42 Magic squares
‘A Mathematician’, wrote G.H. Hardy, ‘like a painter or a poet, is a maker of patterns.’
Magic squares have very curious patterns even by mathematical standards. They lie on
the border between heavily symbolled mathematics and the fascinating patterns loved
by puzzlesmiths.
A magic square is a square grid in which distinct whole numbers are written
into each cell of the grid in such a way that each horizontal row and each vertical
column, and each diagonal add up to the same number.
Squares with just one row and one column are technically magic squares but
are very boring so we’ll forget them. There is no such thing as a magic square
with two rows and two columns. If there were it would have the form shown.
Since the row additions and the column additions should be equal, then a + b =
a + c. This means b = c, contradicting the fact that all the entries must be
distinct.
The Lo Shu square
As 2×2 squares do not exist, we’ll look at 3×3 arrays and attempt to construct
one with a grid. We’ll start with a normal magic square, one where the grid is
filled out with the consecutive numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9.
For such a small square it is possible to construct a 3×3 magic square by the
‘trial and test’ method, but we can first make some deductions to help us along.