190 UNIOURSAL PROBLEMS [CH. IX 



The first problem is to go "all round the world," that is, 

 starting from any town, to go to every other town once and 

 only once and to return to the initial town; the order of the 

 n towns to be first visited being assigned, where n is not 

 greater than five. 



Hamilton's rule for effecting this was given at the meeting 

 in 1857 of the British Association at Dublin. At each 

 angular point there are three and only three edges. Hence, 

 if we approach a point by one edge, the only routes open to 

 us are one to the right, denoted by r, and one to the left, 

 denoted by I. It will be found that the operations indicated 

 on opposite sides of the following equalities are equivalent, 



lrH = rlr, rl"r=lrl, 11*1 = 1*, rl'r = l*. 



Also the operation ! 8 orr* brings us back to the initial point: 

 we may represent this by the equations 



Z« = l, r» = l. 



To solve the problem for a figure having twenty angular 

 points we must deduce a relation involving twenty successive 

 operations, the total effect of which is equal to unity. By 

 repeated use of the relation l 2 = rl s r we see that 



1 = J« = M» = (rl'r) I' - {rl*¥ = {r (rl'r) I)* 



= {i*l*rl}* = {r> (rl 3 r) IrVf = {t'I'tM}'. 



Therefore {rt 8 (W) 2 } 2 = 1 (i), 



and similarly {Z V (Zr) 2 } 2 = 1 (ii). 



Hence on a dodecahedron either of the operations 



rrrlllrlrlrrrlllrlrl ... (i), 



lllrrrlr Ir lllrrrlrlr... (ii), 



indicates a route which takes the traveller through every town. 

 The arrangement is cyclical, and the route can be commenced 

 at any point in the series of operations by transferring the 

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

 point at which we begin is determined by the order of certain 

 towns which is given initially. 



