62 



1)1si:()Vi;hy 



There is an excellent problem, neither a catch nor 

 a faJlacy, but really perfectly straightforward, which a 

 reader may successfully try on his friends — if they 

 have not heard of it previously. It illustrates the well- 

 known fact that an arithmetical problem is frequently 

 not just what one may hastily suppose. The problem 

 is this : 



A man is engaged at a salary which commences at 

 the rate of (say) £400 per annum, and is asked which 

 he will prefer, a yearly rise of £40 or a half-yearly rise 

 not of £20, but of £10. Which should he select ? 



Most people will say at once, the former ; but they 

 are wrong. For with the first choice the man would 

 receive £400 in the first year, and with the second 

 choice £200 + £210. He has thus £10 more at the 

 end of his first year. During the second year his first 

 choice would give him £440, his second choice £220 

 + £230. Again he has £10 more, and this continues 

 each year. 



Geometrical recreations consist chiefly of fallacies, 

 paradoxes, and games of position, and it will be 

 sufficient for me to illustrate them by taking a typical 

 example of the latter known as the Hamiltonian Game.* 



' W. W. Rouse Ball in Malhcmatical liecrealioiis, 1919, p. 189. 



This game was invented by Sir William Hamilton, 

 and consists in the determination of a route along the 

 edges of a regular dodecahedron which will pass once 

 and once only through every angular point. As a 

 dodecahedron is a solid figure, it is consequently 

 difficult to use for this game, and for the purposes of 

 solution the solid may be conveniently represented by 

 either of the diagrams of Fig. i. 



To make the problem really a game, Hamilton gave 

 each point lettered in the diagram the name of a town, 

 so that the game consists in starting at any town and 

 going " all round the world," visiting every town once 

 only, and ending up at the point of departure. The 

 game is varied by compelling the solver to visit certain 

 towns in a certain order first before going on to the 

 others. Not more than five towns should be included 

 in this restriction. 



The problem is attacked in the following way : At 

 each angular point there are three edges (three lines in 

 the diagram). As we approach a point there are only 

 two routes open to us after passing it — the one leading 

 to the right is denoted by r, that to the left by I. If 

 we go to the left; then, after passing a town, to the right; 

 then, after passing a second town, to the left, we denote 

 the operation bj' Irl ; twice to the left and once to the 

 right by l-r, and so on. Five times in succession to the 

 left, or to the right brings us back to where we started. 

 These facts may be represented by the equations 

 /5 _ J y5 _ J jt yvQi also be found from the figure 

 that the following pairs of journeys lead to the same 

 place : IrH and rlr ; rl^r and Irl ; IrH and r' ; rl^r and i'. 



Now, in our problem we have to move through 

 twenty points, ending up where we began. There are 

 thus twenty successive operations, the total effect of 

 which is equal to unity. We have, therefore, to find 

 an expression which has twenty terms in r and / which 

 will be equal to unity. This expression is obtained 

 thus : By making repeated use of the relation /* = rl^r 

 we find that 



I = /^ = IH^ = r?r /' = (rl^f = {r(W'r)/l» ' 

 = [r'^l^rl)- = {r-rPrlrl'r = {r' I' rlrl » 

 .-. {rU^rl)-\^= I. 

 Similarly {l^r^{lrY}-= 1. 



Each of these has twenty terms. 



Hence to go " all round the world " our directions 

 are ; 



r r r III rlrlrrrHlrlrl, 

 or 



lllrrrlrlrlllrrrlrlr. 



The first set of directions means " go to the right till 

 the next town is reached, then go again to the right 

 till the next town is reached, then again to the right, 

 then to the left, to the left again, etc.," and similarly 

 for the second. 



