Y

8	37	78	29	70	21	62	13	54	5
7	6	38	79	30	71	22	63	14	46
6	47	7	39	80	31	72	23	55	15
5	16	48	8	40	81	32	64	24	56
4	57	17	49	9	41	73	33	65	25
3	26	58	18	50	1	42	74	34	66
2	67	27	59	10	51	2	43	75	35
1	36	68	19	60	11	52	3	44	76
0	77	28	69	20	61	12	53	4	45
	0	1	2	3	4	5	6	7	8

       Magic Squares and Labyrinth seed Forms                                                                                                                                                                                                                                  

In the Bridges 2025 workshop, we were given a method of     constructing an odd order magic square and shown its relationship to a labyrinth seed form.  We used the example of a 9 x 9 square.  We were left with the question of why the magic square and labyrinth were so related.  I will not address that question here.                                  

 What I will do is give a formula for the entries in the magic square. We number the rows from bottom to top, 0 to 8, and the columns, left to right, 0 to 8, and regard the rows and columns and diagonals as wrapping around (so as to make a torus). One successively enters into the cells all the numbers from I to 81, beginning at the cell directly below the centre cell and proceeding along the diagonal that                                                                                                             X

slopes down to the right, until it fills in all 9 cells of the diagonal,  and then one moves down two cells, and continues along that diagonal, and so on, yielding the magic square at the right.                                      

Let f (x , y) be the entry in the cell, (x, y).

Let MOD (A, B) be the remainder when A is divided by B.

f (x. x + 1) = MOD (36 – 9 x, 81)          (with some abuse of notation, cell (4, 5) is taken to be 81 instead of 0)

Beginning at cell (x, x + 1), and moving up column x, two cells at a time reduces the entry each time by 10 Mod 81. Thus, 

f (x, x + 1 + 2k) = MOD (36 – 9x – 10 k, 81)

Setting y = MOD (x + 1 + 2k, 9) and solving for k, yields:

 f (x, y) = MOD (36 - 9x – 10 * MOD (5(y – x – 1), 9), 81).

One can easily generalize this magic square construction and this formula to any odd order square.  There are many interesting patterns in this array of numbers.

Martin Levin

mdlevin_public@msn.com