CH. IX] UNICURSAL PROBLEMS 181 



We proceed to consider each of the reduced figures E and 

 F. First take E, and in it let us suppress the node 4. For 

 simplicity of description, denote the two paths 2 by f3 and 

 j8', and the two paths 4 3 by 7 and 7'. Then we can couple /3 

 and 7, as also /3' and 7', or we can couple /3 and 7', as also y8' 

 and 7 : each of these couplings gives the figure 0. Or we can 

 couple /S and $', as also 7 and y': this gives the figure H. Thus 

 e = 2g + h. Each of the figures Q and H has only two nodes. 

 Hence by the formulae given above, we have # = 2 . 3 . 2 = 12, 

 and A = 2. 2. 2 = 8. Therefore e = 2# + fc = 32. Next take 

 the figure F. This has a loop at 4. If we suppress this 



( U -4 ) 



Figure Q. Figure H. Figure J. 



loop we get the figure J, and /= 2j. But the figure J, if 

 we couple the two lines which meet at 4, is equivalent 

 to the figure G. Thus /=2j = 2# = 24. Introducing these 

 results we have a = 6e + 3/= 192 + 72 = 264. And therefore 

 JV=15 . 2 B . a = 126720. This gives the number of possible 

 arrangements in line of a set of 15 dominoes. In this solution 

 we have treated an arrangement from right to left as distinct 

 from one which goes from left to right : if these are treated 

 as identical we must divide the result by 2. The number of 

 arrangements in a closed ring is 2 6 <z, that is 8448. 



We have seen that this number of unicursal routes for a 

 pentagon and its diagonals is 264. Similarly the number for 

 a heptagon is h = 129976320. Hence the number of possible 

 arrangements in line of the usual set of 28 dominoes, marked 

 up to double-six, is 28 . 3' . h, which is equal to 7959229931520. 

 The number of unicursal routes covering a polygon of nine 

 sides is n = 2 17 . 3" . 5 a - 711 . 40787. Hence the number of 

 possible arrangements in line of a set of 45 dominoes marked 

 up to double-eight is 48 . 4" . n*. 



* These numerical conclusions have also been obtained by algebraical 

 analysis : see M. Reiss, Annali di Matematica, Milan, 1871, vol. v, pp. 63—120. 



