I commence by describing some arithmetical recreations. The interest excited by statements of the relations between numbers of certain forms has been often remarked, and the majority of works on mathematical recreations include several such problems, which, though obvious to anyone acquainted with the elements of algebra, have to many who are ignorant of that subject the same kind of charm that mathematicians find in the more recondite propositions of higher arithmetic. I devote the bulk of this chapter to these elementary problems.

Before entering on the subject, I may add that a large proportion of the elementary questions mentioned...

I devote this chapter to the description of some arithmetical fallacies, a few additional problems, and notes on one or two problems in higher arithmetic.

I begin by mentioning some instances of demonstrations* leading to arithmetical results which are obviously impossible. I include algebraical proofs as well as arithmetical ones. Some of the fallacies are so patent that in preparing the first and second editions I did not think such questions worth printing, but, as some correspondents expressed a contrary opinion, I give them for what they are worth.

First fallacy. One of the oldest of these, and not a...

In this chapter and the next one I propose to enumerate certain geometrical questions, puzzles, and games, the discussion of which will not involve necessarily any considerable use of algebra or arithmetic. Most of this chapter is devoted to questions which are of the nature of formal propositions; the next chapter contains a description of various amusements.

In accordance with the rule I laid down for myself in the preface, I exclude the detailed discussion of theorems which involve advanced mathematics. Moreover (with one or two exceptions) I exclude any mention of the numerous geometrical paradoxes which depend merely on...

Leaving now the question of formal geometrical propositions, I proceed to enumerate a few games or puzzles which depend mainly on the relative position of things, but I postpone to chapter x the discussion of such amusements of this kind as necessitate any considerable use of arithmetic or algebra. Some writers regard draughts, solitaire, chess, and such-like games as subjects for geometrical treatment in the same way as they treat dominoes, backgammon, and games with dice in connection with arithmetic: but these discussions require too many artificial assumptions to correspond with the games as actually played or to be interesting....

A polyhedron is a solid figure† with plane faces and straight edges, so arranged that every edge is both the join of two vertices and a common side of two faces. Familiar instances are the pyramids and prisms. (A pentagonal pyramid has six vertices, ten edges, and six faces; a pentagonal prism has ten, fifteen, and seven. See figure 7 on plate i.) I would mention also theantiprism,‡ whose two bases, though parallel, are not similarly situated, but each vertex of either corresponds to a side of the other, so that the lateral edges form a zigzag. (Thus a...

A chess-board and chess-men lend themselves to recreations, many of which are geometrical. The problems are, however, of a distinct type, and sufficiently numerous to deserve a chapter to themselves. A few problems which might be included in this chapter have been already considered in chapter iv.

The ordinary chess-board consists of 64 small squares, known as cells, arranged as shown below in 8 rows and 8 columns. Usually the cells are coloured alternately white and black, or white and red. The cells may be defined by the numbers 11, 12, etc., where the first digit denotes the number of...

AMagic Squareconsists of a number of integers arranged in the form of a square, so that the sum of the numbers in every row, in every column, and in each diagonal is the same. If the integers are the consecutive numbers from 1 ton², the square is said to be of thenth order, and it is easily seen that in this case the sum of the numbers in every row, column, and diagonal is equal to$\frac{1}{2}n({{n}^{2}}+1)$. Unless otherwise stated, I confine my account to such magic squares – that is, to squares formed with...

This chapter and the next are concerned with the branch of mathematics known as Topology or Analysis Situs, which differs from Geometry in having no connection with the idea of straightness, flatness, or measurement. Here every oval is equivalent to a circle, every spheroid to a sphere; in fact, no distinction is made between any two figures derivable from one another by the kind of transformations that are familiar as crumpling and stretching, without tearing or joining. (One’s thoughts turn naturally to indiarubber.) But topology does distinguish between a solid sphere and a hollow sphere, and between either of these...

I propose to consider in this chapter some problems which arise out of the theory of unicursal curves. I shall commence withEuler’s Problem and Theorems, and shall apply the results briefly to the theories ofMazesandGeometrical Trees. The reciprocal unicursal problem of theHamilton Gamewill be discussed in the latter half of the chapter.

Euler’s problem has its origin in a memoir* presented by him in 1736 to the St. Petersburg Academy, in which he solved a question then under discussion, as to whether it was possible from any point in the town of Königsberg to...

In this chapter we return to the kind of problems that arose at the end of chapter vi. In the nineteenth century such problems were only of recreational interest, but more recently they have proved useful to statisticians. One important type of combinatorial design became available in 1856, when von Staudt discovered the possibility of a geometry involving only finitely many points. His work was forgotten for forty years, but the idea has become immensely fruitful.

A projective plane. A lady wishes to invite her seven friends for a series of dinner parties. Her table provides room for exactly three...

I propose to discuss in this chapter the mathematical theory of a few common mathematical amusements and games. I might have dealt with them in the first four chapters, but, since most of them involve mixed geometry and algebra, it is rather more convenient to deal with them apart from the problems and puzzles which have been described already; the arrangement is, however, based on convenience rather than on any logical distinction.

The majority of the questions here enumerated have no connection one with another, and I jot them down almost at random.

I shall discuss in succession theFifteen,...

Among the more interesting geometrical problems of antiquity are three questions which attracted the special attention of the early Greek mathematicians. Our knowledge of geometry is derived from Greek sources, and thus these questions have attained a classical position in the history of the subject. The three questions to which I refer are: (i) the duplication of a cube – that is, the determination of the side of a cube whose volume is double that of a given cube; (ii) the trisection of an angle; and (iii) the squaring of a circle – that is, the determination of a square...

At rare intervals there have appeared lads who possess extraordinary powers of mental calculation.* In a few seconds they gave the answers to questions connected with the multiplication of numbers and the extraction of roots of numbers, which an expert mathematician could obtain only in a longer time and with the aid of pen and paper. Nor were their powers always limited to such simple problems. More difficult questions, dealing for instance with factors, compound interest, annuities, the civil and ecclesiastical calendars, and the solution of equations, were solved by some of them with facility as soon as the meaning...

The art of writing secret messages – intelligible to those who are in possession of the key and unintelligible to all others – has been studied for centuries. The usefulness of such messages, especially in time of war, is obvious; on the other hand, their solution may be a matter of great importance to those from whom the key is concealed. But the romance connected with the subject, the not uncommon desire to discover a secret, and the implied challenge to the ingenuity of all from whom it is hidden have attracted to the subject the attention of many to...