Mental Exercises

Mental exercise can greatly increase the ability and speed to comprehend new concepts and solve problems. Remember just like physical exercise your ability to solve problems will atrophy if you do not perform the exercises regularly and consistently. It would be beneficial to create a training plan akin to training in physical exercise. The best advice would be to perform these exercises everyday, or at least every other day. One could also use these exercises as “cross training” for other fields like mathematics or physics.

 In addition, just like exercise, it will take time to see results, so be patient and consistent. If you want to improve your performance on timed tests, it is crucial that you time yourself on practice tests and focus on improving your score within the required time. Do not forget those who have used these techniques from childhood will have a tremendous advantage over those who have not. Speed and accuracy are crucial for timed tests so you must speed train to ensure a good result. (Though the speed train link is for runners they have many good ideas we can apply to training for timed tests, scroll down to “A lesson from Ondiek”.)

In addition, it would be beneficial to create your own mental exercises. We can look towards sports; there exists an exercise for nearly every skeletal muscle in your body. Certain exercises target a broad range of muscles to build strength while others target very specific muscles. You can create mental exercises that target a broad range of mental faculties to build overall performance like speed, concept comprehension, and problem solving ability, while other mental exercises will target only a very specific portion of your mind like memory, calculation, visualization, etc. The possibilities are endless.

The exercises should be perform in the order listed, emphasis should be placed one the first 3 exercises as they are the most important.

We are going to focus on several types of mental exercise

More important than any mental exercise is the belief that you can change. Another important quality is the ability to continue trying to change, despite setbacks. Your mental attitude is possibly the most important mental exercise of all, for if you believe that you can't improve, that you can't change then it has the same effect as not being able to change. So to guard against these thoughts it is best to have some source of motivation, something that will help you proceed through the everyday process of slow improvement, and these are motivational audio programs. Some of best are listed below:

Napoleon Hill: One of the first motivational speakers, his style is reflective of the early 20th century United States.

Zig Ziglar: One of the best motivational speakers,

The first mental exercise enhances your concentration.

There are several ways to develop your concentration, one way would be to obtain a concentration hypnosis tape by effective learning systems, these tapes are used by the US government and top schools. The other methods are listed on my concentration exercises page.

Concentration is probably the most important (and most difficult to master) trait of most problem solvers, the better your concentration the easier and faster it will be to solve problems.

The second mental exercise enhances your memory

Improving one’s memory is important because we use it every time we solve problems. You need memory to remember important equations, relationships, concepts, notation, rules, similar problems, and much more. A good memory is important in the problem solving process as well, the better your memory the easier it is to keep track of all the rules, notation, and equations. There are many good memory systems out there one of the best is by Harry Loryane, his book The Memory Book is a classic. Another great memory book is How to Develop a Super Power Memory. (Download)

The third mental exercise is learning how to think critically.

Critical thinking is crucial to problem solving; it gives us the basic structures to reason effectively in any situation. Anyone can learn how to think critically, but it requires practice and should be applied in all situations to develop the habit of thinking critically. Probably the best website on critical thinking is The Teaching Company has a great lecture series on critical thinking and effective reasoning.

Critical thinking is the basis for solving problems but the techniques and strategies involved in solving problems are just as important (especially in math and physics). Critical thinking is best used to guide our reasoning and to correctly identify, analyze, and state a problem, but to actually solve a problem one needs problem solving techniques.

The fourth mental exercise is solving problems.

This might sound strange but one of the best introductions to problem solving is through SAT prep books. Most of them are self-contained, meaning they review all the mathematics you need to solve a wide range of problems and they contain many exercises to build your problem solving skills. One of the best SAT prep books is: How to prepare for the SAT 1 published by Barron’s. I would start with the SAT prep book then move on to the books listed below.

Obtain one or some of the fine problem-solving books listed on my Book List. I suggest you first obtain How to Solve It by George Polya then move on to The Art and Craft of Problem Solving by Paul Zeitz. Do not forget about the solutions in the back of the book. After you have worked through the problem solving book(s) you can further develop your problem solving ability by solving problems in general. There are many great books dedicated strictly to IQ type problems, logic problems, math problems, etc.. Work through both the problem solving books and your problems book using the techniques listed under How To Learn Higher Math and Do Proofs. One should work through at least one or two problems from those books everyday, the more consistent you are the better your ability to solve problems (if you work through enough IQ type problems you can literally increase your IQ).

Sterling publishers (look under puzzles and games) has a wide selection of puzzle books many of them IQ type puzzles. If you are looking for problem solving techniques for IQ type puzzles then you should look at aha! Insight and aha! Gotcha by Martin Gardner. For more problem solving books look at my booklist page.

If you prefer solving physics problems, please visit this Physics Olympiad of visit the official international physics Olympiad page. Also, for additional practice please view the Harvard physics problem of the week.

The more problems you solve the greater your ability to solve problems in any specific area of study. You can never know how the elements of one problem can relate to the elements of another problem. In addition, you learn techniques helpful in solving problems in your area of study.


The fifth mental exercise is to practice learning or understanding new concepts

Another mental exercise will be to "understand" or "learn" a new idea, technique, or event everyday.

Wikipedia features a different article from its archives everyday, it would be a good source for learning new ideas. The New York Times Guide to Essential Knowledge is a great way to stay informed and learn new ideas. Also, if you have the time, reading the Great Books of the Western World is one of the best ways to learn the ideas, the processes, and the reasoning that gave rise to modern western thought. Most public libraries have copies of these books.

Philosophical ideas are also a good source of new and challenging ideas; there are many websites about philosophy, Philosophy Talk is a great radio show about philosophy. Talk radio is a great way to learn something new if one is short on time here is a list of some of the best:

US Programs FM

Foreign Programs

Forum from KQED

Radio 4 Programmes (they have lots!)

Focus from WILL

DW-World from Deustche Welle

On Point from WBUR

WRN Worldwide Radio Broadcasts

Berkeley Groks from KALW (link fixed) For an extensive list of public radio talk programs.


Book TV on C-SPAN (Not really radio, but high quality.)


For AM US programs please visit They have an extensive list of AM radio shows in the US.  If you live outside  the United States some online websites may not allow you listen, simply click on "all streams" and locate an online webstream without the "cc" initials.

Each type of radio program has different topics and different approaches to the topics they cover. They explore many subjects often with world-class experts. In addition, one can order The Teaching Company’s series on philosophy and challenge your mind everyday. If you want to explore the philosophy of religion, The Modern Scholar’s series on the philosophy of religion is quite good, and well balanced.

We should try to "learn" as many ideas from as many different fields so that our minds will be in habit of understanding and learning new things.

The sixth mental exercise is called image streaming. I have modified it a bit to better suit our purposes.

First sit back in comfortable chair, close your eyes, focus on any problem or series of problems you have solved before, and describe aloud, in extreme detail, every step you took to solve it. Visualize every step as if you were writing it down on a piece of paper or on an imaginary chalkboard in your mind. Talk to a tape recorder or another person. You can extend your image stream to include other images, if so describe it as if you were there. Describe the color, the smell, the taste, the touch, and the temperature of whatever you visualize, in extreme detail. Afterwards listen to your image stream and take note of how you go about solving problems and how you can improve.

Visualization is a central part of problem solving both in math and physics; the more you strengthen your ability to visualize in detail, the stronger your ability to visualize mathematical notation and physics problems.

The seventh mental exercise is “time reversing”

Take any situation you may encounter, like standing in line at the bank, driving down the freeway, walking around campus and ask yourself; how did this situation look one minute before, two minutes before and so on. Or you can focus on one specific part of the situation like a specific car on the freeway and think, where was this car one minute before, two minutes before, and so on.

The eighth mental exercise is pattern construction/finding

Look for patterns in or construct patterns for nearly everything you see. Whether it be license plate numbers, architecture, words in magazines, pictures in magazines, almost anything.

The ninth mental exercise is “word play”

Most of the exercises outlined above are heavily focused on improving mathematical type thinking, I thought I should include at least one non-mathematical exercise. Take the words of any sentence and try to create as many new sentences from those words as you possibly can in the shortest amount of time. Also, you can take a set of random words and try to create as many different sentences as you can in the shortest amount of time.

The tenth mental exercise is a generalization of re-deriving.

Take any mathematical or physical concept and ask your self how you could or would have derived/created such a concept knowing only what people knew back when it was derived/created. Think rationally and in extreme detail.

 Most proposed theories in the history of physics have been wrong. You should be very careful to observe all prior theories, assumptions, and experimental evidence when trying to re-derive the theory, basically follow the critical thinking steps. Also you may want to become your own best critic. Remember the final arbiter of any theory you may re-derive or any theory you may create is experiment. Physics is not like math; we cannot prove things with the certainty of proving Fermat’s last theorem, so make sure to review the following suggestions; how to avoid being a crackpot. Also, you may want to ask yourself Am I a Quack!

I hear this repeatedly from accomplished physicists: before you can think outside the box you must learn to think inside the box. This means you should learn and master the traditional theories of physics before you start developing new theories of physics.

This exercise is very helpful to people in the sciences, it helps build our ability to think creatively and rationally. In addition, it helps us to think like the great minds of history.

Take any sufficiently complex man made object and ask yourself how could it work without looking inside the object. For every guess, you make think about its effect on the behavior of the object and compare to the actual behavior. Create a theory about the inner workings of the object and design experiments to test your theory.

This exercise is very helpful to people in the sciences, it helps build our ability to form plausible theories given only experimental data.


Another great exercise is to emulate the thought processes of great minds in history. Read biographies about the lives of great minds, in addition read books about how great problems in history were solved or how successful theories were created.

Make it a goal to have at least one new idea everyday; this will get you to be more creative. Create your own mathematical or physical problems and try to solve them. Get in the habit of solving little problems all day, for instance finding the quickest way around town, in a building, how to increase the horsepower in your car, increase the reception in your radio, anything that pops into your mind.

Never mind what others do; do better than yourself, beat your own record from day to day, and you are a success. - William J.H. Boetcker

No man ever made a great discovery without the exercise of the imagination. - George Henry Lewes

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