Approach Problems on Their Own Terms

When people use their imagination to develop new ideas, those ideas are heavily structured in predictable ways by the properties of those existing categories and concepts. This is true for scientists, artists, inventors, politicians and business people. Consider the following accident which was reported in The American Railroad Journal in 1835:

“As a train was approaching the depot at Paterson, an axle of the leading car gave way, which overturned that and the following two cars. None of the passengers were injured, though they felt the shock by the concussion. Mr. Speer, the conductor, a very industrious and sober man, was seated on the car at the break, and unfortunately was crushed to death under the load.”

Mr. Speer was the only casualty. What factors contributed to his untimely death? Certainly there was the immediate cause–the breaking of the axle and the overturning of the cars–but there is a more subtle cause as well. Note that Mr. Speer was riding on the car, not in it, and that none of the passengers, who were inside, were hurt. Why was he not in the car? What in the world was he doing on top of the car? Speer’s death was the result of a design flaw that required conductors to ride on the outside of cars.

This flaw is an example of the phenomenon of structured imagination. Early designs for railway cars were heavily influenced by the properties of the stagecoach, the most common vehicle of the day. The first railway cars were little more than stagecoaches with wheels on tracks, with no central aisle and designed so that conductors had to ride outside on running boards. The idea of a central aisle was considered odd and even unsanitary, based on the notion that it would become one long spittoon. Finally, as was true of stagecoaches, the brakes were located on the outside and were operated by the conductor who was seated on the top front of the car.

We would not consider the developers of the railroads to be unimaginative people. On the contrary, they were visionaries who saw the railroad as the transportation of the future long before other people took the idea seriously. Yet, even after a number of conductors had been killed, there was strong resistance to designing the railway cars so conductors could ride safely inside. As late as 1866, according to the Railroad and Engineering Journal (1887), 72 trainmen were killed in falls from cars in Massachusetts, New York, and Michigan alone.

What this suggests is that even highly creative individuals and the ideas they develop are susceptible to the constraining influences of structured imagination. Their idea of a design for a railway car was heavily influenced by what they knew, understood, and were most familiar with–the stagecoach. Even Thomas Edison’s idea for an electric lighting distribution system is an example of an idea that was the result of a structured imagination. His reliance on the existing gas distribution system at the time led to his stubborn reliance on the problematic procedure of running wires underground, just as gas mains ran underground. More recently, the fact that many modern computer terminals display exactly 80 columns of text is a direct outgrowth from the era when we literally fed data into computers by way of 80-column punch cards.

With creative thinking, one generates as many alternative approaches as one can. You consider the least obvious as well as the most likely approaches. It is the willingness to explore all approaches that is important, even after one has found a promising one. Einstein was once asked what the difference was between him and the average person. He said that if you asked the average person to find a needle in the haystack, the person would stop when he or she found a needle. He, on the other hand, would tear through the entire haystack looking for all the possible needles.

Please solve the following challenge. The challenge is to draw no more than four straight lines which will link up all nine dots without lifting your hand from the paper. What is your solution?














The first time a person tries to solve this puzzle they are stymied. This is because of our perception of the arrangement of the dots as a box or square. Once perceived as a box, most people will not exceed the imaginary boundaries of the imaginary box and are unable to solve the puzzle.

There is nothing in the challenge statement that defines the arrangement as a box and nothing demands the lines must be drawn within the box, but people make the assumption based on past experiences with boxes and find the puzzle difficult.

The answer involves drawing a line that goes beyond the limitations of the imagined box. This is where the cliché “Think out of the box” comes from. To solve it, you have to start the line outside of the imaginary box. The nine-dot puzzle was popularized by William North Jayme, a direct-mail copywriter who was hired by Esquire magazine in 1958 which wanted to abandon its unwholesome image for a more sophisticated one. Mr. Jayme came up with the ”puzzle letter”: an envelope with nine dots on it and a challenge to the recipient to connect them using no more than four uninterrupted lines. The enclosed letter showed that to do so, one had to go outside the box. Or, in other words, you had to break normal thinking patterns, something that the new Esquire said it could help modern men do. The letter was a phenomenal success, Esquire’s image was changed overnight and the subscriptions poured in.


Over time the puzzle became synonymous with creative thinking and the phrase “thinking outside the box” has now become a cliché for creativity. A cliché because the puzzle has become commonplace and most people remember the solution from their past experience with it. When the brain recognizes the pattern and we solve the problem it seems like a new insight has been sparked.

Education has conditioned us to think of one right answer to a problem. When we have an answer we stop thinking. Creative geniuses, on the other hand, are constantly searching for more and better alternative answers and never stop with their first good idea.

APPROACH A PROBLEM ON ITS OWN TERMS. If you approached the problem with the perspective of “How can I connect all nine dots………? you create the impression that there is only one right answer. The wording limits your thinking. Approaching the puzzle with this perspective “In what ways might I connect all nine dots ……………..?” liberates the mind to think more freely about alternatives.

You are now encouraged to imagine the following perspectives of the lines, the dots and the puzzle as a whole. Looking at the puzzle from the perspective of the lines gets you thinking of the number of lines, the lengths of the lines, and the width of the lines.

THREE LINES. For instance, there is no requirement that you must use four consecutive straight lines. Why not three, two or even one line? When linking things such as dots, we are used to linking the centers and our first attempts are to draw lines through the centers of the dots. This is another false assumption based on past experiences. The way to link up the dots is to have the line just touch the dots as illustrated.


ONE LINE. Another dimension of the puzzle is the instrument used to draw the line. Most people assume you must use a pencil or pen and draw a normal-sized line. But there is nothing in the challenge statement that prohibits the person from using an alternative instrument. One solution is to use a wide paint brush, dip it in paint and connect all nine dots with one straight continuous wide swipe.









LENGTH OF LINE. There is no limit on how long you can draw the straight line. So another solution is to draw the line around the world three times intersecting and linking up all nine dots.


This, of course, is impossible to do. Your brain is a dynamic system that evolves its patterns of activity rather than computes them like a computer. It thrives on the creative energy of feedback from experiences real or fictional. You can synthesize experience; literally create it in your own imagination. The human brain cannot tell the difference between an “actual” experience and a fantasy imagined vividly and in detail. In the nine dot puzzle, we take the impossible solution of going around the world three times and imaginer the essence of this idea into a solution that is realistic and practical as illustrated.


Or you can adjust place the paper with the dots on the center of a turntable and replace the needle at the tip of the recording arm with a tiny pen. Turn on the turntable and the pen will draw a line through all nine dots.


GET RID OF THE BOX. Now, let’s look at from the perspective of the way the dots are arranged. There is nothing that prohibits us from rearranging the dots, so another one line solution is to cut out the dots and tape them into one straight line and draw one line straight through.


REARRANGE. And another is to fold the puzzle over and over following the below steps until you have one line of dots and then link them with one straight continuous line.


Another way to change the pattern is to cut the dots out and line them up in a row. Then punch a pencil through the centers linking all nine dots.


The more ideas you generate, the more connections you make. These connections and their associated ideas often spark new ideas and new questions. The creative mind synthesizes all that is created and goes beyond them to create more creative products. For example, the above idea of stabbing a pencil through the cut out dots triggers another idea. That is to rumple up the puzzle into a small wad of paper and punch a pencil through the wad. You may have to do this several times, but probability being what it is, sooner or later, you will punch the pencil through the dots linking them together.


In genius, there is a patience for the odd and the unusual avenues of thought. This intellectual tolerance for the unpredictable allows geniuses to bring side by side what others had never sought to connect. An unusual and imaginative solution is to widen the dots with a pencil so that each dot touches the adjacent dots. Now the nine dots are linked together with no lines.


The playful openness of creative geniuses is what allows them to explore unthinkable ideas. Once Wolfgang Pauli, the discoverer of electron spin, was presenting a new theory of elementary particles before a professional audience. An extended discussion followed. Niels Bohr summarized it for Pauli’s benefit by saying that everyone had agreed his theory was crazy. The question that divided them, he claimed, was whether it was crazy enough to have a chance of being correct. Bohr said his own feeling was that it wasn’t crazy enough.

Logic hides in Bohr’s illogic. In genius, there is a tolerance for unpredictable avenues of thought. The result of unpredictable thinking may be just what is needed to shift the context and lead to a new perspective.

Another unusual solution is to light a match and burn the paper with the puzzle into a pile of ashes. Then carefully form the ashes into one straight line.


Within a short time, we came up with a quantity of solutions because we approached the problem on its own terms, looked at the problem from several different perspectives, did not settle for the first good idea, did not censor ideas because they looked silly or stupid and consequently created several ideas, thought unconventionally, changed the way we looked at the puzzle, worked with the essence of the problem and used our imagination.



(Michael Michalko is the author of Thinkertoys: A Handbook of Creative Thinking Techniques; Cracking Creativity: The Thinking Strategies of Creative Geniuses; Thinkpak: A Brainstorming Card Deck, and Creative Thinkering: Putting Your Imagination to Work.


  1. Great article Michael!

  2. This gives us a practical approach to solve any problem.
    Slice the issue into as many whole bits,as possible.
    Then -‘ connect the dots.’

Comments are closed.