Archive for the ‘Great Minds’ Category

Optical Illusions

Wednesday, November 16th, 2011

Only those who attempt the absurd will achieve the impossible. I think it’s in my basement… let me go upstairs and check.”

– M.C. Escher (1898 – 1972)

I have never met anyone who did not enjoy a great magic trick. It is simple to see why. There is nothing more delightful than being playfully fooled while, at the same time, trying to figure out how it could happen so quickly and cleverly. You feel as though your mind’s eye is blinking at the speed of light to catch up to the image or magic in front of you. And it is. To me, optical illusions are magic. Take a good look at the painting by Rob Gonsalves below and you will instantly know what I mean.

Optical Illusion - Cathedral of Commerce

“Cathedral of Commerce”
Copyright Rob Gonsalves (2004)

Optical illusions are enjoyed universally. They can be appreciated on several different levels. From shear fun and silliness to mathematical concepts to fine art, optical illusions have delighted readers from ages 5 to 105. Optical illusions are used in psychology and neuroscience to investigate how our minds interpret different images. It extends into philosophy where errors of perception are studied in terms of the schools of realism and how our minds perceive objects. It occurs in nature when the setting sun mystically becomes three or four times larger while resting on the horizon. In reality, it is the same size as always.

Unless you think that optical illusions are something that mysteriously jumped off the canvas in the 1960s with the paintings of Victor Vasarely, Bridget Riley and M.C. Escher, know that people have been fascinated and entertained by these for thousands of years.

The history of optical illusions has an irregular, up and down pathway to the present. There are long periods of apparent inactivity, or at least little evidence of it in the form of art or architecture. The earliest documented evidence appears in the form of a coin found on the Island of Lesbos around 2500 B.C. In the picture of the metal coin below, you can see an image of two bulls facing each other.   If you take a second look, it can also be perceived as a wolf’s face looking straight ahead.

Optical Illusion - Coin found on Lesbos

© Gianni A. Sarcone,, Italy

Portraying three dimensional objects on a two dimensional canvas is the backbone of both Western art and simple illusions. Taking this one step further, artists began to intentionally deceive the viewer into believing his or her fictional creation was real, however briefly. In essence, you had a “double dip” illusion: three dimensions on a two dimensional canvas with the visual trickery incorporated into that canvas. There is evidence to suggest intentional illusions were first used in ancient Greece and later picked up by the Romans. While other forms of optical illusions became popular beginning in the 14th century, this form of art vanished for a time and reappeared in the Netherlands in the 17th century. This art form came to be known as Trompe l’oeil, a French term meaning “trick the eye.” The ancient Greeks were aware of the effects of illusion in art even through their architecture.  They made the columns of buildings like the Parthenon bulge so that from a distance they would look perfectly straight. They understood that the proper slant of roofs also had an illusory effect on observers by giving the appearance of the building standing more erect than it actually was.

There is little mention of optical illusion art from ancient times through the Dark Ages. It seems to have reappeared in full force during the Renaissance.  This era marks the beginning of many different types of illusory art. During this period, artists began to explore and fully comprehend linear perspective. The artists of this time were the first to see that parallel lines can merge to give the feeling of depth. They used different hues, shading and size of images in a manner that made their paintings more realistic than their predecessors. They explored new techniques to show and distort perspective so that the brain might interpret an image in two completely different ways. In the late 1400s Leonardo da Vinci is given credit for an art form known as anamorphosis or “slant art.”  It requires the viewer to either occupy a certain vantage point to be able to see the image properly or employ the use of reflection in a conical, cylindrical or pyramidal mirror to get the right view. As an example, in its original form on canvas, the image is distorted so significantly that the viewer cannot even offer a guess as to what it is until it is viewed from the perfect angle where the image reveals itself as the artist intended. Anamorphosis became extremely popular and was a means where questionable material or politically sensitive messages could be hidden. It remains quite popular today. An example of this type of art that combines elements of a puzzle with magic is provided below.

Optical Illusion - Anamorphosis

Title and artist unknown. Believed to be from the 18 th century.
From Mathematics. David Bergamini, Time, Inc., New York, 1963. 200 pp. Illus.

The other type of illusory art that became popular during the Renaissance is called “double imagery” or “double meaning” and it was made popular by Giuseppe Arcimboldo, an Italian painter who did most of his work in the mid to late 1500s. He painted portrait heads made from fruits, animals, vegetables, fish and other objects and molded them together in such a manner that the portrait was a highly accurate image of the person. If you moved in for a closer look, you could see the myriad of objects he painted to comprise the portrait. His work was considered so revolutionary that some art critics thought his creations to be a manifestation of mental illness. But Arcimboldo was simply an immensely talented individual who embraced this illusionary art form. Evidence of his sanity exists in the form of his traditional art work.  While his traditional art was lost through the years, his double meaning work reappeared in the early 20th century.  These paintings had a tremendous influence on surrealist artists like Salvador Dali.  He still has a strong influence today.

Optical Illusion - The Librarian Giuseppe Arcimboldo’s “The Librarian” – 1562

During the late 19th and early 20th centuries, double meaning or metamorphic art again became popular and was used extensively on postcards and in advertisements.  Scores of illustrators created these visual puns which often featured dark themes and subject matter including skulls, devils and other taboo topics.  One classic example from George A. Wotherspoon titled “Society, a Portrait” features a gentleman escorting a woman on each of his arms. Closer inspection of the portrait, however, reveals that the man’s chest and arms also resembles the head of a donkey.

Optical Illusion - Society, a Portrait George A. Wotherspoon’s “Society, a Portrait”
Courtesy of and

No other artist has contributed more to the popularity of optical illusions and mathematical art than M.C. Escher. Maurits Cornelis Escher was born at the end of the 19th century in Leeuwarden, Holland. He initially studied architecture but switched to the arts where he became proficient in drawing and wood carving. As a young man in his twenties, he traveled extensively through Europe and during this time visited the Alhambra, a Moorish castle in Spain. He was fascinated how the Moors segmented the plane into repeating periodic shapes with colors. The Moors were forbidden by their religion to use living things in their art, and Escher thought how dynamic congruent patterns of birds, fish, humans and other living things might appear when tiled together. Thus began his work in tessellations, the symmetrical way of tiling a plane, and the broad spectrum of different and sometimes shocking images he would go on to create during his lifetime.  Escher was one of the first artists to explore the art concept of infinity on a plane in detail – more than parallel lines disappearing on the horizon. This led to his famous Circle Limit series of wood engravings which became some of his most popular work.

By the 1950s M.C. Escher had become famous in Europe but was still relatively unknown in the United States. He wrote The Regular Division of the Plane in 1958, which discussed some of the math behind his work. This is when he commented that, “Mathematicians have opened the gate leading to an extensive domain.” His work was the first in history to build the bridge between mathematics and art, fortified by solid mathematical constructs. The beauty of all this is that an individual can enjoy the fun and pleasure of his creations and explore the mathematical applications at the same time. Symmetry, periodic tiling of the plane, platonic solids, hyperbolic geometry, topology, group theory and crystallography can all be found and studied in Escher’s diverse body of work.

Escher’s popularity was just reaching critical mass when he died in 1972. Today he is more popular than he ever was during his life. Bookmarks, calendars, T – shirts, posters and more can readily be found across the retail landscape in virtually every country.

In the 1960s the term “op art” was coined in an article appearing in Time magazine. This came to be the catch-all phrase for optical art and illusions of all kinds in the years following. After a major exhibit at the Museum of Modern Art in New York, which received significant coverage in 1965, op art became hugely popular with the general public. Images of op art were seen on television, in newspapers, graffiti, album covers, clothing and advertising of all kinds. Hungarian born Victor Vasarely is often cited as the father of this movement with Bridget Riley, Yaacov Agam and Josef Albers as other notables. Salvador Dali, although known by most as a surrealist painter, created some excellent op art paintings that are still popular today.  Much detail has been given to discussing the aspects of what constitutes op art, but the number one feature is trompe l’oeil. Op art is a form of abstract art where lines, space and color are integrated into symmetrical forms. The lines are simple and repetitive with sometimes unusual shading and coloring that give the art its beauty and excitement. Often, some of the art seems to have an almost uneasy feeling of depth while at the same time possessing a pleasing sense of simplistic beauty.

Op art saturated everything for about 5 years and even though its intensity leveled off, it is still popular today. One of the reasons is that computers are a perfect medium for designing and creating op art style imagery. One can see evidence of this on the Internet and on many websites that use op art as logos. Of course, the masters did their creations by hand, which is amazing when you consider what goes into creating even one of these images.

Today’s optical illusion landscape is an amalgamation of the aforementioned types along with a new generation of artists and their creations. With the ability to now create illusions using a computer and a variety of commercially available software, they take on a new dimension not available to those in the past.  Static images can now be made to move, or at least appear to move as we all know that a photograph cannot actually be in motion, right?  Perhaps you should judge for yourself.

Optical Illusion - courtesy of Paul Nasca Courtesy of Paul Nasca (2010)

One of the interesting aspects of current day op art is that there is even more emphasis on illusions and their interrelationship with math, science, neuroscience, graphic design and the whole broad spectrum of the arts. As evidence of this, the 2010 winner of the Best Illusion of the Year Contest is a mathematical engineer who created a now well-known illusion of balls rolling uphill. An article appearing in the May/June 2011 issue of Scientific American MIND discusses the contest and offers the following: “Whereas scientists once created illusions from simple lines and shapes and artists focused on making eye-popping illusions, the overlap between science and art is now greater than ever. Scientists are using graphic design tools to make their illusions more artistic, and artists have grown more knowledgeable about the neuroscience behind the magic.”

During the 1970’s, two men simultaneously (but unknown to each other at the time) launched a barrage of clever design creations on the American public.  Scott Kim and John Langdon each believed they had invented these designs independently of one another.  These designs would later be dubbed “ambigrams” in the early 1980s by Douglas R. Hofstadter.

An ambigram is a form of art that presents two or more different words within the same physical space through a mix of illusion and symmetry.  The different words can be viewed by looking at the design from another perspective (such as upside down or reversed).  There are several different types of ambigrams, but popular categories of this art include rotational ambigrams (a design that can be rotated by some angle to present a different instance of a word), mirror image ambigrams (a design that can be read when reflected in a mirror) and figure-ground ambigrams (a design that has one word embedded within the spaces of another word).

Peter Newell is given credit as the creator of the first ambigram – his designs were pictures that could be inverted.  His book “Topsys & Turvys”, released in 1893 featured pictures on each page that could be turned upside down to reveal a completely different (but recognizable) image.  The last page of his book featured a drawing that reads “THE END” when viewed in one direction and “PUZZLE” when viewed upside down. These types of puzzles remained popular in both England and the United States throughout the early 1900’s.  Books, postcards, magazines and posters were dedicated to this genre and enjoyed by both kids and adults.

Optical Illusion - Topsys and Turvys Ambigram Peter Newell – Ambigram from Topsys and Turvys (1893)

Kim was the first to put motion to his ambigrams by spinning several of his creations to reveal their duality.  Langdon made ambigrams even more popular when his designs were incorporated into the plot of Dan Brown’s bestselling novel Angels & Demons.  Brown thought so highly of Langdon he named the hero of his book Robert Langdon in his honor.  Originally, both men made all their designs by hand. This would change with the increased use of computers and it was a natural evolution for both men to augment even more brilliant creations using digital enhancements.  Computers have allowed this type of illusion to have an unlimited panorama for new creations and one can only imagine what lies ahead.

Optical Illusion - Teach-Learn Scott Kim’s “Teach-Learn” (2000)

It should go without saying that artists will continue to push the boundaries of illusionary effects going forward.  In fifty years time, an entirely new breed of illusions may exist that we never knew to even be possible.  And people will love them because we all enjoy being fooled.  Some illusionists are mathematicians fascinated with symmetry and some are vision researchers in search of answers.  Others are graphics designers and painters who desire to push the boundaries of what can be done on a flat surface or canvas.  Ultimately, however, they are all masters at creating art that both fascinates and deceives.

Terry Stickels

Fort Worth, Texas


1) Mathematical Circus by Martin Gardner, Knopf Publishing, 1979. Chapter 1, Optical Illusions.

2) The Playful Eye, by Julian Rothstein and Mel Gooding, Chronicle Books LLC, 2000

3) Scientific American MIND, 10 TOP ILLUSIONS by Susana Martinez – Conde and Stephen L. Macknick, May/June 2011

4) Masters of Deception by Al Seckel, Sterling Publishing, 2004

5) Visual Illusions in Art and Science by Dr. Pascale Michelon, appearing in Blog of SharpBrains, February 11, 2011 ( article includes an excerpt of Sleights of Minds: What the Neuroscience of Magic Reveals about Our Everyday Deceptions by Stephen L. Macnick and Susana Martinez – Conde, published by Henry Holt and Co., LLC, 2010 )

6) M.C. Escher – The Official Website,

7) M.C. Escher – Life and Work, THE COLLECTION, National Gallery of Art,

8) Giuseppe Arcimboldo – The Complete Works,

9) Victor Vasarely  Biography- Leader of the Op Art Movement, Masterworks Fine Art,

10) Brainy Quote,


12) Ambigram Magazine –


Thursday, January 6th, 2011

Imagine you are the head of Human Resources at a large, successful high tech company. Your job depends on hiring the best and brightest minds in the world to produce new software and related products at an almost exponential pace. Your personal success depends on not only hiring better candidates than the competition, it must be accomplished within budget guidelines. You need employees who are “can’t miss” winners. Management has given you free reign to find these people.  A reliable system must be in place to identify exceptional employees.

How are you going to proceed?

One way is through the use of puzzles. What? Puzzles? Since the early 2000s more and more companies, high tech companies initially and companies across the spectrum now , are using puzzle questions to identify future employees who can demonstrate critical thinking at the highest levels. Microsoft was one of the first large corporations to use puzzles to identify critical thinking skills. While that may be true but many smaller companies were already zeroing in on this method using logic-oriented questions which were one step away from puzzles. These tests were standardized; puzzle tests are not. The theory is if puzzles are constructed correctly, you will know all you need about the critical thinking abilities of the solver. Companies also utilize these types of questions to note the emotional responses of how candidates respond under pressure – intense focus required under strict time limitations. The other interesting aspect of this process is companies are dropping some of the usual criteria previously required for one to land a job. For the types of positions requiring large brains, they are not worried about social skills and from what college the applicant graduated. They want high-end thinkers. Period.

How did this evolve?  The answer may be as easy as saying Bill Gates and the “Petals Around the Rose.” At one of the early computer conferences in 1977 Gates and Paul Allen, co – founders of Microsoft, were intrigued by the “ Petals ” game. A group of fellow computer attendees were headed home with Gates and Allen when they introduced the game to the pair. Gates was said to be so fascinated he would not leave the puzzle alone until he found the solution. Gates saw a whole new use for puzzles from this day forward. Want to see how you fare with the puzzle? A more detailed look at the puzzle can be found at

Now, many companies use puzzles for job applicants and it has proved to be a reliable identification process. It has become big enough business that  author William Poundstone wrote a book titled How Would You Move Mount Fuji? Microsoft’s Cult of the Puzzle: How the World’s Smartest Companies Select the Most Creative Thinkers. (Little, Brown and Company, 2003). In this fun, informative book, Poundstone not only gives a history of this type of thinking behind the interviews but puzzles with possible solutions in the Answer Section. Notice I said “possible” answers. The reason for that is the puzzles used by companies seem to fall into two categories. One set of questions is more open-ended and allows for the candidates to discuss possible options and approaches for their solutions. Sometimes these questions seem impossible to answer.

Difficult Interview Brainteasers

Here are examples of these more “ open ended “ type questions found on numerous sites on the internet: 1) You are locked in an empty room with a working refrigerator as the only object in the room with you. The room is very warm and you are looking for a way to cool the room. What can you do? 2) What weighs more on the moon than on earth? How many garbage men are there in California? 3) And, of course, the book title question – How would you move Mount Fuji? (solutions are at the end of this article) These are not trivial. These are real questions and the blinking cursor is shining on the job applicants to unfurl their brilliance in front of people who might employ them and at a good salary.

The second group of questions are puzzles which separate the pretenders from the contenders. These are difficult, logic-based brainteasers with one answer and one answer only. No prior education or training is required, just the individual’s deductive/inductive reasoning abilities but under pressure. These are puzzles few people can answer and the process of solving them is just as important to the interviewer as the solutions. In other words, it might be okay to arrive at the wrong solution but if you processed the information in the right direction, it counts favorably. There are examples of these puzzles on the internet: actual puzzles that appeared on some of the interview tests. The problem with that is twofold. First, those puzzles are now removed from the interviewing process at all companies since the companies are quite aware the public has easy access to them. Secondly, the internet sites use the same handful of example puzzles again and again – so once you’ve solved them, you’ll need more. Practice, practice, practice. It’s no different than Peyton Manning working on his passing game for hours. The more puzzles you see and attempt, the better chance you have reaching their solutions.  It also helps to gain confidence and become more relaxed in a testing environment.

For those interested in working on more difficult puzzles, there are some good resources.  Several internet sites provide excellent material. has a broad spectrum of brainteasers of different varieties and at different levels. These are good “ warm – up “ puzzles. Macalester College has a math forum with archived puzzles and math problems which  are world class ( Nick’s Mathematical Games is an excellent resource ( as is with puzzles by Mike Shackleford ( Typing Difficult math brainteasers or or  just plain Difficult Brainteasers into any of the search engines will provide some good sites.

There are additional resources and books listed at the end of this article.

To help the cause, I have created new brainteasers presented here for the first time. These are in line with the types you will encounter in an actual interview. No higher mathematics is required.

Stickels’ Original Brainteasers

Here are five puzzles for you to consider.  I’ll give the answers with explanations at the end of the article. Try your luck at these to see if Bill Gates will be knocking on your door. Here’s a tip that may help. Approach these in the spirit of challenging fun and take the attitude that these puzzles are not bigger than you  . . . or your mind. In other words, you want to be a “ mind ” warrior and conquer these. Do not accept defeat. You’ll need that type of attitude during an interview. Have fun.

Puzzle 1
You die and the devil says he’ll let you go to heaven if you beat him in a game. He sits you down at a round table. He divides a large pile of quarters in two so you both have the same amount of quarters in your own pile. He says, “Ok, we’ll take turns putting quarters down, no overlapping allowed, and the quarters must rest on the table surface. The first guy who can’t put a quarter down loses. You cannot shift or try to squeeze any quarter into a space that moves another quarter.” The devil says he wants to go first.

You realize if the devil goes first, he probably has a strategy to win. You haven’t had time to think this through yet, so you ask for some time to consider your options. He grants you 15 minutes. At that end of that time you know how to beat him…but you also know you must go first for your strategy to win. You convince him to let you go first, saying you really didn’t have enough time to consider a strategy so you should at least go first. What is your winning strategy?

Puzzle 2
There’s an old puzzle that asks, “If a hen and a half lays an egg and a half in a day and a half, how many hens will it take to lay 6 eggs in 6 days? The answer is 1.5 hens. Here’s the new puzzle: at this rate, how many eggs will one hen lay in one day?

Puzzle 3
You are standing outside a 30-story building holding two identical glass spheres. You are told that either sphere, if dropped from the roof, would shatter upon hitting the earth, but that it would not necessarily break if dropped from the 1st story. Your task is to identify the lowest possible floor from which you can drop a ball and break it.

In the general case, what is the smallest number of drops required to guarantee that you have identified the lowest story? NOTES: 1) Both balls have the same minimum breakage story; 2) You only have two balls to use. If one breaks, it cannot be used for the rest of the experiment.

Puzzle 4
If the stack of cubes below was originally 5 x 5 x 5, is A, B, C, D or E the missing piece from the broken cube? Note: All rows and columns run to completion unless you actually see them end. The blocks seen in the five choices are to be placed back onto the cube upside down.

Puzzle 5
Using the numbers 5, 6, 7, and 8 once and only once, what is the largest number that can be created by “stacking” the numbers in exponential form?


You may stack the numbers any way you choose in any combination to achieve the largest number.

What is the second largest number?

Answers to “impossible” interview brain teasers:

1: The refrigerator puzzle
If you said open the door to cool the room, it wouldn’t be the best answer. In fact this will make the room even warmer by sucking the heat of the room into its cooling compartment and then pumping it through the coils on its back. A refrigerator is really a heat pump and it takes the heat away from the refrigerator and pumps it into the room. One good response would be to unplug the refrigerator and then open the door. It won’t result in a big cool down but it is the best you can do. This is a stretch but if you can move the refrigerator ( easily ) to a location in the room where the rising heat will move toward the vents higher on the wall, this will help a little.

2: The weight of an object on earth vs. moon puzzle
One of the keys here is to realize the earth has an atmosphere and the moon does not. That means that a balloon filled with helium would weigh nothing on earth but since there is no atmosphere on the moon, it would have a weight. Don’t get weight and mass confused here. The mass doesn’t change but the weight does. There is another aspect of this puzzle that might give the job candidate extra credit. If he or she said that the experiment would be impossible to do, you would probably be held in high regard. The reason is no inflatable balloon can exist in a vacuum and would explode immediately when introduced into that vacuum.

3: Garbage men in California
If you tried all kinds of mathematical or arithmetic gyration to come up with the answer, you are probably on the wrong track. One answer would be to call the state agency overseeing garbage collectors and ask them. Another good response is to call to ascertain if they are represented by unions. If so, call the union headquarters and ask them. Finally, if you are really quick on your feet and understand the importance of humor you might just give him or her a number. Remember, they are looking for an answer. Your position should be you are going to give them an answer. Give them a “ballpark” number and then say nothing more. If the interviewer asks how you arrived at your figure, your response should be (with a smile on your face), “When you hire me, I’ll tell you.” Human resource professionals love well-placed confidence.

4: How would you move Mount Fuji?
Read the book. I’m not going to take away Mr. Poundstone’s thunder.

Answers to the “five original puzzles”:

Puzzle #1
Put the first quarter exactly in the center of the (circular) table.

Next, for each quarter the opponent places, place one directly opposite it. That is, place it so that the center of the table is halfway between your piece and the opponent’s previous piece.

This will generate a completely symmetric layout of quarters on the table. This means that whenever the opponent selects a free space to place a quarter in, the space opposite is guaranteed to be free as well. Since you are always guaranteed an open space, you will never lose with this strategy.

Puzzle 2
The rate per hen per day is 2/3 egg. If a hen and a half lays an egg and a half in a day and a half, then one hen lays 1 egg in 1.5 days. 1.5 days is 3/2 days.

a) 1 hen X 3/2 days X rate per hen per day = 1 egg
b) 3/2 rate per hen per day = 1 egg
c) Multiply each side by 2/3…

Rate per hen per day = 2/3 egg

Puzzle 3
To test this, start from floor number 8 and drop a ball. If it doesn’t break then go to floor 15. If it doesn’t break there go to floor 21. If it doesn’t break there, go to floor 26. If it doesn’t break there, go to floor 30. If it doesn’t break there you have to test floors 27, 28, and 29. You have taken 8 drops to test this.

Now, take the other case: suppose you drop one ball on floor 8 and it breaks. You now have to test floors 1 through 7 to find out if that ball will break. That, also, totals 8 as does each testing stage between any of the two consecutive floors you’ve chosen. Notice how the difference between each set of consecutive floors decreases by 1…8, 15, 21, 26 and 30? An easy way to find the minimum number of drops for any floor is to add 1+ 2 + 3 + 4 +…until the sum reaches the number of the top floor. As I demonstrated in the first sentence, once the number has been met (even if the last added number surpasses the top floor number when adding it, it is taken to that whole number to determine the minimum number of attempts) that is the solution: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36 so the threshold of 30 has been passed after the 8 is added. Therefore, it takes a minimum of 8 drops.

Puzzle 4
Figure D is the missing piece from the broken cube.

Puzzle 5
The largest number is approximately 10 to the 4.5 millionth power. It is the largest number by a long way.

1) 5^6^7^8

2) 6^5^7^8

Further reading:

William Poundstone is the author of over 11 books. A few of his most popular are Carl Sagen: ALlife in the Cosmos ( Henry Holt Publishing, 1999 ) . Prisoner’s Dilemma, John von Neumann, Game Theory, and the Puzzle of the Bomb ( Doubleday Publishing, 1992 ) Labyrinths of Reason, Paradox, Puzzles, and the Fraility of Reason ( Doubleday Publishing, 1988 ) and Big Secrets, The Uncensored Truth About All Sorts of Stuff You Are Never Supposed to Know). He has written for Esquire, Harper’s, the Economist and has been nominated twice for the Pulitzer Prize.

More books:

John Kador’s HOW TO ACE THE BRAIN TEASER INTERVIEW ( by McGraw – Hill, 2004 )

ADVANCED APTITUDE TESTS by Jim Barrett and HOW TO PASS ADVANCED APTITUDE TESTS by Jim Barrett (both by Kogan Page Publishing. The first was published in 2004. The second was published in 2002)


BRAIN BUILDING IN JUST 12 WEEKS by Marilyn vos Savant and Leonore Fleischer, published by Bantam Books, 1990

Two books not specifically designed for interviews but can be used for mental warm ups for sharpening spatial/visual skills –

OFFICER CANDIDATE SCHOOL TESTS by Rod Powers ( Barron’s Educational Series, Inc. 2006)


Two books by Martin Gardner – Aha! Insight published by Scientific American, 1978 and Aha! Gotcha, published by W.H. Freeman, 1982 ( originally Scientific American, Inc. 1975)

View more resources under the heading of LINKS.

The Great Book of Mind Teasers & Mind Puzzlers by George J. Summers, published by Sterling Publishing. (The Great Book of Mind Teasers & Mind Puzzlers is a combined edition of Mind Teasers, 1977, and Mind Puzzlers, 1984, both published by Sterling Publishing and both written by George J. Summers ).

Dover Publications has a catalog of some of the older puzzle books that can be viewed and ordered at Many of these puzzle books offer excellent brainteasers.

Puzzle Power! Mindstretch

Monday, September 27th, 2010

The most important capacity we possess that separates us from our peers and competitors is our ability to think.  The way we express that thinking often says volumes about us from the first sentence on.  Not only are we judged immediately by the way we communicate, people inevitably start drawing conclusions about our thinking aptitude.  It stands to reason, then, that the better our thinking skills, the better chance we stand to be successful.  This naturally prompts the question, “Is there anything we can do to increase or enhance our thinking skills and mental flexibility?”

The answer is yes, there is.  Recent findings in the field of neuroscience and cognition conclusively show that new, more powerful brain connections can be created with “mind-stretching” activities.  And what might those be?  Interestingly, some of the strongest evidence finds games and puzzles at the top of the list.

One of the best examples of this can be found in Minnesota with the nuns of Mankato, many of whom are now in their eighties and nineties.  These remarkable women, featured in Time Magazine, are energetic, bright, and in excellent physical condition. They follow a rigorous routine of both mental and physical exercise.  Two of their favorite activities they credit for their mental acuity are games and puzzles.  Not only have they slowed down the aging process, but they keep adding to their mental acumen and flexibility.

The ongoing research in the cognition field by notable scientists such as Drs. Denise Park, Fred Gage, Jeffrey Macklis, and John Ratey points clearly to the power of actively exercising your brain in much the same manner as we work out physically: fun challenges from different approaches.

“Brain cells actually thicken when you solve puzzles and play games.”
-Gene Cohen, Former Director
National Institute of Mental Health Center on Aging

The idea of solving puzzles to increase your mental abilities is not a new concept—and   it has been taken seriously by several heavyweight thinkers.  For example, Charles Sanders Peirce, considered by many to be one of America’s great philosophers and mathematicians, was convinced that insufficient teaching methods used in schools might end up categorizing some bright children as poor students.  His solution was to introduce puzzles into the curriculum.  He filled three notebooks with novel ways of using puzzles, games, and toys to introduce various concepts.  He often asked teachers to let him instruct a group of youngsters who detested mathematics and seemed incapable of learning it.  In one case, two of those students went on to lead their school as the best mathematicians—after a mere ten lessons.

” A good math puzzle, paradox, or magic trick can stimulate a child’s imagination much faster than a practical application  . . .  and if the game is chosen carefully, it can lead almost effortlessly into significant mathematical ideas.”
– Martin Gardner

“In a recent study it was found that adults with hobbies that exercise their brains—such as reading, puzzles, and games such as chess—are 2.5 times less likely to have Alzheimer’s disease.”
-Dr. Robert P. Friedland, Associate Professor Neurology
Case Western Reserve University, School of Medicine

Many researchers now believe that education is less important in maintaining a healthy brain than the habit of staying mentally active as you age. A 2003 study reported an association between frequent participation in cognitively stimulating activities ( such as reading, doing puzzles, visiting museums ) and a reduced risk for Alzheimers.

Dr. John Ratey’s book, A User’s Guide to the Brain, offers compelling evidence supporting the efficacy of using puzzles and games to boost mental flexibility.  He writes, “Activities that challenge your brain actually expand the number and strength of neural connections devoted to the skill.”  He goes on to say, “We always have the ability to remodel our brains.  To change the wiring in one skill, you must engage in some activity that is unfamiliar–novel to you but related to that skill–because simply repeating the same activity only maintains already established connections.  To bolster his creative circuitry, Albert Einstein played the violin.  Winston Churchill painted landscapes.  You can try puzzles to strengthen connections involved with spatial skills….”

Progressive organizations are now recognizing the benefits of a workforce of critical and creative thinkers.  Author and lecturer Michael Michalko’s best-selling book, Thinkertoys, has been labeled “the business book of the nineties.”  What sets it apart from other business books?  It is the use of Michalko’s Thinkertoys—activities in puzzle formats that steer the brain into discovering new ideas, offering alternative approaches to decision making, and increasing one’s creativity.

Such corporations as AT&T, PricewaterhouseCoopers, IBM, United Airlines, Chevron, Intel, and Liberty Mutual now hold problem-solving sessions utilizing puzzles of all varieties: logic, spatial/visual, word puzzles, and board games.  These companies realize the importance of the skills and tools these puzzles bring, by engaging participants in rigorous, thought-provoking activities.

More than ever, America’s corporations are demanding better critical and creative thinkers.  Having impressive degrees from prestigious universities is not the main prerequisite for success as it once was.  Thinking skills are.  Winston Churchill was on the money with his famous quote, “All the great empires of the future will be empires of the mind.”

And recent research continues to support the thesis that “exercising the brain” is vital to maintaining mental functions at a high level.  Three examples:

  • Neuropsychologist Elkhonon Goldberg, author of The Wisdom of Paradox. has developed a mental fitness program for his aging patients, with computer puzzles for strengthening the mind’s acuity and “muscles”—with positive results.
  • In his book, Train Your Brain, Japanese neurologist Ryuta Kawashima says the key to staving off negative mental effects of aging is a daily five-minute regimen of reading aloud and doing simple math problems.  The idea is to “work out the brain much like we work out the rest of our bodies.”
  • Subjects in a UCLA study showed substantially improved metabolic changes in the brain area linked to working memory when they made four simple lifestyle changes:
    1. Eating five small meals rich in Omega-3 fatty acids, whole grains, and antioxidants
    2. Brisk walks
    3. Stretching and relaxation exercises to counter stress
    4. Brainteasers and games

In summary, you will find that as you solve puzzles, your mind’s flexibility will indeed improve.  And that the more you do, the better problem solver you will become.  And remember this essential fact: your overall success in life will depend in large measure on your ability to think creatively in dealing with the wide-ranging problems that lie in store for each of us—every day!

Author Terry Stickels is an internationally acclaimed creator of puzzles and speaker on mental flexibility.  His work appears in papers and magazine worldwide and he has authored 27 books.  For more information, he can be reached at his website: or 817-542-0102.

Below are some tips and thoughts on thinking you may enjoy  . . .   make it part of your Thinker’s Tool Kit.

Addendum: Tips and Quotes

Tip: Keep learning and thinking in as many different areas as possible.  Make it a point to read an article on a topic you’ve previously had no interest in…exercising new connections for the mind.

Quote: “As with our muscles, we can strengthen our neural pathways with brain exercise, or we can let them wither.  The principle is the same: Use it or lose it.”
-Dr. John Ratey
From A User’s Guide to the Brain

Tip: Solutions can often be reached just as easily by eliminating extraneous choices.

Quote: “It is an old maxim of mine that when you have excluded the impossible, whatever remains, however (apparently) impossible, must be the truth.”
Sherlock Holmes

Characteristic of Great Thinkers:
Great thinkers are always looking for the next mental challenge, even if it means repeated failures. They have an ability to let criticism and failures fall off their backs.

Quote: “I think and think for months and years.  Ninety-nine times the conclusion is false, the hundredth time I am right.”
Albert Einstein

Characteristic of Great Thinkers:
It’s not a coincidence that clever minds also seem to have a quick wit.  It seems high level thinkers, from scientists to comedians, are able to generate a different perspective on things that can result in some hilarious humor.  Don’t take yourself so seriously that you can’t laugh at your own expense now and then.  The legendary math and science writer, Martin Gardner, writes, “Psychologists are not sure, but studies of creative thinking suggest some sort of relationship between (mental agility) and humor.”

Quote: “I would never join a club that would have me as a member.”
Groucho Marx

Characteristic of Great Thinkers:
Many, if not most, great thinkers have taken their time in arriving at solutions.  In our fast-paced world, you may want to grant more leeway to those who take a little longer to arrive at a solution that affects you.  It may pay big rewards.

Quote: “The strongest of all warriors are these two—time and patience.”
Leo Tolstoy

Tip: Here are some ways top-notch problem solvers look at a new mental challenge:  Turn the problem inside out, upside down, and sideways. Spin, twist, stretch, and break it down. Play with it (both literally and figuratively).  Put the problem into different frames of reference to view it.

Quote: “Innovation comes from creative destruction.”
Yoshihisa Tabuci, CEO
Normura Securities

Characteristic of Great Thinkers:
When closing in on a solution, keep your emotions and personal prejudices out of it—and check your ego at the front door.

If the solution requires an emotional component, the proper weighting of that can be assigned at the end of the solution process.  Work to keep an open mind.

Quote: “The most characteristic feature of stupidity is not the ability to think, or lack of knowledge, but the certainty with which ideas are held.”
Dr. Edward deBono

Tip: Make a written legend of the data, facts, and information you are given.  Put it into some semblance or order that is easy for you to manipulate.  The mere act of writing or typing can often trigger the mind into action.

Quote: “The best way to have consequential thoughts is to write them down.”
E.B. White

Tip: When everything fails in arriving at a solution, try one or all of the following:
1.    Get completely away from the problem temporarily.
2.    Talk it out with someone else (it worked for Sherlock).
3.    Go for a long walk.

In regard to Tip #3, there are several stories of great thinkers having that AHA! moment during a long walk (physicists Einstein and Penrose being two).

The brain seems to be able to subconsciously process problem information in a more relaxed manner while engaged in another activity.

As a side note, the stories of “walkers” have two interesting features to them: most of the walking was done at night and alone.

Quote: “The creative process cannot be summoned at will or even cajoled by sacrificial offering.  Indeed, it seems to occur most readily when the (logical) mind is replaced, and the imagination is roaming freely.”
Morris Kline, Scientific American

Tip: Unless you are a columnist or television commentator, leave cynicism alone.  However, healthy, respectful skepticism is a must for growth and success.  In business, for example, if a proposition is not well grounded and supported by logic and evidence, be skeptical…and be ready to seek out new approaches.

Quote: “In order to seek truth, it is necessary once in a course of our life to doubt as far as possible all things.”
Rene Descartes

Characteristic of Great Thinkers:
They are never content with just solving the current problem at hand.  They mentally keep creating new alternatives, and are constantly asking the question “what if”?  They have an insatiable desire to always go to the next level.

Quote: “Opportunity does not come to those who wait.  It is captured by those who attack.”
General Douglas McArthur

Tip: Don’t let technology be a roadblock in your thinking pursuits.  Brainpower and flexibility are ultimately your best tools.  Remember that Einstein’s “technology” was pencil on paper.

Quote: “You already possess everything necessary to become great.”
Crow Indian Proverb

Tip: After you’ve reached a solution, always plug your answer back into the problem to see if it fits all the parameters you were asked to consider. Then put different solutions back into the problem…for checks and balances.  You may find more than one answer.

If you are having difficulty arriving at any solution, try a cross section of arbitrary solutions.  That may help to narrow the range of possibilities.

Quote: “The most erroneous stories are those we think we know best—and therefore never scrutinize or question.”
Stephen Jay Gould

Characteristic of Great Thinkers:
Great thinkers prefer to do their most serious thinking alone.  This may seem at odds in the business world, where committees, boards, and teams are prevalent and encouraged.  The smart organization realizes there is a way to accommodate both situations with great advantage to their strategic initiatives.

Quote: “My idea of a board meeting is when I see my face in the mirror when I shave.”
Warren Buffett

Characteristic of Great Thinkers:
Sometimes, people with great minds are seen as unreasonable.  They have thought out a situation through more levels than their associates, causing those around them to draw inaccurate conclusions about them and their solutions.  So be it.  To them I say: Keep thinking and forging ahead of the pack!

Quote: “The reasonable man adapts himself to the world.  The unreasonable man persists in trying to adapt the world to himself.  Therefore, all progress depends on the unreasonable man.”
George Bernard Shaw

Martin Gardner: America’s Mathemagician

Monday, June 7th, 2010

By Terry Stickels – June 7, 2010

When he was writing his Mathematical Games column for Scientific American magazine, the Washington Post called his office “the mathematical center of the earth.” His book Relativity for the Million (later reprinted as The Relativity Explosion, Random House, 1976) is still considered to be the best explanation of Einstein’s theories. Yet, consider this: He’s never taken a college course in mathematics or physics.

His expertise is not “limited” to the numbers and symbols that envelop our universe; his mental dexterity has led him to write brilliantly in the fields of magic, puzzles, philosophy, pseudoscience, and children’s literature. There are few, if any, limits on the talents of omniheurist (solver of all problems) and American treasure Martin Gardner. In all, there are 80 plus books, countless articles and columns, each meticulously researched and perspicuously written, that stand as a glowing testament to his lifelong pursuit of knowledge, truth – and fun.

Growing Up
Martin Gardner was one of three children born to Dr. James and Willie Wilkerson Spiers Gardner of Tulsa, Oklahoma, on October 21, 1914. His mother was a devout Orthodox Methodist and the lessons of the church were not lost on young Martin. His formal Protestant upbringing would lead to a lifetime fascination with religion.

Gardner’s father was a geologist and lover of magic, so it followed naturally that Martin became fascinated with science and magic at an early age. By the time Gardner was in high school, he was a regular contributor to The Sphinx, a magazine dedicated to magic.

The College Years
Because of his love of math and physics, Gardner’s original plans were to attend California Institute of Technology, where he intended to major in physics. Because of a chain of events, most of which were out of his control, Gardner instead majored in philosophy at the University of Chicago.

He was an excellent student, graduating Phi Beta Kappa in 1936. His experiences and studies during these years would lay the foundation of what would be a relentless dedication to the scientific method: seeking conclusions based on logic and evidence.

It was also during this period that Gardner began to question Methodist fundamentalism; those long-held views were crashing head-on into the scientific philosophy he found at the university. Out of this experience would come one of his most critically acclaimed books, The Flight of Peter Fromm, (William Kaufman, 1973). The healthy skepticism he developed during his undergraduate days would serve as one of the driving forces of his literary life.

The Beginning of a Great Writer
After graduation, he worked for a short time as a writer for the Tulsa Tribune and then in the public relations department at the University of Chicago.

In 1941, he enlisted in the Navy and was a yeoman on a destroyer escort until the end of World War 2. After the war, Gardner returned to the University of Chicago, took an occasional graduate course in philosophy, and began his career as a freelance writer. He sold his first humorous story to Esquire magazine. Red Skelton happened to read it. When Skelton mentioned the column on the air, Gardner’s literary career was out of the starting blocks. Esquire bought a dozen of Gardner’s stories over the next few years, and he moved to New York, where he could have ready access to the excellent research material found in the city’s main public library.

New York would be Gardner’s home until 1981. He was married to Charlotte Greenwald in 1952, and they raised two sons in their home on the Hudson. In the early ’50s, Gardner’s boyish enthusiasm and sense of wonderment, guarded by his keen mind, led him to explore a broadening spectrum of interests. With his skeptic’s drawn sword, he found the world of pseudoscience a perfect enemy to engage in battle.

Martin’s 1st Book
A literary agent talked him into writing his first book, Fads and Fallacies in the Name of Science (originally named In the Name of Science, republished by its current name in 1957 by Dover). The book is a tough yet amusing attack on pseudo-scientific topics such as ufology, phrenology, ESP, and dianetics. Gardner would write several more books on pseudoscience and eventually become one of the founding fathers of the Committee for the Scientific Investigation of Claims of the Paranormal, which publishes a quarterly titled The Skeptical Inquirer. Composed of magicians and scientists, the group made huge strides into debunking many of the claims made by “psychics.” The committee is still active in those endeavors today.

At the time Gardner was writing Fads and Fallacies, he was also contributing editor for the children’s magazine Humpty Dumpty. He spent eight happy years with the magazine and discovered that his love of word play was equal to that of math and science. It would lead to more than 20 books, essays, and light verse. Gardner wrote stories, poems, and games for Humpty Dumpty. His claim to fame, though, was the “cutout” pages he created. He encouraged his young readers to damage the pages with ingenious exercises involving coloring, pasting, folding, and “cutouts.” Little did he know that this desire to wreck paper would lead to a 25-year relationship with Scientific American.

Gardner and Scientific American
Gardner had written one article for Scientific American on Logic Machines in 1952, featuring a cardboard punch-out sheet. His big break came in 1956 when he wrote a story for Scientific American on Hexaflexagons, structures folded out of paper. The inventors of the flexagons were a group of Princeton students that included an eventual Nobel Prize winner, physicist Richard Feynman. The publisher of Scientific American, Gerry Piel, was impressed. He asked Gardner if he had enough material to write a monthly column on recreational mathematics. Gardner assured him that he had plenty of material, and a columnist was born.

The only problem was Gardner did not own a single math book. He rushed out and bought as many books on “fun math” as he could and subscribed to a dozen math journals. “The fact that I don’t have any formal training in math has worked to my advantage. If I don’t understand what I’m writing about, my readers won’t either. I have to do a lot of work on each column,” Gardner said. Gardner retired from Scientific American in 1981. His entire work for the magazine now resides in a 14-volume collection.

Annotating Alice
In the 1960s and ’70s, Gardner’s prolific writing produced books and articles at a seemingly exponential rate. In addition to his math games and science books, he started a series of “annotated” works. His most successful, The Annotated Alice, (C.N. Potter, 1960), is a footnoted commentary on Lewis Carroll’s Alice’s Adventures in Wonderland and Through the Looking-Glass. Fifty plus years after its release, it still sells more than 40,000 copies a year.

In a 1981 interview with Omni magazine, Gardner said, “I have always found Lewis Carroll a fascinating person. He was a professional mathematician whose books are filled with all kinds of logical and mathematical jokes. The chapters of Through the Looking-Glass actually are based on a series of chess moves.” This led to The Annotated Casey at the Bat (C.N. Potter, 1967), which gained widespread fame when it was reprinted in Sports Illustrated. To date. there are eight annotated Gardner books, the latest being The Annotated Night Before Christmas (Random House, 1991).

Fun through Anonymity
As if these efforts were not prodigious enough, there is more. Apparently, Gardner has written other books both anonymously and under pseudonyms. Only the author knows for certain the reasons for this, but given Gardner’s penchant for play, one might suspect he’s having a little fun.

In Confessions of a Psychic, an “anonymously” written booklet, some of “psychic” Uri Geller’s secrets are revealed. In Gardner’s “Science: Good, Bad and Bogus” (Prometheus, 1981), he refers to Confessions in a footnote stating, “It contains the most detailed explanations to date of Geller’s methods.”

More Than Just a Writer
Through the years, Gardner has become friends with the leading mathematicians, physicists, and magicians of the world, often seeking their assistance and offering his. His devotees are legion. Conferences are held to discuss his work, but Gardner still considers himself an amateur. He once turned down the “Scientist of the Year” award from a prestigious university, remarking, “They were fooled by my writing into thinking I’m a highly trained, true mathematician.”

Although Gardner may consider himself an amateur, there is ample evidence to suggest otherwise. Example: In the early 1950s, Gardner wrote a paper titled “Is Nature Ambidextrous?” In this paper, he “considered the possibility that someday a basic law of nature might prove to be left-right asymmetric.” Five years later, his hypothesis was proven to be true, which led him to write “an entire book about mirror­ reflection symmetry” (now published as The New Ambidextrous Universe, W.H. Freeman).

Staying Out of the Limelight
Like Lewis Carroll, Gardner abhors fame, politely refusing all requests for public appearances. He will do an occasional magic trick for friends – especially mathematicians and scientists, who, according to Gardner, are the easiest to fool. In fact, it seems that too many scientists have been bamboozled in areas that should fall under the magic category, he asserts. It dismays Gardner to no end how reputable professionals can become prey to the “quackery of parapsychology” and other pseudoscientific phenomena. Many of his essays and books recount stories of onetime scientific heroes who have been taken in by “bad science.”

Of course, Gardner’s truth-seeking mission spans beyond the scientific community. From Shirley MacLaine in The New Age: Notes of a Fringe Watcher (Prometheus, 1988) to Conan Doyle in Science: Good, Bad and Bogus (Prometheus, 1981): If he perceives a phony, he’ll get them. Gardner knows better than most that it takes a conjurer to catch one.

Gardner can cut through nonsense like few others. Never condescending to the reader, he states his points clearly. The weight of the evidence he brings to bear is hard to dismiss, even if you happen to be skeptical of a skeptic.

Martin Gardner’s Passing
Gardner moved to Norman, Oklahoma, in 2002, to be near his son James. He lived in an assisted living facility and continued his work until the day he died, which was May 22, 2010. He was 95 years old. Martin had seen this article and thought I had covered most of the essential things about his life. As I remember, he gave me one or two sentences of approval with no emotion. He had a difficult time getting overly interested in articles covering him in any manner. He really was that modest. When I went to see him in Norman he enjoyed discussing the things most people like to discuss: the economy, politics, weather, food… but his interest would intensify if I told him I had a new puzzle or asked if he had read the most recent article about Roger Penrose, Richard Feynman or anyone who had anything to say about math, science, puzzles, critical thinking, etc. He would like it even more if I or anyone could make him laugh while telling a story. If I went on and on about the genius of this man and his prolific accomplishments, he wouldn’t like it. So, this article is done and in closing I would like to encourage you to discover Mr. Gardner on your own as I did years ago. It will be a treat you can enjoy over and over.