## Easy to construct, 18×18 magic square.

May 15, 2008 | General | By: Mark VandeWettering

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.