240 



Kansas Academy of Science. 



Equal sums are obtained from this square by adding- to- 

 gether four numbers in each of the following regular ways : 

 1. — By lines — ways. 



Each of the columns (fig. 1) 4 



Each of the lines (fig. 2) 4 



Every diagonal line, entire or broken (figs. 3, 4) 8 



2. — By squares — 



The four corners and the four centrals (fig. 5) 2 



Any four contiguous numbers in a square (figs. 5, 6, 7) 8 



Corresponding corners of quarters (fig. 8) 4 



3. — By rectangles — 



Linear pairs opposite lines (figs. 9, 10, 12) 3 



Columnar pairs in opposite columns (figs. 9, 11) 3 



4. — By rhomboids and trapezoids — 



Columnar pairs at opposite ends of alternate columns (fig. 13), 4 



Linear pairs at opposite ends of alternate lines (fig. 14) 4 



Opposite numbers in alternate lines (fig. 15) 4 



Opposite numbers in alternate columns (fig. 16) 4 



Total regular ways of adding 34 52 



Each of these squares yields the equal sum of 34 in 52 regu- 

 lar ways, as here shown, taking the first one as an example : 



EIGHT LINES. RECTANGLES. RHOMBOIDS. 



1 + 14 + 11+5 = 34 1 + 14 + 15+ 4= 4 1 + 15 + 16+ 2 = 34 



15 + 4 + 5 + 10 = 34 14 + 11 + 4 + 5 = 34 14 + 4 + 3 + 13 = 34 



6+ 9 + 16+ 3 = 34 11+ 8+ 5 + 10 = 34 11+ 5+ 6 + 12 = 34 



12+ 7+ 2 + 13 = 34 1 + 15+ 8 + 10 = 34 8 + 10+ 9+ 7= 34 



1 + 15+ 6 + 12 = 34 15+ 4+ 6+ 9 = 34 1 + 14 + 16+ 3 = 34 



14+ 4+ 9+ 7= 34 4+ 5+ 9 + 16 = 34 15+ 4+ 2 + 13 = 34 



11+ 5 + 16+ 2= 34 5 + 10 + 16+ 3 = 34 6+ 9 + 11+ 8= 34 



8 + 10 + 3 + 13 = 34 15 + 6 + 10 + 3 = 34 12 + 7 + 5 + 10 = 34 



DIAGONAL LINES. qtifil'^Yi 9 = 34 TRAPEZOIDS. 



1+4 + 16 + 13=34 16+3+2 + 13=34 1+8+9 + 16 = 34 



14+5+3 + 12 = 34 filiol s + 13=S4 15 + 10+7+2 = 34 



11 + 10+6+7 = 34 1 + 14 + 12+7 = 34 6+3 + 14 + 11=34 



8 +.15+ 9+ 2=34 14 + 11+7+2=34 12 + 13+4+5=34 



1 + 10 + 16+7 = 34 11+8+2 + 13=34 1 + 12+5 + 16 = 34 



14 + 15+3+2=34 1 + 11+6 + 16 = 34 14+7 + 10+3 = 34 



11+4+6 + 13=34 14+8+9+3 = 34 11+2 + 15+6 = 34 



8+ 5+ 9 + 12 =34 15+ 5 + 12+ 2 = 34 8 + 13+ 4+ 9 = 34 



4 + 10 + 7 + 13 = 34 _, ^ , , „, ,- 



]^ _j_ 8 + 12 + 13 = 34 Total sums of 34 = 52 



Rule. — To fill a square according to this scheme consider the 

 eight numbers in one of the trial arrangements to represent 

 the eight couplets of the series; apply the numbers in con- 

 secutive order to the upper ends of the coupling-arrows in the 

 scheme; double each number in turn and place the product in 

 the cell corresponding with the point of the arrow ; subtract 1 

 from that product and place in the cell corresponding to the 

 tail of the arrow (represented by a circle) . The result will be 

 a perfect square. 



From each of these six squares fifteen other squares ap- 



