CH. IX] UNICUBSAL PROBLEMS 175 



This at once reduces to the second theorem. Let A and Z 

 be the two odd nodes. First, suppose that Z is not a free 

 end. We can, of course, take a route from A to Z; if we 

 imagine the branches in this route to be eliminated, it will 

 remove one hook from A and make it even, will remove two 

 hooks from every node intermediate between A and Z and 

 therefore leave each of them even, and will remove one hook 

 from Z and therefore will make it even. All the remaining 

 network is now even: hence, by Euler's second proposition, 

 it can be described unicursally, and, if the route begins at Z, 

 it will end at Z. Hence, if these two routes are taken in 

 succession, the whole figure will be described unicursally, be- 

 ginning at A and ending at Z. Second, if Z is a free end, 

 then we must travel from Z to some node, Y, at which more 

 than two branches meet. Then a route from A to Y which 

 covers the whole figure exclusive of the path from Y to Z can 

 be determined as before and must be finished by travelling 

 from Y to Z. 



Fourth. A figure having 2ra odd nodes, and no more, can 

 be described completely in n separate routes, n being a positive 

 number. 



If any route starts at an odd node, and if it is continued 

 until it reaches a node where no fresh path is open to it, this 

 latter node must be an odd one. For every time we enter an 

 even node there is necessarily a way out of it; and similarly 

 every time we go through an odd node we use up one hook in 

 entering and one hook in leaving, but whenever we reach it 

 as the end of our route we use only one hook. If this route 

 is suppressed there will remain a figure with 2n—2 odd nodes. 

 Hence n such routes will leave one or more networks with 

 only even nodes. But each of these must have some node 

 common to one of the routes already taken and therefore 

 can be described as a part of that route. Hence the com- 

 plete passage will require n and not more than n routes. It 

 follows, as stated by Euler, that, if there are more than two 

 odd nodes, the figure cannot be traversed completely in one 

 route. 



