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BULLETIN OF THE 



numbers at the angular points of the square as well as the middle 

 squares all count the same number 34. There are also in the fig- 

 ures many more parallelograms whose sum is equal to 84. 



In the second figure there are 12 ways in which the 5 numbers sura 

 up to 65, but beyond this fact there appears to be no special sym- 

 metrical arrangement of the numbers. 



The following figures appear to fulfil the conditions of symmetry. 



Fig. i. 

 Fig. 3. 



The special characteristic of these two figures is, that any two 

 figures whatever similarly placed with reference to the centre will 

 always amount to the same number, in the first to 17, and in the 

 second figure to 26 ; hence, in the first figure there will be 8 separate 

 lines which add up to 17 ; this of course will only be by consider- 

 ing 6 and 11 as a difierent line to 1 and 16, and the same for the 

 other diagonal. If now any one of these lines is considered as the 

 diagonal of a parallelogram, and any one of the remaining lines 



8x7 

 the other diagonal, we shall have — ^ — = 28 parallelograms simi- 

 larly situated with respect to the centre, the sum of the numbers at 

 the angular points being equal to 34 ; it will be observed that 

 these parallelograms include the limiting case when the diagonals 

 coincide with the two straight lines 1, 6, 11, 16, and 4, 7, 10, 13 ; 

 they also of course include the central square and the 4 angular 

 points of the whole square, besides these 28 there are the 4 squares 

 into which the squares may be divided also the 4 horizontal and 4 

 vertical lines and 8 more rectangles, viz: 1, 14, 7, 12 ; 15, 6, 4, 9, 

 and 6 others from the other sides, making altogether 48 straight 



