I only have time for a quick post tonight, so let me direct you to one of my favorite math videos. It’s of Arthur Benjamin, a mathematician at Harvey Mudd College in California. Art is also a professional magician, and is especially well known for his skill as a lightning calculator. The video is fifteen minutes long, but very enjoyable.
Just so we’re clear, his calculations are not tricks. He really is doing what it looks like he’s doing. The only portion of the video that could be described as a trick is the part where he determines the missing digit of a seven-digit number after the audience member tells him the other six. That’s based on an elementary principle in number theory, which I’m sure some commenter will enjoy explaining.