Ultramagic Squares of Order 7

Coordinates (x,y)
The 49 cells of an order-7 magic square are adressed by two coordinates x and y. Cell (x,y) or shortly xy is situated in row x and column y. The rows and columns were counted from 1 to 7. The upper lefthand cell has the adress (1,1). (Only for special purposes like transposition-formula we also use coordinates x, y from 0 to 6.) The square on the left shows the adress of each cell.
The blue centre-cell has the adress (4,4) or 44. I call the 24 yellow cells axis-cells and the 24 green cells filling-cells. It is of great importance to divide the square without centre-cell in these two segments.

Natural numbers n(x,y)
It is usual to fill the cells of a 7x7-squares with consecutive natural numbers from 1 to 49. I use this way for printing squares but not for calculating squares. The square on the left is self-complementary and pan-magic. n(x,y) is the number in cell (x,y). For example n(6,5) = 12 in this square. n(4,4) = 25 in each self-complementary 7x7-square. Two numbers located symmetrically about the centre-cell sum up to 50. This could be expressed by the equation n(x,y) + n(8-x,8-y) = 50.

Integers i(x,y)
The cells of the square on the left are filled with whole consecutive numbers from -24 to 24. This square is equivalent to the above square.
i(x,y) = n(x,y) - 25 is the integer in cell (x,y).
This numeration has many benefits for the calculation of odd-order magic squares.
i(4,4) = 0 in each self-complementary 7x7-square.
Integers with the same absolute value a(x,y) = |i(x,y)| are located symmetrically about the centre-cell:
i(x,y) = -i(8-x,8-y) or a(x,y) = a(8-x,8-y).
The integers in each column, row or diagonal add up to zero.
You get a homogeneous system of equations.
There are derived equations that contain only 6 variables. The sums of several 6-cell-triangles are zero in each self-complementary pan-magic order-7 square. See for instance the upper righthand triangle of the square on the left:
8 + 5 - 9 - 11 + 3 + 4 = 0

summary cells equations transpositions improvements
results programs files conclusions samples

Walter Trump, Nürnberg, Germany, (c) 2001-04-25 (last modified: 2004-06-27)