I’ve been reading a 1978 book called The Art of Problem Solving, by Russell L. Ackoff. (Used copies sell for $7 including shipping on Amazon). At the beginning of the book, Ackoff tells about the time his school-age daughter came home with a familiar puzzle. On a piece of paper, there are nine dots arranged in three rows of three, like this:
The challenge is to put a pen on the paper and, without lifting the pen, draw four straight lines that go through all nine dots. You probably remember this puzzle, because it’s been around for a long time. Ackoff remembered it, too, but he’d forgotten the solution. But he came up with another way to solve it. He folded the paper so two rows of dots touched each other and used a felt pen to draw through both rows with one stroke. Then he unfolded the paper and drew two more lines to cover the rest of the dots.
When Ackoff’s daughter went to school the next day, she started to show the teacher the solution her father came up with. As soon as she began folding the paper, the teacher stopped her and said she was not allowed to fold the paper, even though the instructions didn’t say she couldn’t.
“This is how creativity is suppressed, although usually not so overtly,” writes Ackoff. “The teacher made it clear to her class that the objective of the assignment was not to find a solution to the problem, but to find the solution she knew and could pretend to have discovered on her own. She had no interest in any other solution.”
Ackoff defines a puzzle as “a problem that we usually cannot solve because we make an incorrect assumption that precluded a solution.” In the nine dots puzzle, some people assume that they aren’t allowed to draw lines outside the grid. Even more people don’t think about folding the paper.
With Ackoff’s definition in mind, here’s a puzzle from Martin Gardner’s book, My Best Mathematical and Logic Puzzles:
The Colliding Missiles
Two missiles speed directly toward each other, one at 9,000 miles per hour and the other at 21,000 miles per hour. They start at 1,317 miles apart. Without using pencil and paper, calculate how far apart they are one minute before they collide