Magic Square

Extensions and Variations.

Though strictly speaking the magic square should be composed of a natural sequence of num bers from I to n' this condition is sometimes dispensed with. Any arithmetical progression will serve, as in fig. 19, where the numbers in the series used have a common difference of 2. But squares of the 3rd order may be formed with any nine numbers if we can write them thus: I 2 3 7 8 9 13 14 15 where the horizontal differences are all alike and the vertical dif ferences also all alike but not necessarily the same as the hori zontal. The arrangement is shown in fig. 20. This applies equally to the higher orders. Thus fig. 21 is constructed from the numbers I 2 3 4 6 7 8 9 II 12 13 14 16 17 18 19 and is a Nasik square.

A general solution for squares of the 4th order in which any numbers (all different but not necessarily consecutive) are used the product of these four numbers, 210, to each of the nine num bers. The result is the nine consecutive composite numbers 212 to 220 inclusive with which we can form a magic square. Yet a square with a smaller constant may be formed, as in fig. 22, with the numbers 114 to 122 inclusive. But it would be difficult to find empirically (if possible) a lower solution for the 4th order than we get by the rule, using the numbers 510512 to 510527 inclusive.

Subtracting, multiplying and dividing magic squares have also of late been investigated, and examples of the 3rd order are given in figs. 23, 24 and 25. In fig. 23, a subtracting square, you get a constant 5 by subtracting the first figure in a line from the second and the result from the third. The operation may, of course, be performed in either direction hut in order to avoid negative num bers it is more convenient simply to deduct the middle number from the sum of the two extreme numbers. In fig. 24, which is a multiplying square, the constant 216 is obtained by multiplying together the three numbers in any line. In the dividing square

(fig. 25) the constant 6 is obtained by dividing the second num ber in a line by the first in either direction and the third number by the quotient. But again the process is simplified by dividing the product of the two extreme numbers by the middle number. This question has been dealt with in Amusements in Mathematics.

Other extensions of the problem such as magic polygons, magic cubes, magic circles and magic stars have also at tracted the attention of mathemati cians. Attempts have also been made to construct a magic knight's tour. The knight has to be played once to every square of the chessboard in a complete tour, or non-re-entrant path, numbering the squares visited in order, so that when completed the square shall be magic with the constant 26o. The nearest approach to a solution is shown in fig. 26 given by C. F. Jaenisch in Applications de l'Analyse Mathematique au feu des Echecs (St. Petersburg, 1862). Unfortunately it is not a perfect magic square because the diagonals are incorrect, one adding and the other 256—requiring only the transfer of 4 from one diagonal to the other. Probably a perfect solution is impossible but no rigid proof of this has been obtained. Magic squares have been devised for construction with dominoes and playing cards; also with coins and postage stamps, where one is restricted, of course, by the limitations of issue, coins and stamps of certain con venient values not being in existence. These interesting questions are of a puzzle character and often call for considerable mathe matical analysis, though, as a man said of Paradise Lost, they "prove nothing" and lead nowhere. They are therefore strictly speaking of no direct mathematical importance.