Every entry is a chain of digit groups. Each group comes from a layer: an n×n array of symbols in which every row, every column and both diagonals contain exactly the same symbols. Writing each symbol as a fixed group of allowed digits and joining the groups gives n² different numbers whose rows, columns and diagonals all have the same sum, whatever digits are chosen.
Seen in water the rows come in reverse order and 2 ↔ 5. This only rewrites each symbol with other digits, so the water reflection is again a magic square. For the same reason the digit families of one generation are digit substitutions of each other (for example 1 → 8, or 0 → 2 and 1 → 5), as in the papers.
Layers used: orders prime to 6 (5, 7, 11, 13, 17, 19, 23, 25, …) take cyclic Latin squares, and the squares are pandiagonal; orders 4, 8, 9, 16, 27, 32 take finite fields; orders such as 20 = 4 × 5 or 28 = 4 × 7 take products; the remaining orders (10, 12, 14, 15, 18, 21, 22, 24, …) take a smaller order in s × s blocks, the block layers being found by computer search. Orders 3 and 6 cannot be made fully magic this way for all digits. Order 6 therefore gets semi-magic squares (rows and columns), found by computer search. Order 3 is taken from a complete computer search: with entries of up to 4 digits, water-reflection magic squares of order 3 exist only for (0,1,3,8), (2,3,5,8), (0,2,3,5,8), (1,2,3,5,8) and (0,1,2,3,5,8); the other digit families get semi-magic squares from two orthogonal Latin squares of order 3, as in the papers.