122 CHESS-BOARD RECREATIONS [CH. VI 



76, and 82. Analogous problems with other pieces will suggest 

 themselves. 



There are endless similar questions in which combinations 

 of pieces are involved. For instance, if queens are put on the 

 cells 35, 41, 76, and 82 they command or occupy all but two 

 cells, and these two cells may be commanded or occupied by a 

 queen, a king, a rook, a bishop, or a pawn. If queens are put 

 on the cells 22, 35, 43, and 54 they command or occupy all but 

 three cells, and two of these three cells may be commanded by 

 a knight which occupies the third of them. 



Re-Entrant Paths on a Chess-Board. Another problem 

 connected with the chess-board consists in moving a piece in 

 such a manner that it shall move successively on to every pos- 

 sible cell once and only once. 



Knight's Re-Entrant Path. I begin by discussing the 

 classical problem of a knight's tour. The literature* on this 

 subject is so extensive that I make no attempt to give a full 

 account of the various methods for solving the problem, and 

 I shall content myself by putting together a few notes on some 

 of the solutions I have come across, particularly on those due 

 to De Moivre, Euler, Vandermonde, Warnsdorff, and Roget. 



On a board containing an even number of cells the path 

 may or may not be re-entrant, but on a board containing an 

 odd number of cells it cannot be re-entrant. For, if a knight 

 begins on a white cell, its first move must take it to a black 

 cell, the next to a white cell, and so on. Hence, if its path 

 passes through all the cells, then on a board of an odd number 

 of cells the last move must leave it on a cell of the same colour 

 as that on which it started, and therefore these cells cannot be 

 connected by one move. 



* For a bibliography Bee A. van der Linda, Qeschichte und Literatwr det 

 SchachspieU, Berlin, 1874, vol. n, pp. 101 — 111. On the problem and its 

 history see a memoir by P. Volpicelli in Atti delta Reale Accademia dei Lincei, 

 Rome, 1872, vol. xxv, pp. 87 — 182 : also Applications de I'Analyse MatMmatique 

 au Jeu des E,checs, by C. F. de Jaenisch, 3 vols., Petrograd, 1862-3; and 

 General Parmentier, Association Francaise pour I'avancement des Sciences, 1891, 

 1892, 1894. 



