Associative magic square

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.

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.

There are, up to rotation and reflection, 48 distinct 4 × 4 associative magic squares, four of which are given here.

There are 48,544 distinct 5 × 5 associative magic squares, up to rotation and reflection, such as the two below.

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.

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