CH. IX] 



UNICURSAL PROBLEMS 



189 



Using these formulae we can find successively the values of 

 A 1 ,A i , .,., and £ X) .B,,,.... The values of A n when n = 2, 3,4, 

 5, 6, 7, are 2, 4, 9, 20, 48, 115; and of B n are 1, 2, 5, 12/33, 

 90 



I turn next to consider some problems where it is desired 

 to find a route which will pass once and only once through 

 each node of a given geometrical figure. This is the reciprocal 

 of the problem treated in the first part of this chapter, and is 

 a far more difficult question. I am not aware that the general 

 theory has been considered by mathematicians, though two 

 special cases — namely, the Hamiltonian (or Icosian) Game and 

 the Knight's Path on a Chess-Board — have been treated in 

 some detail. 



The Hamiltonian Game. The Hamiltonian Game consists 

 in the determination of a route along the edges of a regular 

 dodecahedron which will pass once and only once through 

 every angular point. Sir William Hamilton*, who invented 

 this game — if game is the right term for it — denoted the 

 twenty angular points on the solid by letters which stand for 

 various towns. The thirty edges constitute the only possible 

 paths. The inconvenience of using a solid is considerable, 

 and the dodecahedron may be represented conveniently in 



perspective by a flat board marked as shown in the first of 

 the annexed diagrams. The second and third diagrams will 

 answer our purpose equally well and are easier to draw. 



* See Quarterly Journal of Mathematics, London, 1862, vol. v, p. 305; or 

 Philosophical Magazine, January, 1884, series 5, vol. xvn, p. 42 j also Lncas, 

 vol. II, part vii. 



