40 Prof. Tait on Listing's Topologie. 



If we suppose the pieces to be originally arranged in cir- 

 cular order, with two contiguous blank spaces, the law of 

 this process is obvious. Operate always with the penultimate 

 and antepenultimate, the gap being looked on as the end for 

 the time being. With this hint it is easy to generalize, so 

 as to get the nature of the solution of the corresponding 

 problem in any particular case, whatever be the number of 

 coins. It is also interesting to vary the problem by making it 

 a condition that the two coins to be moved at any instant shall 

 first be made to change places. 



(13) Another illustration, commented on by Listing, but 

 since developed from a different point of view in a quite 

 unexpected direction, was originated by a very simple ques°< 

 tion propounded by Clausen in the Astronomische Nachrichten 

 (No. 494). In its general form it is merely the question, 

 " What is the smallest number of penstrokes with which a 

 given figure, consisting of lines only, can be traced ? " No 

 line is to be gone over twice, and every time the pen has to be 

 lifted counts one. 



The obvious solution is: — Count the number of points in 

 the figure at each of which an odd number of lines meet. 

 There must always be an even number of such (zero included). 

 Half of this number is the number of necessary separate 

 strokes (except in the zero case, when the number of course 

 must be unity). Thus the boundaries of the squares of a chess- 

 board can be traced at 14 separate pen-strokes; the usual 

 figure for Euclid I. 47 at 4 pen-strokes; and fig. 5 at one. 



(14) But, if 2n points in a plane be joined by 3n lines, no 

 two of which intersect; so that every point is a terminal of 

 3 different lines, the figure requires n separate pen-strokes. It 

 has been shown that in this case (unless the points be divided 

 into two groups, between which there is but one connecting 

 line, fig. 7) the 3n lines may be divided into 3 groups of n 

 each, such that one of each group ends at each of the 2n points. 

 See fig. 6, in which the lines are distinguished as a, /3, 01*7. 

 Also note that a /3 a (3 &c, and ay ay &c. form entire cycles 

 passing through all the trivia, while fly (3y &c. breaks up into 

 detached subcycles. 



Thus, if a Labyrinth or Maze be made, such that every inter- 

 section of roads is a Trivium, it may always be arranged so 

 that the several roads meeting at each intersection may be 

 one a grass-path, one gravel, and the other pavement. To 

 make sure of getting out of such a Labyrinth (if it be pos- 

 sible) , we must select two kinds of road to be taken alter- 

 nately at each successive trivium. Thus we may elect to take 

 grass, gravel, grass, gravel, &c, in which case we must either 



