An antimagic square is an 
array of integers from 1
to 
such that each row, column, and main diagonal produces a different sum such that
these sums form a sequence of consecutive integers.
It is therefore a special case of a heterosquare.
It was defined by Lindon (1962) and appeared in Madachy's collection of puzzles (Madachy
1979, p. 103), originally published in 1966. Antimagic squares of orders 4-9
are illustrated above (Madachy 1979). For the 
square, the sums are 30, 31, 32, ..., 39; for the 
square they are 59, 60, 61, ...,
70; and so on.
Let an antimagic square of order 
have entries 0, 1, ..., 
, 
, and let
 |
be the magic constant. Then if an antimagic square of order 
exists, it is either positive with sums ![[M(n)-n,M(n)+n+1]](/images/equations/AntimagicSquare/Inline9.svg)
, or negative with sums ![[M(n)-n-1,M(n)+n]](/images/equations/AntimagicSquare/Inline10.svg)
(Madachy 1979).
Antimagic squares of orders one, two, and three are impossible. In the case of the 
square, there is no known method
of proof of this fact except by case analysis or enumeration by computer. There are
18 families of antimagic squares of order four. The total number of antimagic squares
of orders 1, 2, ... modulo the full group of symmetries (reflection, rotation, complementation,
and exchanges) are 0, 0, 0, 299710, ... (OEIS A050257;
Cormie).
Madachy (1979) and Abe (1994) asked for methods of constructing antimagic squares of every order. Recently, J. Cormie and V. Linek have developed general
constructions for squares of order 
for all 
, as well as for bordering antimagic squares.
