In Part 1 (read that first), I mentioned that if the square's side was divisible by 3, the resulting square will only be semimagic, but can become magic if the average of all of the numbers in the square is moved to the center. It turns out that there are many other interesting properties for this kind of square. For example, the 9x9 square looks like this:
This is only a magic, rather than a panmagic, square, but in addition to that, some, but not all, of the pandiagonals add up to the magic constant - in this case, 369. These pandiagonals are illustrated in yellow.
Notice that there are a few squares where the pandiagonals intersect. If these squares were moved to the center, the pandiagonals that intersected them would become diagonals, and the square would remain magic. The squares where the pandiagonals intersect are highlighted in green.
Notice how these make a magic square of their own, with a magic constant of 123 - 1/3 of the original square's magic constant!
Here are a few other properties of this particular square:
1. The pandiagonals that do not add up to 369 add up to either 288, which is 369-(9^2), 360, which is 369-9, 378, which is 369+9, or 450, which is 369+(9^2).
2. Every 3x3 square, including those that wrap around, also adds up to one of these five sums (288, 360, 369, 378, or 450).
3. When the highlighted squares are displaced by some amount, they add up to one of the four totals described in #2.
What other magic series can you find in this square?
Here are a few other properties of this particular square:
1. The pandiagonals that do not add up to 369 add up to either 288, which is 369-(9^2), 360, which is 369-9, 378, which is 369+9, or 450, which is 369+(9^2).
2. Every 3x3 square, including those that wrap around, also adds up to one of these five sums (288, 360, 369, 378, or 450).
3. When the highlighted squares are displaced by some amount, they add up to one of the four totals described in #2.
What other magic series can you find in this square?