66 



KANSAS ACADEMY OF SCIENCE. 



Sums Less than 34 and Negative Sums. 



In forming these sums, the rule is just the same as for forming special sums, 

 namely, to take for the extremes of the series two numbers whose sum equals 

 half of the number desired; but in this case it presupposes the necessity of ta- 

 king for the lower numbers of the series minus or negative numbers. A few 

 squares, to show the manner of their formation, will be sufficient to give a good 

 insight into this : 



No. 69. No. 70. No. 71. No. 72. 



Sums = 30. Sums = 17. 



These squares are perfectly harmonic. 



Sums = 16. 



Sums = — 2. 



Numbers of Series. 



The simple arithmetical series 1 to 16, adding in the squares 34, is only one. 

 That is to say : there is only one series of positive integers that can be found, 

 without repetition, that will, when the numbers are arranged in a square, add 

 34 in each line, column, diagonal, etc. The same is true of the series adding 36 ; 

 there is only one series. There are three series adding 38, and three adding 40. 

 Thus, the series adding 38 are: 



The number of series adding in square 42 and 44 is 7 of each ; the number of 

 series adding 46 and 48 is 13 of each ; the number adding 50 and 52 is 23 for each ; 

 adding 54 and 56 is 37 ; adding 58 and 60 is 57 ; the number of series that will 

 each separately add in square 62 or 64 is 83. Thus the number of series that will 

 produce a given number constantly rises in an ever-increasing ratio ; and when 

 we come to series adding in square 100, we find there are no less than 895 series, 

 and of series adding 102, there are 1,085. 



The ratio of increase is not irregular, but is peculiar. The mode of forming 

 the series is similar to that of forming any figurate series, except that each suc- 

 cessive sum, until the series of increase is reached, is duplicated or written twice. 

 This is shown by braces, thus: 



111 1 



Increase 

 Series complete 



2 



112 2 4 4 6 6 9 9 12 12 16 16 20 20 



1 2 4 6 10 14 20 26 35 44 56 68 84 100 120 etc. 



1 3 7 13 23 37 57 83 118 162 218 286 370 470 590 etc. 



Sums 34 38 42 46 50 54 58 62 66 70 74 78 82 86 90 

 Possibilities. 



It has been shown that by the different arrangements and transpositions 

 more than 6,000 squares (6,144) may be constructed from a single series (1 to 

 16) according to the first four schemes; and it is presumed that at a low esti- 

 mate at least four times as many more may be made from the remaining 48 

 schemes that are possible as shown in the schedule, thus making upward 



