Only the digits 0, 1, 2, 5 and 8 are used, drawn in seven-segment form. Turned upside down (180°) each of them is itself; in a mirror 0, 1 and 8 are themselves while 2 and 5 change into each other.
Upside-down: the square is turned by 180° — the last row comes first, each row is read backwards and each number is read backwards. Mirror looking: the columns come in reverse order, each number is read backwards and 2 ↔ 5. Water: the rows come in reverse order and 2 ↔ 5 (it is the mirror image of the upside-down square).
Every entry is a chain of digit groups, one group per layer; in every layer each row, each column and both diagonals contain the same symbols. Turning the square sends rows to rows, columns to columns and diagonals to diagonals, and reading a number backwards only rewrites each group with other digits — so the upside-down, mirror and water squares are magic again, and so are the digit families of one generation (digit substitutions of each other, as in the papers).
When each Latin square fits in one digit (order 4 with four or five digits, order 5 with 0, 1, 2, 5, 8) the generator also writes palindromic 3-digit entries a·b·a: the outer digits come from one Latin square and the middle digit from the other; reading a number backwards leaves it unchanged, so all four views stay magic. Orders 4, 5, 8 and 10 also get three-layer 3-digit entries a·b·c: three layers built algebraically (over GF(2) for 4 and 8, over the integers mod 5 for 5 and 10) in which every row, column and diagonal holds each digit equally often, all n² entries different. Only 125 three-digit numbers can be written with 0, 1, 2, 5, 8, so 3-digit cells stop at order 11; orders 6, 7, 9 and 11 cannot be balanced this way.
Layers used: cyclic Latin squares for orders prime to 6 (pandiagonal squares), finite fields for 4, 8, 9, 16, 27, 32, products such as 20 = 4 × 5, and s × s blocks found by computer search for the other orders. Order 6 gets semi-magic squares (equal rows and columns in all four views), since it cannot be built fully magic with this method. Order 3 comes from a complete computer search (2-, 3- and 4-digit cells): no square is magic in all four views, but every upside-down magic square (magic upright and turned 180°) has as its mirror image a mirror magic square (magic in the mirror and in water), and in all four views rows and columns stay equal. With (0,1), (0,8), (1,8), (0,1,8), (2,5), (0,2,5) and (1,2,5) order 3 gives only semi-magic squares.