In this post, we will see the book Mathematical Games And Pastimes (Popular Lectures In Mathematics Vol 10) by A. P. Domoryad.

# About the book

The greater part of this book is devoted to classical games. The first few chapters deal with various systems of notation and with certain topics in the theory of numbers, the knowledge of which is necessary for the understanding of the theory of various mathematical games. But for some readers these chapters might be interesting in themselves. The theory of some isolated games is presented fairly fully here; in other cases only results are given; and reference is made to sources, where proof of these results can be found. Side by side with classical pastimes, the book devotes much space also to “contemporary” pastimes quick reckoning, re-cutting of figures, construction of curves, and models of polyhedra. Deserving particular attention are the problems which admit a practically inexhaustible or even infinite number of solutions (“Construction of parquets”, “Construction of pleasing patterns”, etc.).

Here, everybody, by applying persistence and inventiveness, can attempt to obtain interesting results.

Whereas such classical pastimes as, for example, constructing “magic squares” may be enjoyed by a comparatively narrow section of people, the cutting out of, say, symmetrical figures in paper, the construction of pleasing patterns, searching for numerical curiosities, by not requiring any mathematical preparation, might give pleasure to both amateur and professional mathematicians. The same can be said about pastimes requiring knowledge confined to that obtained in the 8th to 10th classes of the secondary school (construction of parquets, of interesting curves and borders, etc.).

In group activities it is possible to arrange competitions in making up original parquets, in the construction of curves and borders, in obtaining attractive symmetrical figures cut out of paper, and so on. Each participant in such competitions can dazzle with his inventiveness, accuracy of execution, or artistry of colouring the figures obtained. Such collective activity can be rounded off by compiling an album or by organizing an exhibition of the best items. Many pastimes and even single problems may suggest to the amateur mathematician themes for independent investigations (the use of knight’s moves instead of the “short” moves of the fook in the “game of 15”, the search for interesting identities — see § 37 —, the generalization of the problem about tourists — problem No. 13 in §37 —and so on). On the whole, this book caters for readers with mathematical knowledge within the limits of the 9th and 10th classes of the secondary school, even though the greatest part of the material is accessible to pupils of the 8th class, and some topics — even to school- children of the 5th and 6th classes. Many chapters can be used by teachers of mathematics for extracurricular activities.

Various categories of readers can use the book in various ways: persons not particularly fond of mathematics can become acquainted with curious properties of numbers or figures, without going into the fundamentals of the games and pastimes, and taking for granted single propositions; amateur mathematicians are advised to study certain parts of the book with pencil and paper, solving the problems given and answering the questions posed. § 38 gives answers to the problems to be found in the text, questions and hints towards their solution and also proofs of certain of the theorems mentioned in the text. References to the appropriate section of § 38 are given in small figures between ordinary brackets.

References to books in which the reader may find a more detailed discussion of the topics touched upon are given by a number enclosed in square brackets. This number refers to the corresponding entry in the bibliography at the end of the book.

The book was translated from Russian by Halina Moss and was published in 1963.

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You can get the book here.

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# Contents

- Various systems of Notations 1
- Some facts from the Theory of Numbers 10
- Congruences 15
- Continued Fractions and Indeterminate Equations 20
- Pythagorean and Heronic Triples 31
- Arithmetical Pastimes 33
- Numerical Tricks 37
- Rapid Calculations 45
- Numerical Giants 57
- Games with Piles of Objects 61
- Meleda 71
- Lucas’ Game 75
- Solitaire 77
- The “Game of Fifteen” and Similar Games 79
- Problems on determining the Number of ways of reaching a goal 86
- Magic Squares 97
- Euler Squares 105
- Pastimes with Dominoes 107
- Problems Connected with Chessboard 109
- Making up Timetables 120
- The “Problem of Josephus Flavius” and similar ones 124
- Pastimes connected with Objects Changing Places 127
- Simplest methods of Constructing Pleasing Patterns 136
- Regular Polygons from Rhombi 142
- The Construction of Figures from Given Parts 145
- The Construction of Parquets 149
- Re-cutting of Figures 158
- The Construction of Curves 166
- Mathematical Borders 188
- Models of Polyhedra 193
- Pastimes with a sheet and strips of paper 202
- The Four Colour Problem 207
- Drawing Figures at one stroke of the pencil 211
- Hamilton’s Game 215
- Arranging Points on a Plane and in Space 219
- Problems on a Logical Nature 222
- Rag-Bag 232
- Notes and Answers to Problems 246

Bibliography 297