An associative magic square is a magic square for which every pair of numbers symmetrically opposite to the center sum up to the same value.
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3 × 3
The Lo Shu Square, the unique 3 × 3 normal magic square, is associative, and as such is the only 3 × 3 associative magic square. The square is shown below in Frénicle standard form. All rows, columns and diagonals sum to 15 and all pairs symmetrically opposite the center sum to 10.
4 × 4
There are, up to rotation and reflection, 48 distinct 4 × 4 associative magic squares, four of which are given here.
5 × 5
There are 48,544 distinct 5 × 5 associative magic squares, up to rotation and reflection, such as the two below.
Larger sizes
There are no 6 × 6 (or 10 × 10) associative magic squares. Other than when there are none the precise number of associative magic squares at sizes bigger than 5 × 5 is unknown. The following table, due to Walter Trump, gives the sizes up to 10, followed by examples of 7 × 7 and 8 × 8 tables.