Puzzles for Programmers and Pros

6.1. Order the Ages

You might encounter elimination puzzles like this one in Mensa-style books. As mental calisthenics, they have a lot to recommend them.

Andrew, Carol, Jessica, Luke, and Tommy are sitting around a circular table. Carol is 12 years older than her neighbor to the left. Tommy is five years older than his neighbor to the right. Jessica is 14 years older than her neighbor to the left. Luke is five years younger than his neighbor to the left. By age, from youngest to oldest, they are ordered as follows: Luke, Tommy, Andrew, Jessica, and Carol. Luke is 16 and Carol is 40. The total of the ages is 135. In which order are the people sitting (starting with Tommy and then proceeding in clockwise order) and what are their ages? First try to do this on your own; when you've solved it or are ready for help, read on.

Start with the most specific facts: Luke is 16 and Carol is 40, totaling 56. This implies that the ages of Tommy, Andrew, and Jessica add up to 79 (135 - 56 = 79). Further, we know that Tommy and Luke are neighbors because Luke's left neighbor is 21, so if that's not Tommy, then both Tommy and Luke's left neighbors are 21 or less (because Tommy is second youngest after Luke). This leaves 79 - 42 = 37 left for the last person. But Carol is 12 years older than her neighbor to the left, so someone must be 28 years old. Hence, in clockwise order we must have Tommy, ...