Source: Asked to me by Nigel Coldwell (now posted on his blog)
Problem: Two creepers, one jasmine and other rose, are both climbing up and round a cylindrical tree trunk. jasmine twists clockwise and rose anticlockwise, both start at the same point on the ground. before they reach the first branch of the tree the jasmine had made 5 complete twists and the rose 3 twists. not counting the bottom and the top, how many times do they cross?
Solution: My solution posted on Nigel's Blog here. Slightly different solution posted by Gaurav Sinha (1996 CSE IITK passout, now at Indian Revenue Service) in comments! A more general argument by Aaditya Ramdas (CMU Grad Student - CSE IITB 2009 Alumnus - Ex Tower Research Analyst) in comments!