170 



CHAPTER IX. 



TJNICUR8AL PROBLEMS. 



I propose to consider in this chapter some problems which 

 arise out of the theory of unicursal curves. I shall commence 

 with Euler's Problem and Theorems, and shall apply (the results 

 briefly to the theories of Mazes and Geometrical Trees. The 

 reciprocal unicursal problem of the Hamilton Qame will be 

 discussed in the latter half of the chapter. 



Euler's Problem. Euler's problem has its origin in a 

 memoir* presented by him in 1736 to the St Petersburg 

 Academy, in which he solved a question then under discussion 

 as to whether it was possible from any point in the town of 

 Kb'nigsberg to take a walk in such a way as to cross every 

 bridge in it once and only once and return to the starting point. 

 The town is built near the mouth of the river Pregel, 

 which there takes the form indicated below and includes the 

 island of Kneiphof. In the eighteenth century there were 

 (and according to Baedeker there are still) seven bridges in 

 the positions shown in the diagram, and it is easily seen that 

 with such an arrangement the problem is insoluble. Euler 

 however did not confine himself to the case of Kbnigsberg, but 

 discussed the general problem of any number of islands con- 

 nected in any way by bridges. It is evident that the question 



* ' Solutio problematis ad Geometriam situs pertinentis,' Commentarii 

 Academiae Scientiarum Petropolitanae for 1736, Petrograd, 1741, vol. viii, 

 pp. 128 — 140. This has been translated into .French by M. Ch. Henry; see 

 Lucas, vol. I, part 2, pp. 21 — 33. 



