By Edward Glassman, PhD | Posted 5/4/11 | Updated 1/4/22

"R&D people experience a great deal of fun when creative. Not so much HA-HA fun as A-HA fun."

Many people who know the following puzzle think there it has only one answer. If you agree, prepare for a surprise. If you have seen it before, solve it anyway; shift the paradigm and generate another quite different solution.

Here are nine magic dots.

The problem: Draw **four** connecting straight lines that will touch all nine dots only once without lifting your pen or pencil from the paper.

If you have done it before, find a totally different solution. Spend at least five minutes before reading further…

If you did not solve the problem, what do you see when you look at the nine dots?

If you see a square, or two triangles, or some geometric figure, then you probably blocked yourself by assuming boundaries that do not exist and staying within mind funnels that kept you inside the lines.

**A HABIT THAT SPOILS R&D CREATIVE THINKING:** We often **assume boundaries** that may not exist. We stay within the lines. We think within the rules. We use unstated, phantom criteria. We use past company policies and attitudes as guidelines on how to do things without checking it out with others. We don't shift paradigms without permission.

You can connect these nine dots with **four** straight lines by moving outside the boundaries as shown here:

Do you like this answer? Do you think it elegant, the only one possible? Actually, the biggest assumed boundary of this problem comprises the unwarranted assumption that only one answer exists. In fact, you can find dozens of completely different answers to this problem; the one above constitutes a quick fix, the first adequate answer! How can we shift the paradigm and find the others?

We will use a very important, advanced creative thinking procedure I call **'forced withdrawal,'** in which we forget the original problem and work to solve a distant version
of it. In that way we may find new paradigms, new perspectives, and new solutions. The first forced-withdrawal we shall consider is...

Here are the same nine magic dots.

The problem: This time use **three** connecting straight lines that touch each dot only once. If you don't solve it, try to discover the mind funnels, assumed boundaries, unwarranted assumptions, and unstated criteria that block you.

**First:** What do you see when you look at the nine dots? I hope you kicked the habit of seeing a square or some other geometric figure. This time one block comes from seeing the nine dots on a piece of paper. For some solutions to this three-line problem, you need to perceive the nine dots as existing in space, because the lines will leave the paper.

**Second:** Did you assume that the lines must go through the center of the dots? This unwarranted assumption also blocks you.

**Third:** How do you define a dot? In school, I learned that a dot represents a point in space with no dimension: without length, width or height. Those circles I call dots have length and width. Is that fair? Well, in real life, dots have length and width, and come in all sizes. On billboards, dots grow to the size of your head, and on clown costumes, polka dots fit the size of your shoe. So include reality in your definition of dots, lest you fall victim to another spoiler of creative thinking.

**A HABIT THAT SPOILS R&D CREATIVE THINKING:** We use **restricted definitions** that limit our mind funnels and our creative thinking. We stay stuck in old paradigms.

With expanded boundaries, clarified assumptions, and unrestricted definitions, we can solve the 9 dot, 3 line problem in this way:

**Go off the paper, if necessary.**

The first line touches the side of the first dot tangentially, passes through the center of the second dot, and touches the side of the third dot tangentially. Extend the line as far as necessary, even off the paper, so the second line can do the same to the middle row of dots, and similarly for the last line and the third row of dots.

There's another solution based on non-Euclidean geometry which postulates that parallel lines meet at infinity. Using this mind funnel, the answer consists of three parallel lines, each of which touches a different row of dots, and then all three lines connect at infinity, a neat paradigm shift.

**A True Story:** In a creative thinking workshop, one participant said she discovered this solution based on non-Euclidean geometry, but discarded it because she thought it unfair. This solution lay outside her comfort zone.

**A HABIT THAT SPOILS R&D CREATIVE THINKING:** We only express 'fair' ideas, even before we select one. Don't let **fairness** spoil creative thinking in your head.

Here's another forced-withdrawal with the same nine magic dots.

The problem: Use **two** connecting straight lines that touch each dot only once.

Think it impossible? Check your assumed boundaries, unwarranted assumptions, unstated criteria, restricted definitions, mind funnels, and paradigms.

One block to this problem lies in your restricted definition of a line. In school, teachers define a line as a series of connected points that have only one dimension, length. In real life, lines have width. Look at traffic lines in the center of the road or lines of buses approaching an intersection. Again your habit of restricting definitions blocks you and led to the unwarranted assumption that you could use only thin lines.

Here's one answer to the 9 dot, 2 line problem: A wide line and a narrow line!

Try one last **forced-withdrawal** with the same nine magic dots.

This time use only **one** straight line that touches them all. Find at least 15 answers before you continue reading.

Actually hundreds of acceptable solutions exist. The few solutions here will trigger new paradigms and mind funnels, and whet your appetite for more.

- Use one wide line that touches each dot.
- Run a large 3-dimensional line down through the nine dots from above so it passes through the paper, and touches each dot.
- Fold the paper so you can draw one line that touches each dot. (Did you assume you could not fold the paper?)
- Cut the paper so each dot is on a separate piece. Line up the pieces so one line touches each dot. (Did you
**assume**you couldn't cut up the paper?) - Twist the paper into a cone and draw a straight line that spirals around the surface of the cone and touches all nine dots. (Did you
**assume**you couldn't twist the paper into a cone?) - Put the paper with the nine dots on the equator of the earth and carefully draw a straight line that circles the earth enough times so it touches each dot. Or, put the paper on the edge of the universe and have your straight line circle the universe until it touches each dot. (Did you
**assume**you could not use fantasy? Note we expanded our mind funnels from nine dots in a box to the edge of the universe). - Write "ONE" over the top row of dots, "STRAIGHT" over the middle row of dots, and "LINE" over the bottom row of dots. You touched the dots with the words: "ONE STRAIGHT LINE." (Did you
**assume**you could not use words?) - Draw the line on the thin edge of the paper. View the nine dots through this side line.
- Move a straight line, like the windshield wiper on a car, and touch all dots. (Did you assume you couldn't move the line, or that the line had to touch all the dots at the same time?)
- Cut the line into 1,000 pieces and sprinkle it over the nine dots touching them all. (Did you
**assume**you couldn't cut up the line?) - Cut the paper so there is one dot on each piece of paper. Line up the dots on top of each other. Push a pencil through all the dots. You not only touched all the dots with one straight line, but you also annihilated the dots and the problem. About time, I'd say.
- Wait. Here's another solution to jolt your mind. Imagine you sit in my creative thinking workshop and I merely say: "Touch each dot with only one straight line." Not write it, so you could see the spelling, but you only hear the words. One solution: bring in the king of beasts (or a picture of one) and cover the nine dots with one straight 'lion.' Or how about nine people named Dot eaten by one straight lion?
- I can't resist even more bizarre solutions. Change the dots into clothespins and hang them on one straight clothesline. (Did you assume that you could not convert the dots or the line into something else?)
- Or change the dots into tennis balls and play tennis with them until all have touched the tennis net made from one straight line. Or change the line into the shadow of a sundial so it will eventually touch all the dots as the sun moves across the sky. Or convert the straight line into a sunbeam and use a glass prism to break it up into many colored lines that touch all nine dots. Had enough?

In my workshops on creative thinking, I always hear new and different solutions from the participants. See how many new solutions you can discover. The Nine Magic Dots can help the creative climate of your mind. You should realize by now that this puzzle represents a metaphor for problems at work. You can learn a lot from these nine magic dots.

Excerpted from the R&D Creativity and Innovation Handbook ©2011 by Edward Glassman. All rights reserved.

Edward Glassman, PhD, was the President of the Creativity College®, a division of Leadership Consulting Services, Inc., and Professor Emeritus of the University of North Carolina at Chapel Hill, where he headed the Program For Team Effectiveness And Creativity. ...