Christian Boyer wrote:
Here are some other nice formulas by R. Holmes, The Magic Magic Square, The Mathematical Gazette, December 1970, p.376:
6 1 8 7 5 3 2 9 4
618^n + 753^n + 294^n = 816^n + 357^n + 492^n (rows) 672^n + 159^n + 834^n = 276^n + 951^n + 438^n (columns) 654^n + 132^n + 879^n = 456^n + 231^n + 978^n (diagonals \) 852^n + 174^n + 639^n = 258^n + 471^n + 936^n (diagonals /)
True for n = 1 and 2. True again if you delete the first digit in each number. True again if you delete the second digit in each number. True again if you delete the last digit in each number.
I don't have the reference by Holmes, but this is also true for n = 1 and 2 if you just take either only the first, second or third digit in each number. In fact, there are then only 2 nontrivial (i.e. different numbers on both sides) equations: {4,9,2}={8,1,6} and {4,3,8}={2,7,6}, which are the outer vertical and horizontal lines in the square above. {a,b,c} denotes here a^n+b^n+c^n (n=1,2). --- Ok, I just found something magic out, which might be not so magic at all, anyway you can get infinitely many equations that way... instead of base 10 you can use any base (smaller or larger than the largest number in the square) for the calculations: in the first above example: (6*b^2 + 1*b + 8)^n + (7*b^2 + 5*b + 3)^n + (2*b^2 + 9*b + 4)^n = (8*b^2 + 1*b + 6)^n + (3*b^2 + 5*b + 7)^n + (4*b^2 + 9*b + 2)^n for (n=1,2). --- Ok, I just played around, and this works also for Duerer's 4x4 magic square 1 15 14 4 12 6 7 9 8 10 11 5 13 3 2 16 in exactly the same form. In all cases I just checked the rows and columns. I hoped that I would work (magically...) also for n=3 in this case, but this isn't true. Just for n=1,2. Also the original case (walk around clock and counter clock wise)
492^1 + 276^1 + 618^1 + 834^1 = 294^1 + 438^1 + 816^1 + 672^1 492^2 + 276^2 + 618^2 + 834^2 = 294^2 + 438^2 + 816^2 + 672^2 492^3 + 276^3 + 618^3 + 834^3 = 294^3 + 438^3 + 816^3 + 672^3.
works as well for any base, and for the 4x4 magic square (for any base) and I conjecture that it works for any nxn magic square. I haven't tried to leave out numbers in other bases and in the 4x4 case. Christoph