178 UNICURSAL PROBLEMS [CH. IX 



By the use of these results, we may reduce any unicursal 

 figure to one in which there are no closed loops of the kind 

 above described. Let us suppose that in this reduced figure 

 there are k nodes. We can suppress one of these nodes, say A, 

 provided we replace the figure by two or more separate figures 

 each of which has not more than k — 1 nodes. For suppose 

 that the node A is of the order 2«. Then the 2« paths which 

 meet at A may be coupled in n pairs in 1.3.5 ... (2ra— 1) 

 ways and each pair will constitute either a path through A, 

 or (in the special case where both members of the pair abut on 

 another node E) a loop from A. This path or loop will form 

 a portion of the route through A in which this pair of paths 

 are concerned. Hence the number of ways of describing the 

 original figure is equal to the sum of the number of ways of 

 describing 1.3.5... (2n — 1) separate simpler figures. 



It will be seen that the process consists in successively 

 suppressing node after node. Applying this process continually 

 we finally reduce the figure to a number of figures without 

 loops and in each of which there are only two nodes. If in one 

 of these figures these nodes are each of the order In it is easily 

 seen that it can be described in 2 x (2« — 1)! ways. 



We know that a figure with only two odd nodes, A and B, 

 is unicursal if we start at A (or B) and finish at B (or A). 

 Hence the number of ways in which it can be described uni- 

 cursally will be the same as the number required to describe 

 the figure obtained from it by joining A and B. For if we 

 start at A it is obvious that at the B end of each of the routes 

 which cover the figure we can proceed along BA to the node 

 A whence we started. 



This theory has been applied by Monsieur Tarry* to deter- 

 mine the number of ways in which a set of dominoes, running 

 up to even numbers, can be arranged. This example will serve 

 to illustrate the general method. 



A domino consists of a small rectangular slab, twice as 

 long as it is broad, whose face is divided into two squares, 



* See the second edition of the French Translation of this work, Paris, 1908, 

 vol. II, pp. 253 — 263 ; see also Lucas, vol. iv, pp. 145 — 150. 



