Take a grid of nine dots:
Connect all nine dots using no more than four contiguous (i.e., without lifting pen from paper) straight lines.
Most have trouble solving this puzzle upon first encounter. There solution depends upon the identification and rejection of a false assumption. The assumption itself is caused by a perception that the problem space is limited to the boundary indicated by the dots; therefore, the solution literally involves thinking outside of the box. (The usual solution uses four lines. There is also a solution that involves rejecting the assumption that a point drawn on paper is only a representation of a zero-dimensional location, rather than a two-dimensional shape the actual height and width of the spot on the paper. Further solutions reject the assumption of Euclidean geometry, or that the line itself is one-dimensional.)
In the puzzle, we can learn from a thorough examination of the puzzle space and the rules. For example, the number of lines that can be drawn through one dot and one dot only is infinite. There are only two ways (though this can be applied multiple places within the puzzle) that a line can connect two and only two dots (one up, one over; one up, two over). There are also only two ways (though, again, this can be applied multiple places within the puzzle) that a line can connect three and only three dots (one over, one over; one up, one over, one up, one over).
In addition, the lines must connect at their ends to solve the problem, because, based on the rules (i.e., not lifting the pencil), one would have to waste a line to backtrack to the middle of a previously drawn line. Any solution involving lines connecting only at 90-degree angles can also be quickly dismissed through trial and error, because so few options exist.
"Just as a solution is sensitive to the proper isolation of the problem, it is also sensitive to proper delimitation (constraints). In general, the more broadly the problem can be stated, the more room is available for conceptualization. A request for the design of a better door will probably result in a rectangular slab with hinges and a handle. Is that what is wanted, or is the problem really a better way to get through a wall?" -James L. Adams, Conceptual Blockbusting
A few factors play into our puzzle space constraint, most notably Gestalt principles of grouping:
- The principle of closure: It is part of our perceptive mechanism to create order from disorder—thus when we see a group of dots, we connect them into shapes—e.g., astronomical constellations and connect-the-dot pictures. Therefore, when we implicitly connect the outer dots into a box, a barrier to extra-dot perception. There is, in effect, an illusory contour enclosing the box.
- The principle of proximity: The space surrounding the nine dots makes the group of dots proximate to each other. In addition, the space surrounding the dots is not defined either geometrically, or linguistically. It is, thus, a puzzle void. One of the test subjects noted that a hint could be provided by drawing a box around the nine dots to include the space surrounding, thus defining it as part of the puzzle space and affecting the perception of proximity.
- The principle of similarity: The dots are all the same size and shape, lending to their perception as a closed group.
Another puzzle that relies on a false perceptual assumption is this one:
How can the same smaller shapes rearranged leave a gap in the bottom version of the larger shape?
The false assumption here is one of perceptual approximation. A puzzler will measure the base and side of what appears to be a right triangle, and confirm that they are equal in the top and bottom shapes. Of course, as we know from the Pythagorean theorem, in a right triangle, the square of the hypotenuse will be equal to the sum of the squares of the other two sides. But the puzzler's mistake is that they've assumed based on the right angle that this is a right triangle—but it isn't a triangle, because the false hypotenuse isn't a straight line. When first presented with this problem, I was working with a group of engineers. When I pointed out that the "hypotenuse" in the two shapes was intersecting the grid at different points, and therefore, wasn't a straight line, they dismissed this as a poor rendering of a valid problem. In fact, I was correct, because the two triangle-like shapes in the larger shapes are right triangles, but they have different slopes; therefore, the first of the two conglomerate shapes is actually a polygon, cleverly disguised as a right triangle.
In both cases, the problem is how one can identify, and therefore validate or invalidate, these hidden assumptions. What is the process of reconciliation that takes place?
The idea that assumptions, whether valid or not, occlude our problem-solving abilities means that we must develop methods of analysis for making implicit assumptions explicit so that they can be evaluated.
When I was in elementary school, a teacher posed a problem—make ten with five nines. The answer she intended was some version of 9 / 9 + 9 - 9 + 9 = 10. I was the first to raise my hand. I drew my solution on the blackboard:
Is this answer "right"? It depends on the use. A problem without a purpose leads to a solution without a use.