UPDATE: Gah! The link below is corrected. And I am reposting the entire thing tomorrow for anyone who misses this. This is off topic, but this online mind-reader was really freaking me out. Check it out, then come back and click through below to get the explanation. I felt dumb when I realized what the trick was.
From the first comment here:
This ‘magic’ is based on multiples of 9. The answer to the 2-digit formula is always a multiple of 9. On the board, the signs change order, but the sign on the multiples of 9 (9, 18, 27, 36, etc) are always identical.