9 Magic Dots: Creative Thinking Puzzle

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

— Edward Glassman

Many people who know the following puzzle think 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 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, two triangles, or some other geometric figure, you may have blocked yourself by assuming boundaries that do not exist. You have stayed inside the lines and created a mind funnel that limits your thinking.

You can connect these nine dots with four straight lines by moving outside the assumed boundaries.

Do you like this answer? Do you think it is elegant or the only one possible? Actually, the biggest assumed boundary is the unwarranted assumption that only one answer exists. There are dozens of completely different answers. The one above is a quick fix, the first adequate answer. How can we shift the paradigm and find others?

One important advanced creative thinking procedure is forced withdrawal. Forget the original problem and work on a distant version of it. This can lead to new paradigms, perspectives, and solutions.

Here are the same nine magic dots.

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

If you do not solve it, consider the mind funnels, assumed boundaries, unwarranted assumptions, and unstated criteria that may be blocking you.

First: What do you see when you look at the nine dots? If you still see a square or another geometric figure, that habit may be blocking you. For some solutions to the three-line problem, you need to perceive the nine dots as existing in space because the lines leave the paper.

Second: Did you assume that the lines must go through the centers of the dots? That is another unwarranted assumption.

Third: How do you define a dot? In school, a dot represents a point in space with no dimension. The circles on the page have length and width. In real life, dots have length and width and come in all sizes. On billboards, dots can be as large as your head. On clown costumes, polka dots can be the size of your shoe. Include reality in your definition of dots rather than letting a restricted definition block your thinking.

With expanded boundaries, clarified assumptions, and unrestricted definitions, the nine-dot, three-line problem can be solved by extending the lines off the paper when necessary.

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 it as far as necessary, even off the paper. The second line does the same with the middle row, and the third line does the same with the last row.

There is another solution based on non-Euclidean geometry, which postulates that parallel lines meet at infinity. Using this mind funnel, three parallel lines each touch a different row of dots, then all three lines connect at infinity. That is another neat paradigm shift.

A True Story: In a creative thinking workshop, one participant said she had found the non-Euclidean geometry solution but discarded it because she thought it was unfair. The solution lay outside her comfort zone.

Try another forced withdrawal with the same nine magic dots.

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

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

One block comes from a restricted definition of a line. In school, a line is defined as a series of connected points with only one dimension, length. In real life, lines have width. Think of traffic lines in the center of a road or lines of buses approaching an intersection.

Again, a restricted definition blocks creative thinking. You may have assumed that only thin lines could be used.

One answer to the nine-dot, two-line problem is simply a wide line and a narrow line.

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

The problem: Use only one straight line that touches them all. Find at least 15 answers before continuing.

Actually, hundreds of acceptable solutions exist. The following examples can trigger new paradigms and mind funnels:

• 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 one line 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 could not cut up the paper?
• Twist the paper into a cone and draw a straight line that spirals around the surface and touches all nine dots. Did you assume you could not twist the paper?
• Put the paper with the nine dots on the equator of the earth and draw a straight line that circles the earth enough times to touch each dot. Or put the paper on the edge of the universe and have your line circle the universe. Did you assume you could not use fantasy?
• Write "ONE" over the top row, "STRAIGHT" over the middle row, and "LINE" over the bottom row. You have 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 and view the nine dots through this side line.
• Move a straight line like a windshield wiper and touch all the dots. Did you assume the line had to touch them all at the same time?
• Cut the line into 1,000 pieces and sprinkle it over the nine dots so that it touches them all. Did you assume you could not cut up the line?
• Cut the paper so there is one dot on each piece. Line up the dots and push a pencil through them. You have touched all the dots with one straight line and annihilated the dots and the problem. About time, I'd say.
• Imagine you hear the words, "Touch each dot with only one straight line," without seeing them written. Bring in the king of beasts, or a picture of one, and cover the nine dots with one straight "lion." Or imagine nine people named Dot eaten by one straight lion.
• Change the dots into clothespins and hang them on one straight clothesline. Did you assume you could not convert the dots or the line into something else?
• Change the dots into tennis balls and play tennis with them until all have touched the tennis net made from one straight line.
• Change the line into the shadow of a sundial so it eventually touches all the dots as the sun moves across the sky.
• Convert the straight line into a sunbeam and use a glass prism to break it into many colored lines that touch all nine dots.

Creative thinking workshops regularly produce new and different solutions from participants. See how many new solutions you can generate.

The Nine Magic Dots can help create a more creative climate in your mind. The puzzle is also a metaphor for problems at work. It reminds us to question assumed boundaries, restricted definitions, unstated criteria, and familiar paradigms when they limit our thinking. •