250 Kansas Academy of Science. 



ODD Sl)MS. 



Perfect squares yielding odd sums cannot be constructed of 

 integral numbers. A half -unit must be added to at least half 

 of the numbers. This can be done in various ways, as (1) by 

 inserting a half-unit in the main gap; (2) by inserting a half- 

 unit in each of the side gaps and main gap; (3) by inserting 

 a half -unit in each of the four notches and main gap ; (4) by in- 

 serting a half-unit in each of the four notches, two side gaps 

 and main gap ; (5) by inserting a half-unit in each of the eight 

 differences and main gap; (6) bj^ inserting a half-unit in each 

 of the eight differences, two side gaps and main gap; (7) by 

 inserting a half-unit in each of the eight differences, four 

 notches and main gap; and (8) by inserting a half -unit in each 

 of the eight differences, four notches, two side gaps and main 

 gap. In either case half the terms will be integral and half of 

 them fractional. This principle is clearly seen in the three 

 following examples, where irregular series beginning with 7 

 are arranged to add 77 in every direction. 



(1) 7 8 10 11 14 15 17 18 20.5 21.5 23.5 24.5 27.5 28.5 30.5 31.5 



(2) 7 9 10 12 13.5 15.5 16.5 18.5 20 22 23 25 26.5 28.5 29.5 31.5 



(3) 7 8.5 10 11.5 14 15.5 17 18.5 20 21.5 23 24.5 27 28.5 30 31.5 



(4) 7 8.5 10 11.5 14 15 17 18.5 20 21.5 23 24.5 27 28.5 30 31.5 



In order to facilitate inspection of the series and the placing 

 of the several terms in a square the series should be arranged 

 each in four sets, thus : 



No. 4 is the same series as No. 3 but differently arranged. 



In the first series a = 7, d = l,n = 2, g = S, G = 2.5; which 

 makes 2a = 14, 8d = 8, 4n = S, 2g = 6, G = 2.5 ; total, 38.5, 

 which is one-half of 77. 



In the second series a = 7, d = 2, n = 1, g = 1.5, G = 1.5; 

 therefore 2a = 14, 8d = 16, in = 4,2g = S, G = 1.5 ; total, 38.5. 



In the third series a = 7, d = 1.5, n = 1.5, g = 2.5, G = 1.5; 

 from which 2a = 14, 8d = 12, 4n = Q, 2g = 5, G = 1.5 ; total, 

 38.5, as before. 



In another arrangement (4) of the third series a = 7, d = S, 

 n = 4, g = —8.5, G = 1.5; which gives 2a = 14, 8d = 24, 

 4n = 16, 2g = —17, G = 1.5. 



