Nine Dots: A Key to Psychological Problems?

Getting trapped by "what we know for sure".

Posted May 17, 2019

In his popular book on creative problem solving, Conceptual Blockbusting: A Guide to Better Ideas, James Adams (2001) follows the nine-dot problem through several increasing challenges. Referring to the dots below, the basic directions to the problem are these:

Puzzle One: Draw no more than FOUR straight lines (without lifting the pencil from the paper), which will cross through all nine dots.

.      .      .

.      .      .

.      .      .

This puzzle is extremely difficult to solve if you see the nine dots as a box that you cannot go outside of with your lines. For those attempting this puzzle for the first time, that is the key clue. Think outside the imaginary box![1] However, before turning to the solution, or going smugly on if you know the solution, first try several more challenges with these dots.

Puzzle Two: Now draw no more than THREE straight lines (without lifting the pencil from the paper) that will cross through all nine dots.[2]

Got that one? OK. Now try this one:

Puzzle Three: Do the same with ONE straight line. [There are several solutions.][3]

Finally, try this last challenge:

Puzzle Four: Do the same thing using ONE POINT. [Again, there are several possible solutions.][4]

In each variation of this puzzle, the solution involves breaking a self-imposed set of logical assumptions. These assumptions are not intrinsic to the problem set out. They are something we impose on them ourselves. Of course, once we accept these assumptions, they set implicit limits on the range and kind of possible solutions we can employ.[5]

When “What we know for sure” is wrong

Readers who read my first post, “When Our Solutions Become the Problem”, might quickly see how Lars and Maria, were trapped in their own nine dots—their assumptions and solutions became their problem. They each needed to step outside their own “neurotic box” so to speak. Once again, this is often easier said than done. The most common cycle in couples is called the “advance-retreat” pattern. One partner advances to draw their mate closer as their partner retreats in the face of those advances. The more the first partner advances or escalates their efforts, the more the second intensifies their retreat and so on.

Breaking the Cycle

Effective problem resolution as in effective psychotherapy involves breaking that cycle and reversing solutions. This is not only true in effective couple therapy but, as we will see in future posts, it is true for all effective psychotherapies. We get trapped in what we know for sure, and we need a trusted guide outside our world views and solution cycles to help us make the kind of changes that work. Each of my next posts will build on these basic ideas. We turn next to finger traps and “Stop It”!

Solutions:

[1] Four Lines: The first puzzle is solved by drawing the first line across the top three dots and beyond them so the second line can be drawn through the third dot on the second row and through and beyond middle dot on the bottom row ending below the first column of dots. The third line is drawn straight up the first column. The fourth line is then drawn diagonally from top left to bottom right to connect the remaining dots. To solve the problem, our lines must go “outside the box”.

[2] Three Lines: This challenge is solved by drawing three lines in a “W” like formation, starting above and to the left of the top left dot. The first line goes diagonally through the left corner of the first column dot the middle of the second dot in that column and the right edge of the bottom dot in that column to a point below the dots. The second line goes up the middle column in the same diagonal way to a point beyond that column to the right. The final line goes back down diagonally through the last column of dots.

[3] One Line: One solution is to use a line so fat it covers all the dots as it goes across them. Another is to bend the paper into a cylinder and draw an angled line that goes continuously around the cylinder and through each row of lines as it passes around. And there are other options.

[4] One Point: Folding or cutting the paper to stack the dots on top of each other allows one to stab a sharp point through them all at one time. And there are other options here too.

[5] Fraser, J. S. & Solovey, A. D. (2007). Second-order change in psychotherapy: The golden thread that unifies effective treatments. Washington DC: APA Books.

References

Adams, J. (2001) Conceptual blockbusting: A guide to better ideas. Cambridge, MA: Perseus.

Fraser, J. S. & Solovey, A. D. (2007). Second-Order change in psychotherapy: The golden thread that unifies effective treatments. Washington, DC: APA Books.