The Lo Shu is the unique magic square of order three up to rotation and reflection. Its eight distinct arrangements use the integers 1 through 9 exactly once and have magic constant 15. The center contains 5. The four corner entries are 2, 4, 6, and 8, while the middle entries on the four sides are 1, 3, 7, and 9. Both groups of four entries sum to 20. The Lo Shu is an associative magic square, but not a panmagic square.
To see why the center must be 5, let its entry be 
. Each of the four pairs of entries opposite each other through
the center sums to 
,
since each pair and the center form a row, column, or main diagonal.
These pairs contain all eight noncentral entries, whose sum
is 
. Therefore
 |
which gives 
.
