CH. Vl] CHESS-BOARD RECREATIONS 125 



re-entrant. This is the case here if p = 52, q = 51. Thus the 

 cells 1, 2, 3, ..., 51; 60, 59, ..., 52 form a re-entrant route of 60 

 moves. Hence, if we replace the numbers 60, 59, . . . , 52 by 52, 

 53, ..., 60, the steps will be numbered consecutively. I recom- 

 mend the reader who wishes to follow the subsequent details 

 of Euler's argument to construct this square on a piece of paper 

 before proceeding further. 



Next, we proceed to add the cells a, b, d to this route. In 

 the new diagram of 60 cells formed as above the cell a commands 

 the cells there numbered 51, 53, 41, 25, 7, 5, and 3. It is in- 

 different which of these we select : suppose we take 51. Then 

 we must make 51 the last cell of the route of 60 cells, so that 

 we can continue with a, 6, d. Hence, if the reader will add 9 

 to every number on the diagram he has constructed, and then 

 replace 61, 62, ..., 69 by 1, 2, ..., 9, he will have a route which 

 starts from the cell occupied originally by 60, the 60th move is 

 on to the cell occupied originally by 51, and the 61st, 62nd, 

 63rd moves will be on the cells a, b, d respectively. 



It remains to introduce the cell c. Since c commands the 

 cell now numbered 25, and 63 commands the cell now numbered 

 24, this can be effected in the same way as the first route was 

 made re-entrant. In fact, the cells numbered 1, 2, . . . , 24 ; 63, 



62, ..., 25, c form a knight's path. Hence we must replace 



63, 62, ..., 25 by the numbers 25, 26, ..., 63, and then we can 

 fill up c with 64. We have now a route which covers the 

 whole board. 



Lastly, it remains to make this route re-entrant. First, we 

 must get the cells 1 and 64 near one another. This can be 

 effected thus. Take one of the cells commanded by 1, such as 

 28, then 28 commands 1 and 27. Hence the cells 64, 63, ..., 28; 

 1,2, . . . , 27 form a route ; and this will be represented in the 

 diagram if we replace the cells numbered 1, 2, ..., 27 by 27, 

 26 1. 



The cell now occupied by 1 commands the cells 26, 38, 54, 

 12, 2, 14, 16, 28; and the cell occupied by 64 commands the 

 cells 13, 43, 63, 55. The cells 13 and 14 are consecutive, and 

 therefore the cells 64, 63, ..., 14; 1, 2, .... 13 form a route. 



