Some strange mathematical sequences are always whole numbers -- until they're not. The puzzling patterns have revealed ties to graph theory and prime numbers, awing mathematicians... . "To me a big part of the allure of math is when you arrive at the same destination by different paths and it seems like something miraculous or deep is going on," Propp said. "The cool thing about these sequences is there are various points of view that explain why you get integers. There are hidden depths there."