How are Christmas and Halloween the same? The intended answer to this question requires you to look at the question from different angles to find the significant relationship between these seemingly unrelated events. In fact, to be a competent problem solver, you often need to be able to look at a problem from multiple angles and find a way to take advantage of a relationship between different parts of the problem that might not be immediately obvious. If the relationship was obvious, there might not be a problem to solve.

I have found over the years that doing different types of puzzles and thinking games often help me to juggle the conditions of a problem around and find that elusive relationship that makes the problem solvable. While I do not believe being able to solve Sudoku puzzles will make you smarter, I do believe that practicing Sudoku puzzle in different ways can help exercise your “cognitive muscles” so that you can more easily reorganize difficult and abstract concepts around in your mind and find the critical relationship between the different parts.

There are several approaches to solving Sudoku puzzles and each requires a different set of cognitive wiring to perform competently. One approach, and one that I see most electronic versions of the puzzle support, involves penciling in all of the possible valid numbers in each square and using a set of rules to eliminate numbers from each square until there is one valid answer. Another approach finds the valid numbers without using the relationships between the “penciled” numbers. Each approach exercises my thought process in very different ways, and I find that switching between them provides a benefit when I am working on a tough problem.

I believe being able to switch gears and represent data in equivalent but different representations is a key skill to effective problem solving. In the case of Christmas and Halloween, rather than looking at the social context associated with each day, looking at the date of each day – October 31 and December 25 can suggest a non-obvious relationship.

I find that many of the best types of puzzles or games for exercising orthogonal thinking engage a visual mode of looking at the problem. The ancient board game of Go is an excellent example. The more I play Go, the more abstract relationships I am able to recognize and most surprisingly – apply to life and problem solving. If you have never played Go, I strongly recommend.

Another game I find a lot of value for exercising orthogonal thinking is Contract Bridge – mostly because it is a game that involves incomplete information – much like real life problems – and relies on the ability of the players to communicate information with each other within a highly constricted vocabulary. Often times, the toughest design problems are tough precisely because it is difficult to verbalize or describe what the problem actually is.

As for the relationship between October 31 and December 25, it is interesting that the abbreviations for these two dates also correspond to notation of the same exact number in two different number bases – Oct(al) 31 is the same value as Dec(imal) 25.

These examples are some of the ways I exercise my orthogonal thinking. What are your favorite ways to stretch your mind and practice switching gears on the same problem?

Tags: Problem Solving

Because math is so fundamental to so many of the problems I work on (set theory, graph theory, combinatorics, linear programming, etc.) I find that Project Euler programming problems help me “think outside the box”.

Many of the problems can be solved with a “brute force” solution, but if an attempt is made to understand the essence of the underlying problem, a more elegant solution can be derived. (And often, the more elegant solution runs – quite literally – orders of magnitude faster).

Also, games like go, checkers, sudoku, etc. are very good at training the mind to look at things in different ways.

Have a good 2011.

I look at the right angle to a problem….