Thinking outside the box
This came up when I was in grade school. the teacher asked us all to do this, and I did it wrong -- which was ironic considering the goal of the test given. I showed my work, explained and defended my answers -- but the smug teacher was mad because I went too far outside the box. Either that or it was pre-teen, "I'm smarter than you" attitude that made him so irate.
When you first look at the problem, the answer seems like 5 is the fewest number. But the tester usually tells you that it can be solved in 4 lines. Then they go away and leave you to ponder this conundrum for a while, and return to give you their smug answer.
Then the tester tells you that since you were thinking "inside the box", you couldn't solve the problem.
I was a little different. When I was given that test, I looked at it for a few seconds and said, "heck, I can to it in 3 lines, or 3 ways to do it in 1 line". Which befuddled the tester, since the obvious proper answer was the one he was told which was 4 lines. How could I do it in 3 lines, or 1 line?
Three lines is easy... just make them long lines in a big Z.
They are dots, not points - so they aren't infinitely thin, nor is the pencil. That was easy -- but he sort of felt it was cheating (even though I was conforming to all the rules set before me).
But then he really wasn't going to like my other three answers in how to do it in one line.
- The first was simple. Split the pencil in half (lengthwise) and then use the exposed lead to smear one really big-fat line across all dots in one swipe.
- The next one was to stop thinking in two dimensions. Wrap the paper in a cylinder (with the dots on the outside), and then like an old phonograph, draw one straight line along the outside surface, in a slowly discending direction, that would eventually bisect all the dots. It doesn't matter how many revolutions it takes, it will eventually cross all the dots, and I'll never have lifted the pencil.
- The third way was a variation on the last. Just fold the paper up so all the dots are aligned over each other, then place the pencil on the first dot and push it through the paper (in one line). This third dimensional line will go through all the dots (literally).... or if your in a less destructive mood, you can just fold it so the edge of the dots are on the edge of the page (but lined up) and draw across the edges.
Ironically, the way I didn't think of solving the problem was the one he was looking for - which is with four lines.
The tester didn't like my answers. Those were all wrong, everyone knows when you're thinking outside the box, you have to do it right.
It seems that I thought too far outside the box.
The point is that the way to think outside the box was not only to question the solution, but to question the problem or tools themselves. If the problem was to get black "lead" on all the dots, then use a fatter line, or fold/manipulate the paper/dots to make the problem easier. Or in other words, always challenging the validity of the question, and whether the right questions are asked. Then question if the limitation are good, or you need to pay attention to them. If you don't like the limitations, then change them.