Mathematical recreations and essays . methodshe enumerated. Taking account of other methods, it wouldseem that the total number of magic squares of the fifth orderis very large, probably considerably exceeding half a million. Product Squares. Before leaving this part of the subject,I may mention that Montucla suggested the construction ofsquares whose cells are occupied by numbers such that theproduct of the numbers in each row, column, and diagonal isconstant. The formation of such figures is immediately de-ducible fi:om that of magic squares, for if the consecutivenumbers, namely 1, 2, 3, &c
Mathematical recreations and essays . methodshe enumerated. Taking account of other methods, it wouldseem that the total number of magic squares of the fifth orderis very large, probably considerably exceeding half a million. Product Squares. Before leaving this part of the subject,I may mention that Montucla suggested the construction ofsquares whose cells are occupied by numbers such that theproduct of the numbers in each row, column, and diagonal isconstant. The formation of such figures is immediately de-ducible fi:om that of magic squares, for if the consecutivenumbers, namely 1, 2, 3, &c., in a magic square are replacedby consecutive powers of any number m, namely m, m^ m^ &c.,the products of the numbers in every line will be magic. Thisis obvious, for if the numbers in any line of a square are a, a\a\ &c., such that Sa is constant for every line in the square,then Ilm* is also constant. Magic Stars. Some elegant magic constructions on star-shaped figures (pentagons, hexagons, &c.) may be noticed in. Figure xiv. A Magic Star. passing, though I will not go into details. One instance willsuffice. Suppose a re-entrant octagon is constructed by the 10—2 148 MAGIC SQUARES [CH. VII intersecting sides of two equal concentric squares. It is requiredto place the first 16 natural numbers on the corners and pointsof intersection of the sides so that the sum of the numbers onthe corner of each square and the sum of the numbers on everyside of each square is equal to 34. Eighteen fundamental solu-tions exist. One of these is given above*. There are magic circles, rectangles, crosses, diamonds, andother figures: also magic cubes, cylinders, and spheres. Thetheory of the construction of such figures is of no value, andI cannot spare the space to describe rules for forming them. Magic Pencils. Hitherto I have concerned myself withnumbers arranged in lines. By reciprocating the figures com-posed of the points on which the numbers are placed we obtaina collection of lines for
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