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Kansas Academy of Science. 



any line, column, or diagonal, whether whole or broken. And 

 besides, an arrow cannot begin and end in the same line, either 

 vertically, horizontally, or diagonally. Only four schemes for 

 perfect squares have so far been constructed. 



Scheme II (fig. 18). — Coupling -arroivs joining alteimate 

 columns and adjacent lines. This scheme is readily trans- 

 formable from scheme I. It is only necessary to transpose the 

 second and fourth lines. The numbers of the coupling-arrows 

 in the trial arrangements are placed as before ; but, instead of 

 governing adjacent lines as before, they are to be applied to the 

 top and bottom lines. This puts 15 in the lower left-hand cor- 

 ner. The model squares resulting from the six arrangements 

 are as follow: 



FIRST ARRANGEMENT. 



SECOND ARRANGEMENT. 



17 4 6 



8 2 5 3 



No. 8. 

 FIFTH ARRANGEMENT. 



14 7 6 

 8 5 2 3 



THIRD ARRANGEMENT. 



7 4 

 2 5 



No. 9. 



SIXTH ARRANGEMENT. 



14 6 7 



8 5 3 2 



No. 10. 



No. 11. 



No. 12. 



Ninety-six additional perfect squares, in every respect equal 

 with the original ninety-six, can be made from these six ar- 

 rangements. 



Scheme III (fig. 19). — Coupling-arrows joining alternate 



