48 



KANSAS ACADEMY OF SCIENCE. 



On arranging the numbers in a circle, the first thing most apparent is that 

 the circle may be divided into three equal parts by the equidistant numbers, 2, 

 5, and 8, the sum of which equals 15; also, that the sum of the three adjacent 

 numbers at the top of the circle equals 15. ( See fig. 5.) If we try the other 

 equidistant numbers we find they will not make 15. We learn then, that the 

 numbers are divided into three sets of three numbers' each, and that both sets 

 and numbers have an initial, a medial, and a final. 



Next, we find that each of the medial numbers may be connected with a pair 

 directly opposite — that is: with the initial of the preceding set, and the final of 

 the following set. (See fig. 6.) 



And, finally, we find that each medial number may be connected with the 

 final of the preceding set and the initial of the following set. (See fig. 7.) 



This makes eight possible sums of 15. The circle complete, which is shown 

 above as three separate circles in order to exhibit the successive steps in the 

 possible additions, is here shown in its entirety: 



Each of the forks (a, b, c, etc.) in the above circle connects three numbers in 

 the margin either by a straight or a curved line. The sum of these three num- 

 bers is in all cases 15. There are eight of these forks. It will readily be seen 

 that to connect the numbers by threes in other ways than as shown, and still 

 have the three numbers add 15, is impossible. 



It may be noticed as a peculiarity that the three medial numbers (4, 5, 6,) 

 form one of the diagonals in the square; and the three numbers of the medial set 

 (2, 5, 8,) form the other diagonal. The lines and columns have each an initial, 

 a medial, and a final. 



Again, in an apparently simpler method, it may be seen by simply arranging 

 the numbers in a square, in regular order, thus: 



