Question

Show that there are no magic squares of order greater than one if we require each row, column and diagonal to have a constant product.

Answer 1

Consider a magic square of or der 2 as follows:

a	b
c	d

First row product = a X b

First column product = a X c

Given: Row product = Column product

So,

we get:

(a X b) = (a X c)

So,

either a = 0

OR

b = c

The option: b = c is not permitted because all elements of magic square should be different.

So,

consider a = 0

Second row product = c X d

Second column product = b X d

Given: Row product = Column product

So,

we get:

(c X d) = (b X d)

So,

either d = 0

OR

c = b

The option: c = b is not permitted because all elements of magic square should be different.

So,

d = 0

This is also not permitted because already a = 0.

Thus, we have proved that there are no magic squares of order 2 if we require each row, column and diagonal to have a constant product.

By extending this argument to any order, we prove that there are no magic squares of order greater than one if we require each row, column and diagonal to have a constant product.