68 



KANSAS ACADEMY OF SCIENCE. 



A. 



B. 



Etc., in manj-, many ways. 



Then next, according to one of the schemes in the Square of Three, add 

 to all the numbers in each of the sections the respective numbers 0, 4, 8, 12, 

 16, 20, 24, 28, and 32. 



Or, divide the thirty-six numbers into four sets of nine each and take one 

 from each set for each of the respective positions. This will be equivalent 

 to adding to all the numbers of each section the amounts 0, 1, 2, 3, 4, 5, 6, 7, 

 and 8, and to the several numbers in each section the amounts 0, 9, 18, and 

 27. Various other methods may be pursued. 



In the above squares, Nos. 1, 2, and 3, it will be seen that not only does 

 each line, column, and diagonal add 111, but that any four numbers taken 

 regularly and at equal distances from the center add 74, also that the trans- 

 verse semi-diagonals in opposite quarters add 111; also transverse tertio-diag- 

 onals add 111. 



These squares bear transposing after a manner similar to the transposi- 

 tions in the square of four. It is possible to transpose so as to bring any 

 required number into the upper left-hand corner, and still preserve the har- 

 mony of the square. 



A few examples are given of transpositions from square No. 3: 



No. 5. 



