Miscellaneous Papers. 



249 



Problem. — Write perfect squares of 4 from varying series, 

 beginning with 1, 2, 5, and 10, in which the differences shall 

 in no case exceed 5, and in which every row and quadrilateral 

 shall sum up 100. 



Solutions. — There are 895 series of integral numbers, any 

 one of which when placed in a perfect square will equal 100 in 

 all its parts. A very large percentage of these have differences 

 not exceeding 5. Here are four : 



(1) 1 6 10 12 15 17 21 24 26 29 33 35 38 40 44 49 



(2) 2 6 9 13 15 19 22 24 26 28 31 35 37 41 44 48 



(3) 5 10 14 15 19 20 21 24 26 29 30 31 35 36 40 45 



(4) 10 11 14 15 19 20 23 24 26 27 30 31 35 36 39 40 



In order to make it easier we arrange the series each into 

 four sets, thus: 



In the first selection a = 1, d = 5, n = 4, g — — 3, G = — 2 ; 

 whence 2a = 2,Sd = 40, 4n = 16, 2g = —6, G = — 2 ; total, 50. 



In the second selection a = 2, d = 4, n = S, g = 2, G = — 2 ; 

 whence 2a = 4, Sd = 32, 4n = 12,2g = 4, G = 2; total, 50. 



In the third selection a = 5, d = 5, % = 5, (/ = — 6, G = — 8 ; 

 whence 2a = 10, 8c^ = 40, 4n = 20, 2g = —12, G = — 8; 

 total, 50. 



In the fourth selection a = 10, d = 1, n = S, g = 4, G = 2; 

 wherefore 2a = 20, Sd = 8, 4w = 12, 2^ = 8, G = 2 ; total, 50. 



Arranging these series in squares according to one or an- 

 other of the twenty-four model squares already given, we have 

 the following: 



(1) 



(2) 



(3) 



(4) 



