TWENTY-SIXTU ANNUAL MEETING. 



47 



Square of Three. 

 The smallest square that can be constructed that will sum up equally in all 

 iti lines, columns, and diagonals, is one of nine places, thus: 



This, though apparently eight different squares, is really only one square pre- 

 sented in eight different aspects by transposition. 

 These schemes show the mode of arrangement : 



*^Cjf. f. 



..^^?- ^.. 



Jfi-^f 3-^ 



This scheme presupposes that the nine numbers are arranged in three sets; 

 and that the initial and final sets are arranged along the broken ari-ows, and the 

 medial set along the diagonal straight arrow. Figure 1, in the scheme above, 

 answers for squares 1 and 8; fig. 2 answers for squares 2 and 6; fig. 3 for squares 

 3 and 7; and fig. 4 for squares 4 and 5. It matters not on which side a beginning 

 be made, nor whether the three numbers of a set increase in the direction of an 

 arrow point or the reverse ( though in the scheme above, the numbers are sup- 

 posed to increase toward the arrow points); but whatever arrangement be made 

 for one set must be observed for all sets in the same square. 



That no more than eight sums of 15 from three numbers each can be ob- 

 tained from these nine numbers may be determined by arranging the numbers 

 in a circle and connecting by lines any three numbers that will add 15, thus: 



~^^3- ^^ 



