Chess, Math, Video Games, and Life

I played a few chess games recently and quickly realized that I should have been playing and studying chess my entire life.  Few activities are as strategically rewarding.  Chess, in some ways, is an allegory for life.  After I played those games, I then quickly relearned the fundamentals of chess using Patrick Wolff’s excellent introduction, The Complete Idiot’s Guide to Chess (second edition).

As a high school student, I developed a deep interest in math and the technical sciences (physics, chemistry) by working through a book of logic puzzles, as I explained here.  Logic puzzles are elegant, self-contained problems that can be worked out with intuition, insights, and logical reasoning.  Math and the technical sciences are similar.  I explored math, physics, and other logic-based subjects extensively on my own, forcing myself to solve any problem I encountered that required ingenuity.  I proved theorems for myself and related them to other learned material.  I developed a visual approach to learning math.  For physics, I used analogies (that I refined further and further) to understand abstract concepts.  (More generally, I often reframed concepts in terms of other concepts.)  I constantly challenged my understanding, refusing to draw conclusions whenever possible.  I cultivated skepticism to improve objectivity.  If my understanding of a concept was flawed, I would take it apart and reconstruct it a better way.

This methodical, exploratory, often tedious approach deepened my understanding and made it much easier to solve difficult problems, even those in related subjects I hadn’t studied.  My understanding of an abstract logical subject would grow with exploration until I encountered an insight, at which point it would make a quantum leap to a higher level of understanding, and so on.  (Aside: medicine is very different because it depends heavily on memorization of detailed material.  Reasoning is important, but it doesn’t matter how well you can think if you don’t have the required knowledge available to you.)

So, in terms of strategies and tactics, the above process, with a few variations, was my sole strategy for acquiring an ever-growing toolkit of tactics.  High school- and college-level problems usually required tactical approaches.  A very difficult problem might require at least one clever perspective switch coupled with a string of tactics.  You could call the entire solution a strategy.  However, if strategy is defined as a plan that makes a solution possible, and tactics are the particular methods used to arrive at a solution, then one’s ability to generate strategies was hardly challenged at all.  In some sense, the strategy was the math class itself and was already thought out for you long in advance!  The tactics were what you learned in the math class.  And then you were presented with clean, well-composed, often elegant problems that you were expected to solve intelligently.  You were on a guided tour that you didn’t control.  (This wasn’t always the case.  In computer science and engineering classes, there was a greater emphasis on novel strategies for more open-ended or complex problems, especially in project-based settings.)

More importantly, there was really no intelligent way to develop one’s ability to come up with good strategies.  And life, though messy and complicated, is all about strategies at every level!  That’s where chess comes in.  As soon as I played a few games of chess, I realized that chess is all about generating and executing effective long-term strategies in the face of extreme complexity.  Chess strategies are what make the clever tactics possible.  Your strategy can change during the course of a game, depending on what your opponent is doing.  So you came up with a good move that’s part of what you think is a good strategy.  Can you come up with an even better move, an even better strategy?  You’re always thinking hard, thinking ahead, thinking as clearly as you can about the many possible moves in front of you, and looking for clever tactics that don’t compromise your long-term plan.  You make a move, your opponent makes her move, and then you often have to reconsider everything all over again.  There are supposedly 10^120 possible moves in a typical game of chess.  As of the time of writing of this essay, chess is still an unsolved game.  You have to learn to think for yourself, to think effectively, to project yourself into future time, and to handle being the master of your own destiny.  (These characteristics hold for any complex strategy game, not just chess.)

What about video games?  Aren’t video games more complicated?  I stopped playing video games in my twenties, after the Nintendo GameCube became passe, so I can’t comment on the current gaming scene.  In my teens and twenties, I enjoyed the more cerebral games when I had the time to play them (which wasn’t often!):  Myst, the Legend of Zelda series, Braid, Portal, and similar games.  (There were strategy-heavy video and computer games on the market, but I rarely played those because they seemed too much like computerized board games.  They’re in the same category as chess and other strategy games as far as I’m concerned.)

These video games were basically guided tours filled with self-contained puzzles.  Your ability to generate strategies wasn’t challenged so much as your ability to come up with clever tactics in a strategically ridiculous “man vs world” type of situation.*  (This also applied to “boss” battles…some of the later games did give you some strategic flexibility in this regard, but it was mostly about figuring out the self-contained tactical puzzles).  Even the large-scale environmental puzzles in these games (e.g., the Legend of Zelda) were tactical in nature, because there was only one way (or at most a few ways) to solve them and you couldn’t move forward until you did so.  Braid, the critically acclaimed video game, is basically a two-dimensional guided tour with some of the most brilliant puzzles in gaming history.

This sort of thing can be quite addictive, because you get a little burst of reinforcement every time you solve a puzzle on your guided tour.  (Aside: many video game puzzles fit neatly into some branch of mathematics.  They are basically math problems in disguise.)  But, wait!  At the highest level, you’re not in control.  You’re not learning how to think for yourself.  You’re not learning how to intelligently construct your own tour.  (In defense of such games, they can certainly teach, directly or indirectly, all sorts of positive things.)

In math, someone comes up with a theorem, someone proves it, and the theorem and its proof are eternal.  In the technical sciences, someone comes up with a law (e.g., thermodynamics) and it stands forever.  This timelessness spoiled me for a long time, because I viewed anything less perfect as inherently less valuable.  This is a flawed viewpoint, however.  Even though timeless ideas may stand forever, their practical value might be quite low, while the value of practical ideas that become obsolete might actually be quite high, because they help the world and can also pave the way to better ideas.  In chess, one cannot come up (at least at this point in time) with a perfect approach.  There are only strategies and better strategies.  Analyses of games written just fifty years ago are already outdated (because computer analysis has found their flaws).  Analogously, life is even more complex.  There is no verifiably perfect approach to life.

This is important because life is all about strategies, at every level, and nobody is going to construct your life (e.g., “tour”) for you.  You’re in charge of planning ahead for yourself and for those dependent on you.  There are innumerable different situations in life that require different strategies (e.g., dating, education, financial planning, debt management, time management, building a family, choosing and succeeding in a career, and so on).  Are you able to come up with effective approaches?

Although life is much more complicated than chess and in some ways very different, I feel that chess is an allegory for life in certain important ways, and that the casual study (not too much!) of chess can be beneficial.  One can rewind a chess game and play it differently at any point to see how it could have turned out.  By playing chess, one learns to persevere, to consider many different points of view and to weigh each against the others effectively.  One practices generating and carrying out effective plans in complex situations.  One learns to make the most of one’s time.  One learns to think “outside the box.”  One learns that the hardest game to win is often the “won game,” as a famous chess master once said.  This has many parallels in life.  One learns that even a game that seems lost can be saved by a clever move (also true in life).  And, finally, the fact that computers are now stronger than humans at chess is a great thing, because it not only means you have a chess partner whenever you want one, but it also means that you can use a computer to help you objectively analyze your decision-making!

*Although I have never been interested in “speed runs” through video games, I realize that speed runs require strategic thinking on a global level.  Speed run experts are probably good strategists.

Update 1/18/2016: I suspect that anyone who masters chess or other strategy games will have a significant advantage at creating computer algorithms (because they are computational strategies).

How I Fell in Love With Mathematics…and Only Then Became Good at It

A paper about the experience of mathematical beauty and its neural correlates was just published in Frontiers in Human Neuroscience.  It reminded me of how I fell in love with mathematics just before my senior year of high school.

Before that time, I found math a little frustrating.  I was decent at it and got good grades, but something critically important about math escaped me.  You could say that I had no trouble with math exercises, but I wasn’t skilled at solving math problems.  I didn’t know it at the time, but I lacked a sufficiently deep understanding of math and problem-solving in general.  All of this changed when I unexpectedly rekindled my childhood interest in logic puzzles.

Near the end of junior year of high school, I saw a curious little book on my English teacher’s bookshelf:  Fantastic Book of Logic Puzzles, by Muriel Mandell and Elise Chanowitz.  I pulled it off the shelf, started reading the back cover, and was immediately taken by such statements as the following:

“The puzzles you’ll meet inside this book are the world’s greatest and most baffling.” (This was not even nearly the case, of course.)

“To solve them, you need a plan of attack and logical reasoning…You’ll go on to solve tougher puzzles than you ever thought possible.” (True!)

It piqued my interest because it reminded me of how I used to enjoy solving logic grid puzzles as a seven- or eight-year-old.  My English teacher made a copy of it for me and I started working through it immediately.

Unlike the exercises I had encountered in math classes, which reinforced facts or blind techniques,  the logic puzzles in this book placed a heavy emphasis on deeply understanding the situations at hand.  You almost certainly wouldn’t solve most of them if you weren’t thinking carefully.

I finished the book, started studying for the SAT, and realized something very important:  I never learned geometry.  I mean, I took geometry in ninth grade (I was now about to start twelfth grade!), and I did “well”, but now that I knew how deep understanding felt, I realized I didn’t actually understand geometry on any significant level.

Nor did I know algebra.  Or trigonometry.  Or precalculus.

So, I sat down and gradually worked through all of these subjects from scratch while studying for the SAT.  I had learned, while solving logic puzzles, to constantly step back and question my assumptions.  I learned to hold only valid assumptions, and as few of them as possible.  I applied these habits of mind as I boldly attempted to prove theorems and other mathematical relationships by myself.  It was hard work, but enjoyable and addictive.

I became a mathematical explorer.  Math was no longer frustrating but fun.  I got into the habit of attempting to solve any test of ingenuity that came my way.  Paul Zeitz, in The Art and Craft of Problem Solving, asserts that this exploratory/investigative mindset is key to gaining skill at solving significant problems.  The problem-solver lays siege to the problem at hand, circling it until an opening is found–until the crux point of the problem is resolved–and then the problem surrenders its secrets to him, often in a trivial way, and is solved.

Gradually, I gained confidence and became skillful at math, doing very well in senior year calculus and in any other class (chemistry, physics) requiring accurate, logical thinking, as well as on standardized exams.  No longer intimidated by math, I switched from biochemistry to computer science after my freshman year of college and deliberately took the hardest math and science classes I could (much harder than entry-level “pre-med” classes), enjoying them greatly and doing very well in them, and, eventually, getting into one of the best computer science doctoral programs in the world.

They say the bumblebee doesn’t know it shouldn’t be able to fly.  I didn’t know I couldn’t become good at math, so I kept at it, and kept surprising myself, until I became pretty good.  I never did become great, but that’s okay.

(Who knows?  Maybe I would have become great if I had not switched careers and had continued to apply the principles of deliberate practice.  Maybe you could become great at some skill that currently escapes you, if you just drop the invisible script of “I don’t have the necessary talent” and get down to business!)