In this normal magic square there are all kinds of symmetries. All the rows,
columns and diagonals add up to 260, as do the ‘bent rows’, one of which we’ve
highlighted. There are many other things to discover – like the sum of the central
2×2 square plus the four corner boxes, which also adds up to 260. Look closely
and you’ll find an interesting result for every 2×2 square.
Squared squares
Some magic squares can have cells occupied by different squared numbers.
The problem of constructing these was posed by the French mathematician
Edouard Lucas in 1876. To date no 3×3 square of squares has been found,
although one has come close.
All rows and columns and one diagonal of this square add up to the magic
sum 21,609 but the other diagonal fails since 127^2 + 113^2 + 97^2 = 38,307. If
you’re tempted to find one yourself you should take note of a proven result: the
centre cell value must be greater than 2.5 × 10^25 so there’s little point in looking
for a square with small numbers! This is serious mathematics which has a
connection with elliptic curves, the topic used to prove Fermat’s Last Theorem. It
has been proved there are no 3×3 magic squares whose entries are cubes or