Making Up Your Own Mind: Thinking Effectively through Creative Puzzle-Solving

Early in a semester, I like to pose this problem to my college algebra students:

Jack is looking at Anne, but Anne is looking at George. Jack is married, but George is not. Is a married person looking at an unmarried person?

This is from the 2010 book What Intelligence Tests Miss: The Psychology of Rational Thought by Keith E Stanovich. From the same book I suggest to the students that they apply “fully disjunctive reasoning”, a phrase I find is a koan-like tool for disruption of quick, reactive thought and nudging students into thinking of categories as a movement toward proof construction. During this time, we work in logic, leaving strictly mathematical topics aside. Such exercises in critical thinking are also germane to this text and can be of benefit to students of mathematics, philosophy, engineering, and, well, life. The promotion line here is, “How you can become better at solving real-world problems by learning creative puzzle-solving skills.”

Devised for those seeking to improve thinking ability, this book requires no expertise in puzzle-solving or in any other specialty area, such as mathematics or logic. Inveterate puzzle-solvers will recognize many chestnuts here dealing with coins, matchsticks and a farmer scheduling a river crossing to avoid undesired consumption. There is also the Hat Puzzle. The prisoners variant is softened by being relocated to a children’s “ball pit” and painted in the color palette of Southwestern University. The author is the president of Southwestern University, and a mathematics professor. Mathematics professor or not, Burger is teaching cogitation and not calculation and studiously avoids mathematical techniques. For instance, in a problem pregnant with Diophantine possibilities he commands: “Solve this puzzle just using practices of effective thinking rather than resorting to doggone algebra.”

Those seeking to improve their “doggone algebra” and especially proof-building skills will many applicable exercises here, such as for learning mathematical induction as a proof technique. From one of the hints:

…create the simplest related challenge you are able. Resolve that easy puzzle and understand its solution in unusual depth. Create a slightly less simple version of the original challenge, resolve that slightly more challenging puzzle, and understand the connection between your two solutions…

The problem at hand here derives from the “Five Couples” handshake problem popularized by Martin Gardner’s in his Knotted Doughnuts, Napier's Bones, and Gray Codes. Burger’s approach in building up from simpler cases applies very well and is the type of consideration generally suggested. This is different from Gardner’s solution, which invokes the Pigeonhole Principle and diagrams the five-couple problem without considering simpler cases. (Burger does not provide solutions but does offer one level of hints in a chapter printed upside down and more explicit suggestions in a chapter of mirror type.) So, even familiar problems may offer different approaches. But what are the effective thinking principles emphasized here?

The five elements of effective thinking are interconnected guideposts to active learning and ongoing growth. They begin and end with the synergistic notions of deep understanding and change. To achieve these two goals, in practice, and create thought-provoking pathways forward, one must include effective failure, the art of creating questions, and an appreciation for the flow of ideas. These interconnected mindsets form the foundation for The 5 Elements of Effective Thinking.

Filed nicely between The New Martin Gardner Mathematical Library and popular critical thinking books, this is an elegant blend of entertainment and enlightenment.

As a teen, Tom Schulte arranged a file of seated relations en chapeau to try to understand the basic Hat Puzzle. He has loved puzzle-solving since, although he converted none of his relatives.