There are exactly eight normal magic squares of order three

Lo Shu uniqueness. Exactly eight admissible parameter pairs give a normal magic square of order three:

where $\mathrm{normalParamCount}(e)$ counts the pairs $(a,c)\in\mathrm{paramSet}(e)$ for which $\mathrm{mkMagic3}(e,a,c)$ is normal.

By magic_three_param_bij the admissible parameter pairs are in bijection with the $3\times3$ magic squares of line sum $15$, so this is precisely the statement that there are eight normal magic squares of order three — the eight images of

under the symmetry group $D_{4}$ of the square. Equivalently: the Lo Shu square is the unique normal magic square of order three up to symmetry.

This is the oldest non-trivial classification in combinatorics, and the reason order three is exceptional: for $n=4$ there are $880$ normal squares up to symmetry, and for $n\ge5$ no classification is known.

Proof. Immediate from magic_three_normal_classify, which characterizes normality by the eight parameter pairs $(2,4),(2,6),(4,2),(4,8),(6,2),(6,8),(8,4),(8,6)$: the defining filter of normalParamSet 5 selects exactly those eight, and they are distinct.