Turbaned, with the penetrating blue eyes of the mountain people of Kashmir, Abdul Samid entered the apartment of the problem-solving twins Cloe and Eli. "Salaam Aleikum," he said as he entered. After accepting offers of tea and exchanging a few pleasantries, he came to the object of his visit.

"The shallow Dal Lake of Srinigar is dotted with lotus flowers," he began. "These broad plants won't support an adult's weight but will support a child's. That is the basis for the Lotus Dance, a ritual dance for eight- to ten-year old girls that is our village's private ceremony. A single chosen girl is dropped off at any lotus she wishes. Her task is to jump on every lotus exactly once. If she succeeds, she brings good luck to our village, and honor to her family and to herself.

"My daughter Jamilla is the chosen girl this year. This figure shows the placement of the lotuses:

"Each circle represents a lotus. Line segments indicate possible jumps. If my daughter is at a lotus X and there is a line segment to another lotus Y, then she can jump from X to Y. If there is no line segment, then the jump is too far."

Problem 1. "Can you tell me where she should start and in which order she should jump on the lotuses to succeed in the Lotus Dance?"

Problem 2. How many alternative paths are there?


1. The key observation is that the rightmost cycle and the upper leftmost cycle each have a single lotus that connects them to the rest of the graph. Therefore Jamilla should start at one of those cycles and end at the other.

2. The above explanation showed that one can start at either the upper leftmost or rightmost cycle and end at the other cycle. For each starting cycle there are two starting points (on either side of the single connecting lotus) and for each cycle there are two ending points. So, starting at the upper leftmost cycle yields four possibilities and starting at the rightmost cycle yields four more. For each of these, there are two ways to go around the bottom cycle, so there are at least 16 possibilities altogether.