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Mathematical Games And Pastimes (Popular Lectures In Mathematics Vol 10) | A. P. Domoryad

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From the wide variety of material collected by various authors under the name of mathematical games and pastimes, there can be extracted several groups of “classical pastimes”, which drew the attention of mathe­ maticians for a long time;
(1) Pastimes, connected with the search for original solutions of problems, which permit a practically un­ limited number of solutions (see e. g. “Magic squares” — ch. 16, “The problem of the chess knight” — ch. 19, etc.). Here, the interest is usually centred on establish­ ing how many solutions there are, working out methods leading to large groups of solutions, or on solutions satisfying some special requirements.
(2) Mathematical games, i. e. games in which two players aim at a definite goal, through a number of “moves” made one after the other in accordance with agreed rules: here it turns out to be possible to pre­ determine the victor for any initial situation, and to indicate how he is to win, no matter what the opponent’s moves are (see e. g. ch. 10).
(3) “Games for one person”, i. e. pastimes in which it is necessary to reach a definite, predetermined goal by means of a number of operations, carried out by the player himself, in accordance with given rules (see e.g. chs. 11-14): here the interest is centred on the con­ ditions under which the goal is reached, and it is required to find the least number of moves necessary to reach that goal.
The greater part of this book is devoted to classical games.
ix
 X
Foreword
The first few chapters deal with various systems of notation and with certain topics in the theory of num­ bers, 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 inven­ tiveness, can attempt to obtain interesting results.
Whereas such classical pastimes as, for example, con­ structing “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 mathe­ maticians. The same can be said about pastimes requir­ ing 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 competi­ tions in making up original parquets, in the construction of curves and borders, in obtaining attractive symmetri­ cal 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 com-

 Foreword xi
piling 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 gene­ ralization 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 mathe­ matics for extracurricular activities.
Various categories of readers can use the book in various ways: persons not particularly fond of mathe­ matics can become acquainted with curious properties of numbers or figures, without going into the funda­ mentals 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 biblio­ graphy at the end of the book.

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