OH. IX] UNICURSAL PROBLEMS 177 



would recompense him for his charity. The scholar died and 

 was honourably buried, and the board was duly exposed. After 

 a considerable time had elapsed, a traveller one day riding by 

 saw the sacred symbol ; dismounting, he entered the inn, and 

 after hearing the story, handsomely remunerated the landlord. 

 Such is the anecdote, which if not true is at least well found. 



As another example of a unicursal diagram I may mention 

 the geometrical figure formed by taking a (2re + l)gon and 

 joining every angular point to every other angular point. The 

 edges of an octahedron also form a unicursal figure. On the 

 other hand a chess-board, divided as usual by straight lines 

 into 64 cells, has 28 odd nodes: hence it would require 

 14 separate pen-strokes to trace out all the boundaries 

 without going over any more than once. Again, the diagram 

 on page 117 has 20 odd nodes and therefore would require 

 10 separate pen-strokes to trace it out. 



It is well known that a curve which has as many nodes as 

 is consistent with its degree is unicursal. 



I turn next to discuss in how many ways we can describe a 

 unicursal figure, all of whose nodes are even*. 



Let us consider first how the problem is affected by a path 

 which starts from a node A of order 2ra and returns to it, 

 forming a closed loop L. If this loop were suppressed we 

 should have a figure with all its nodes even, the node A 

 being now of the order 2(»- 1). Suppose the original figure 

 can be described in N ways, and the reduced figure in N' ways. 

 Then each of these N' routes passes (n - 1) times through A, 

 and in any of these passages we could describe the loop L in 

 either sense as a part of the path. Hence N = 2 (n - 1) N'. 



Similarly if the node A on the original figure is of the order 

 2 (n + 1), and there are I independent closed loops which start 

 from and return to A, we shall have 



N=2 I n(n + l)(n + 2)...(n + l-l)N', 

 where N' is the number of routes by which the figure obtained 

 by suppressing these I loops can be described. 



* See Or. Tarry, Association Francaise pour VAvancement des Sciences, 1886, 

 pp. 49—53. 



B. B. 12 



