is there more ways to get a different 3x3 of magic squares using #'s 1-9?

Question

Can you make more than one 3x3 magic square adding only using the numbers 1-9?....

Answer 1

Discounting rotations and mirror images, there is only one way to make a magic square using the digits 1 through 9.

First, we can prove the 5 has to go in the middle:

Let the numbers in the top row from left to right be a, b, and c. Let the numbers in the middle row from left to right be d, e, and f. (Note e is the middle box.) Let the numbers in the bottom row from left to right be g, h, and i.

We know a + b + c = 15 and g + h + i = 15

We also know a + e + i = 15, b + e + h = 15, and c + e + g =15. Therefore, (a + e + i) + (b + e + h) + (c + e + g) = 15 * 3 = 45. We can rearrange this to (a + b + c) + (g + h + i) + 3e = 45. Since a + b + c = 15 and g + h + i = 15, that reduces to 15 + 15 + 3e = 45, or 3e = 15. Therefore e (the middle box) must equal 5.

Knowing the middle box is 5, try 9 in a corner and you'll notice a problem - the 6, 7 and 8 can't all be placed in the square without the row or column with the 9 exceeding 15. So, the 9 has to occupy a side box. Since we are discounting rotations and mirror images, we can arbitrarily select box b for the 9. This forces the placement of the 1 in box h.

With the 9 and 1 now placed, the 8 and 6 have to go in the row with the 1. So, we'll put the 8 in box g and the 6 in box i. (We still haven't done anything to create a second square if rotations and mirror images are discounted.) That forces the placement of the final four numbers.

Answer 2

The Lo Shu magic square using only 1 through 9 for a 3 x 3 grid is unique.