CH. VI] 



CHESS-BOARD RECREATIONS 



123 



The earliest solutions of which I have any knowledge are 

 those given at the beginning of the eighteenth century by De 

 Montmort and De Moivre*. They apply to the ordinary chess- 

 board of 64 cells, and depend on dividing (mentally) the board 

 into an inner square containing sixteen cells surrounded by an 

 outer ring of cells two deep. If initially the knight is placed 

 on a cell in the outer ring, it moves round that ring always in 

 the same direction so as to fill it up completely — only going 

 into the inner square when absolutely necessary. When the 

 outer ring is filled up the order of the moves required for 

 filling the remaining cells presents but little difficulty. If 

 initially the knight is placed on the inner square the process 

 must be reversed. The method can be applied to square and 

 rectangular boards of all sizes. It is illustrated sufficiently by 

 De Moivre's solution which is given below, where the numbers 



De Moivre'$ Solution. 



Euler's Thirty-six Cell Solution. 



indicate the order in which the cells are occupied successively. 

 I place by its side a somewhat similar re-entrant solution, due 

 to Euler, for a board of 36 cells. If a chess-board is used it 

 is convenient to place a counter on each cell as the knight 

 leaves it. 



The earliest serious attempt to deal with the subject by 



* They were sent by their authors to Brook Taylor who seems to have 

 previously suggested the problem. I do not know where they were first pub- 

 lished ; they were quoted by Ozanam and Moutuola, see Ozanam, 1803 edition, 

 vol. I, p. 178 ; 1840 edition, p. 80. 



