88 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
Remarks on this communication were made by Messrs. CHRISTIE 
and HI.t. 
Mr. G. K. Gitpert made a communication on 
THE PROBLEM OF THE KNIGHT'S TOUR. 
[ Abstract. ] 
The ordinary problem, requiring the knight to traverse the chess- 
board and return to his original position in sixty-four moves, is 
susceptible of very numerous solutions, and is not difficult. Its 
interest is increased by extending it so as to include fields of other 
form and size. 
It is readily shown that a perfect tour is impossible on any field 
containing an odd number of squares. 
A symmetric tour is one divisible into two or more similar parts. 
A tour has bilateral symmetry when one-half, being turned face 
downward upon the other, coincides with it. A tour has biradial 
symmetry when one-half, being rotated through 180° about the cen- 
ter of figure, coincides with the other half. A tour has quadri- 
radial symmetry when its fourth part, being rotated through 90° 
about the center of figure, coincides with the adjacent quarter. 
A tour having bilateral symmetry cannot be devised on a field 
containing a number of squares divisible by four. 
A tour having biradial symmetry cannot be devised on a field 
whose number of squares is divisible by two and not by four. 
A tour having quadriradial symmetry cannot be devised on a 
field whose number of squares is divisible by eight. 
It follows that on square fields the tour is impossible if the num- 
ber of spots on a side is odd; bilateral symmetry is never possible ; 
quadri-radial symmetry is possible only when the number of squares 
on a side is the double of an odd number. The only symmetry 
possible on a chess-board is biradial. 
The above conclusions are deductive. It is determined empiri- 
cally that the smallest square field on which the tour can be exe- 
cuted is that with 36 spots. Upon this field the number of possible 
tours with biradial symmetry is twenty-one, of which five have 
also quadriradial symmetry. 
Remarks on this communication were made by Megsrs. ELLIoTT 
and Hau, who called attention to previous work on the subject. 
