If you take the decimal expansion of 1/19, 2/19… up to 18/19, and write out the first 18 digits of the decimal expansion of each, you get:
0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 | 81 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 | 81 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 | 81 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 | 81 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 | 81 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 | 81 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 | 81 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 | 81 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 | 81 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 | 81 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 | 81 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 | 81 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 | 81 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 | 81 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 | 81 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 | 81 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 | 81 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 | 81 ------------------------------------------------------+ 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81
The sum of the rows, colums, and it turns out, the main diagonals is 81, which means that it’s a magic square. It’s not hard to see why the rows and columns are sum to the same, but the diagonals are a bit trickier. Anyway, just something to think about.