This post continues my effort to write detailed solutions to the problems on this year’s Putnam exam (previous solutions are here and here). More than the actual solutions, I hope that students will find my commentary valuable. When writing these, I try to emphasize the thought processes and the strategies that were used to solve the problem, so that a reader might feel empowered to solve similar problems. I feel like some of the solutions in the MAA’s Putnam Directory are too slick and might reinforce a student’s feeling that they never would have come up with a solution on their own. These solutions attempt to counter that feeling.