Alphamagic Squares

        We've all heard enough of magic square, and the way it works - the sum of the numbers in every row, column and diagonal is the same. The oldest magic square dates back to 23^rd century B.C. when Emperor Yu discovered the pattern
 

on the shell of a sacred turtle in the Lo river. Since then, special categories next to magic squares such as magic triangles, rectangles, circles, stars, antimagic squares, prime-number squares, multiplicitively magic squares, magic cubes, N-dimensional arrays and so on are developed. In the 19^th century, Edouard Lucas devised the general formula for order 3 magic square .
 
  

        While previous problems have only been concerned with increasing the order of squares or creating other 'magic polygons', a new way of playing with this trick arose. In 1985, a book called The Origin of Tree Worship, which was previously misplaced around 1888, was rediscovered. In it was an account of a pilgrimage made by King Mi to a sacred grove in Valley of the Yews, where a magical formula was revealed to him, scored on the bole of the hallowed Li, eldest of yews.   On close examination by word games expert Lee Sallows, it was found out that the magical formula was more than a magic square with a sum of 45. Translated into English,  the number of modern English letters making up the three words used in every row, column, and diagonal is also identical - 21, (coincidentally, the same phenomenon was shown by the runic charm itself), thus creating the new kind of recreational mathematics - alphamagic squares.
 
  

     To create an alphamagic square, it is best to keep two things at hand: the general formula derived by Lucas, and the table of 'natural logs'. The word 'logorithm' was created by Sallows to describe the number of letters contained in the verbal equivalent of a number. When it's 'natural', the language English is used. To form an alphamagic square is then to find one in which both numbers and logorithms satisfy the Lucas formula.

        Here is the guide : Take a center number C, e.g. 15, consider in turn arithmetic triples formed by C and its equidistant neighbors C-1 and C+1, C-2 and C+2,..., C-N and C+N, and so on. Note down those cases in which log(C-N), logC (=7), and log(C+N) also form arithmetic triples. When N = C, we can go no further since C-N = 0. A list of pairs of associated arithmetic triples is then obtained, as in figure 5. Plug in four sets into the matrix, an alphamagic square will be obtained by trial and error. This can be tedious, thus a Pascal program designed by Sallows and implemented by Victor Eijkhout is developed for generating such alphamagic squares.

        This game can be endless. We can generate an infinity of alphamagic squares just by inserting the same prefix to each entry of the fundamental squares. We can also extend the square to higher orders of 4, 5, 6... , or to other languages like French, German, Welsh...etc. And besides, there are still unsolved problems!

1) Does there exist a 3x3x3 alphamagic cube?

2) The logorithmic derivative leads from one square array of numbers to another and so can be iterated to give second logorithmic derivatives, and so on. How far can this process continue with every square begin magic? For the German alphamagic square shown, the answer is forever, because every entry in the second harmonic is 14. But are there any examples of such "recursively magic" squares in which the logorithmic derivative does not have the same entries throughout?
 

        Try the game yourself. Here is a Pascal program for generating alphamagic squares (Turbo Pascal 3)!