Magic Squares

NADA

Hey. I am just wondering I recently solved a magic square using simultaneous equations. I don't know any other way of doing these things and never done them before. Now the solution I got works. As in all rows columns and diagonals add up to 17. But my solution contains many fractions? Is this allowed?

henbane

I'm pretty sure magic squares are composed of integers.

hivizman

henbane wrote: »

I'm pretty sure magic squares are composed of integers.

Usually, magic squares consist of a sequence of integers beginning with 1 (so a 3x3 magic square is made up of the integers 1 to 9). The standard 3x3 magic square is:

8 1 6
3 5 7
4 9 2

Each row, column and diagonal sums to 15.

I don't think that a unique magic square can be produced purely by solving simultaneous equations, because there are eight equations (three rows, three columns, and two diagonals) and nine unknowns (the nine numbers in the magic square - note that the common sum of the rows, columns and diagonals will be one third of the sum of the nine numbers in the magic square).

By the way, the magic square above is not unique, because any linear transformation of the numbers (y = ax + b where x is one of the numbers in the magic square above) will give another magic square.

For example, set a = 1 and b = 2/3 - this gives the following magic square:

8 2/3 1 2/3 6 2/3
3 2/3 5 2/3 7 2/3
4 2/3 9 2/3 2 2/3

The rows, columns and diagonals all add up to 17 in this magic square.

Setting a = 2 and b = 1 gives:

17 3 13
7 11 15
9 19 5

The rows, columns and diagonals all add up to 33 in this magic square.