CH. IX] UNIOURSAL PROBLEMS 191 



Thus, suppose that we are told that we start from F and 

 then successively go to B, A, U, and T, and we want to find 

 a route from T through all the remaining towns which will 

 end at F. If we think of ourselves as coming into F from 

 0, the path FB would be indicated by I, but if we think of 

 ourselves as coming into F from E, the path FB would be 

 indicated by r. The path from B to A is indicated by I, 

 and so on. Hence our first paths are indicated either by III r 

 or by r 1 1 r. The latter operation does not occur either in (i) 

 or in (ii), and therefore does not fall within our solutions. The 

 former operation may be regarded either as the 1st, 2nd, 3rd, 

 and 4th steps of (ii), or as the 4th, 5th, 6th, and 7th steps 

 of (i). Each of these leads to a route which satisfies the 

 problem. These routes are 



FBA UTPONCDEJKLMQRSHGF, 



and FBAJJ TSRKLMQPONG DEJHGF. 



It is convenient to make a mark or to put down a counter 

 at each corner as soon as it is reached, and this will prevent 

 our passing through the same town twice. 



A similar game may be played with other solids provided 

 that at each angular point three and only three edges meet. 

 Of such solids a tetrahedron and a cube are the simplest 

 instances, but the reader can make for himself any number of 

 plane figures representing such solids similar to those drawn 

 on page 189. Some of these were indicated by Hamilton. 

 In all such cases we must obtain from the formulae analogous 

 to those given above cyclical relations like (i) or (ii) there 

 given. The solution will then follow the lines indicated above. 

 This method may be used to form a rule for describing any 

 maze in which no node is of an order higher than three. 



For solids having angular points where more than three 

 edges meet — such as the octahedron where at each angular 

 point four edges meet, or the icosahedron where at each 

 angular point five edges meet — we should at each point have 

 more than two routes open to us; hence (unless we suppress 

 some of the edges) the symbolical notation would have to be 



