . 301 6.5.3 Summary of the difference between the books of 1905 and 1936 . 303 CONTENTS 7 7 Historical transition of the features of diagrams of graph the- ory 305 7.1 Introduction . 305 7.2 Why did Kőnig use diagrams in 1936? . 306 7.3 Diagrams in Kőnig’s treatise of 1936 and their historical back- ground . 308 7.3.1 Appearance of graph-like diagram for the problem of seven bridges of Königsberg . 309 7.3.2 Polygons, dominoes and the introduction of the flexible strings . 313 7.3.3 Polygons and dominoes again: a single diagram and a single way of using it for two distinct problems . 319 7.3.4 Mazes of which the junctions became important . 321 7.4 Tarry’s roles . 324 8 Conclusion 335 8.1 Kőnig’s books on mathematical recreations and that of graph theory . 335 8.2 Diagrams used in texts related to graph theory . 336 8.3 Future works . 338 Bibliography 339 Primary sources . 339 Secondary sources . 358 8 CONTENTS Chapter 1 Introduction This thesis deals with a chapter of the history of graph theory. Its main focus is on the historical transition from mathematical recreations to graph theory, as it can be studied from the various writings by Kőnig Dénes (1884–1944). In this chapter, I will describe a motivation for this research, the historical background of this research, and the significance of this research. 1.1 Motivation for this research Kőnig Dénes (1884–1944) is a Hungarian mathematician born in Budapest. He is now recognised as the “father of graph theory” with his treatise Theorie der endlichen und unendlichen Graphen written in 1936 [121]. In examining publications on Kőnig’s works, I found that he published two books that are apparently not related to graph theory. These two books compose a series entitled Mathematikai mulatságok (Mathematical entertain- ments/amusements), and published successively in 1902 and 1905 [86, 87], when he was still a student. According to Gallai Tibor (1912-1992) [224], a mathematician who was a student of Kőnig, these books were very suc- cessful1. Fortunately in 2007, I found the original version of the second book of 1905, and the recomposed version of the first book of 1902 at the Library of Eötvös Loránd University in Budapest. In the next year, thanks to my Hungarian friend Tóth Imre, deceased in 2010, I also obtained a duplicate of the original version of the first book of 1902. 1These books were recomposed with TypoTEX, and reprinted in 1991 and 1992. The digital version is sold on the Internet. Therefore, it is not very difficult to get the recom- posed versions. However in the recomposed versions, some parts are modified from the original version. It is more serious that the pages of references at the end of the second book of 1905 are totally missing in the digitalized version. 9 10 CHAPTER 1. INTRODUCTION In examining these books, I built a hypothesis that mathematical recre- ations and graph theory are not completely separated in Kőnig’s mind. I suppose that the works of the young Kőnig on mathematical recreations are historically connected to his later works on graph theory. In this thesis, I will try to verify this hypothesis by examining historical documents related to mathematical recreations and those related to graph theory. 1.2 Historical background of this research 1.2.1 Graph theory A historical approach to graph theory was made by Biggs, Lloyd and Wilson in their book Graph Theory, 1736–1936 in 1976 [215]. In this book, important texts of many mathematicians related to graph theory are col- lected and translated into English. Moreover, this book provides a historical approach to graph theory with mathematical complementary commentaries. This aspect is very useful for people studying graph theory. This book dealt with Kőnig’s treatise of 1936 as one of two landmarks of the history of graph theory —the other landmark was Euler’s article on the problem of seven bridges of Königsberg published in 1736 [51]. The authors consider Kőnig’s book of 1936 as “the first full-length book on the subject” (Preface, [215]) and “the first comprehensive treatise on graph theory” (p.219, [215]). 1.2.2 Kőnig’s works As historical approaches to Kőnig’s works, the works of Gallai Tibor and those of Libor Józsefné are remarkable. Gallai Tibor gave biographical notes in Hungarian in 1965 [224]. Gallai listed all the works of Kőnig. He classified Kőnig’s mathematical works into two categories: ones belonging to set theory, geometry or combinatorial topology, and the others belonging to graph theory.