Download Magic Squares in the Tenth Century: Two Arabic Treatises by by Jacques Sesiano PDF
By Jacques Sesiano
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Extra resources for Magic Squares in the Tenth Century: Two Arabic Treatises by Anṭākī and Būzjānī
Sample text
39, as also the magic property of the broken diagonals (M stands for M4 ). 30 Construction of even-order squares 4 M 2 −3 7 M 2 M 2 −8 −2 M 2 1 M 2 −5 6 −7 8 M 2 M 2 −4 3 1 5 M 2 −6 M 2 −2 4 2 M 2 −1 Fig. 41 M 2 −3 −5 M 2 6 M 2 −8 7 −4 3 M 2 8 M 2 −1 2 −7 5 M 2 −6 Fig. 42 Remarks. (1) Fig. 33 is the one we shall adopt for our commentary, being easier to memorize: starting from a corner with 1, the moves successively used are the knight’s, the queen’s, the knight’s (and the same in reverse for the sequence 5, .
The border of Fig. 92 is his result (apart, perhaps, from arrangement within the rows). 55 56 9 56 6 59 55 10 60 5 54 11 4 61 12 53 62 3 13 52 63 2 51 14 1 64 57 57 58 64 2 63 59 6 60 5 Fig. 92 61 4 62 3 51 13 54 7 1 14 53 7 8 52 12 11 10 9 8 58 Fig. 93 In A (Fig. 93), the arrangement is essentially the same, with the corners occupied by the order number and its predecessor; but with the square being filled from within, the ascending sequence found in B be- 53 Bordered magic squares comes a descending one, from 14 on.
Construction of bordered squares of evenly odd orders 1. Particular case of order 6 The general method to be explained here may also be applied to order 6. Put 1 on the bottom, (say) next to the left-hand corner, 2 at the top, 3 on the bottom, 4 on the right, the subsequent two in the upper corners, first left. Putting the next two numbers, 7 and 8, on the left, 9 below and 10 on the right, we shall have filled three cells in each row and the complements will fill the whole border (Fig. 100). 5 36 2 34 28 6 7 30 8 29 27 10 33 4 31 1 35 3 9 32 Fig.