discovi:ry 



63 



It should be pointed out that one may commence 

 at any point in the series of operations by transferring 

 the proper number of letters from one end to the other. 

 Thus, suppose we choose to commence with the ninth 

 direction of the first series, the new series of directions 

 would be : 



rlrrrlllrlrlrr rlllr I, 



or if we commenced with (say) the eighth of the second 

 series, the new series would run : 



rlrlllrrrlrlrlllrrrl. 



The particular direction we commence off with is 

 fixed by the order of certain towns which we must 

 visit initially. 



Suppose, for example, we are told to start at S (see 

 the figure), proceed to H, then to J, then to E, then 

 to D, and then to go once through all the other towns, 

 and return to S. From S to H is either I or ;- depending 

 upon whether we got to S from T or R. From H to J 

 is r, J to E is i, E to D is r. Our directions initially 

 are, therefore, either Mr or rrlr. The first of these 

 combinations occurs in the First Series and the second 

 in the Second. The solutions are, therefore, 



I rl rr rlllr Irlrr rlllr 

 and 



rrlrlrlllrrrlrlrlllr, 



and the orders of the towns visited are respectively 

 S, H, J, E, D, L, K, R, Q, M. N, C, B, F, G, U, 

 .\,0, P,T,S 

 and 



S, H, J, E, D, C, N, O, A, B, F, G, U, T, P, 0, 

 M, L, K, R, S. 



Mr. Rouse Ball, who describes this game in detail, 

 recommends the solver, for convenience and to prevent 

 error, to make a mark at, or put down a counter on, 

 each town as it is reached. 



A geometrical problem of a simpler and better-known 

 kind is the " proof " by dissection that 65 = 64. If 

 the square of side 8 in Fig. 2 be cut along the lines 

 indicated, it may be fitted into a parallelogram whose 

 sides are 13 and 5. 



The paradox depends upon the relation 

 5 X 13 - 8= = I 

 and the fallacy lies in the fact that the four pieces of 

 the cut-up square, when fitted together, do not really 

 make a parallelogram. There is a small diamond- 

 shaped space, the area of \yhich is i, which is apt to 

 be overlooked unless the cutting-up and the fitting 

 together be very accurately carried out. 



A square of side 21 may be similarly cut up and 

 fitted together to form a parallelogram of sides 13 and 

 34 ; and one of side 55 to form a parallelogram of 

 sides 34 and 89. 



The subject of Arithmetical recreations forms part 



of a section of mathematical work which is sometimes 

 regarded as special in character, but whose results 

 permeate the whole of Mathematical study. The 

 Theory of Number is full of paradox and mystery, 

 treating of entities apparently simple in character, but 

 really so complex as to provoke everlasting discussion. 

 Number is the root of all mathematical evil ; it is the 

 source of much mathematical pleasure. The study of 

 its true significance has traditionally belonged to 

 abstract philosophy, but now the mathematician can 

 give the more adequate interpretation. The processes 

 of counting and tallying are amongst the first activities 

 of human intelligence ; and the problems that arise in 

 connection therewith form the subject-matter of the 

 most recent mathematical research. Its domain is 

 vast. A true conception of its nature is fundamental 

 to analvsis. and the modern theory corresponds in a 



remarkable way to the traditional and empirical notions 

 of spatial magnitude. 



Positive numbers present themselves in a dense array 

 stretching from the infinite to the infinitesimal. Imagine 

 them arraj'ed in order of magnitude as we do when we 

 represent them by points on a line. They are so dense 

 that an infinity of them exists between any two of them, 

 however near. But of this vast aggregate certain 

 numbers stand out with great prominence, possessing 

 properties of a special and often mysterious nature. 

 They are human in some of their attributes, for some 

 are perfect and some are amicable ; some lucky and 

 some unlucky. One investigator who spent the greater 

 part of his life in studying them became so familiar 

 with them that he was said to regard each positive 

 integer as a personal friend.' Although this class of 

 positive integers forms the subject-matter of a theory 

 by itself, the interesting fact in connection with them 



' Prof. G. H. Hardy, Some Famous Problems of llie Theory 

 of Xuinbers, 1920. 



