When your boss tells you to think outside the box, he is looking for a spark of creativity. "Don't be constrained by assumptions that may not always be true." The box is a metaphor for those assumptions. This is good advice, for the most part. However, if someone asks you to plan a mission to Mars using new ideas that go beyond the laws of physics, you're not going to get very far. You must temper your out-of-the-box thinking with a healthy dose of reality. Still, the expression has become a commonplace motivational mantra in corporate board rooms, schools, and even government offices. If a problem seems intractable, break out of the existing paradigm and come up with something new. In other words, think outside the box. But where did this phrase come from?
The 9 dots problem asks the reader to connect 9 dots using 4 straight lines, without lifting the pencil off the paper. This puzzle appeared in Sam Loyd's 1914 Cyclopedia of Puzzles, decades before it inspired the corporate "out of the box" slogan in the 1970's. The dots are arranged in a 3 by 3 grid as shown below.
If you could lift your pencil off the page, the problem would be easy. Draw three horizontal lines, each connecting a row of three dots, then draw a vertical line down the left hand side, making a giant letter E. That does the trick, but only if you lift your pencil off the page. Pause for a moment and see if you can connect all 9 dots, using 4 straight lines, in one continuous path. Spoiler is coming up.
If you believe the problem is impossible you're not alone, but that is because you added an implicit assumption that was never stated in the problem description - that your pencil must never stray beyond an imaginary box that encloses the dots. Perhaps you added a border, like this.
Without the border, the problem is easily solved. Imagine the lower left dot is at the origin. This is where your pencil will start. The dots have x and y coordinates of 0 1 and 2, thus the upper right dot is at (2,2). Draw a line from the origin up to (0,3), thus connecting the 3 dots on the left, and bursting through the imaginary box that isn't there, as it travels up to (0,3). The second line slants down and to the right, traveling to (3,0), and connecting two more dots along the way. Next, draw a line back to the origin, and then a fourth line up and to the right to (2,2). There you have it. It's easy, if you think outside the box.
In 1970, the Walt Disney corporation used this puzzle as an in-house training exercise, to encourage its employees to think outside the box. It spread quickly, and was soon used as a motivational tool by corporations and management consultants around the world. "You solved the 9 dots problem by thinking outside the box, now go back to your office and solve your business dilemma the same way." It is not clear that this exercise actually helps people think creatively; and one has to wonder whether these creative ideas, should they arise, would be accepted by upper management in any case. But hey, it's a couple hours that you don't have to work and you still get paid, so just go with the flow.
Soon mathematicians got ahold of the puzzle, and realized that there is an inside the box solution, that is both harder to find and more interesting. First we must define some geometric concepts with a bit more rigor.
A curve is the continuous image of the closed interval [0,1] in a topological space.
Ok, think of it this way. You have a pencil and you are allowed to draw just about anything on your piece of paper, but you can only draw for one minute, and during that minute the pencil must not leave the paper. That is a curve. A circle is fair game, and so is a square. These are closed curves, because the pencil starts and ends at the same place, but a curve need not be closed. Draw the letter S for instance, or just a straight line. A single point is also a curve. Rest your pencil on the paper and don't move it for one minute. Or, leave the pencil still for 10 seconds, draw a squiggly line, stop for another 10 seconds, draw an arc, and so on. A curve can cross itself, like a figure 8, or even backtrack along the path it has already traced. For instance, draw a line from (0,0) to (1,0) for the first 20 seconds, then go back to the origin during the next 40 seconds. The function that realizes this curve, measuring time in seconds, is: x(t) = t/20 as t runs from 0 to 20, and 1 - (t-20)/40 as t runs from 20 to 60. That's a perfectly good function, defining a perfectly good curve. Another function is x(t) = cos(πt/15) and y(t) = sin(πt/15), giving a circle that runs twice around the origin. Many functions and curves are possible.
Are there shapes that aren't curves? Sure there are. Two disconnected lines can't be a curve, because you have to lift your pencil off the page to draw the second line. For a less obvious example, consider the ray that starts at the origin and travels along the positive x axis forever. This is not a curve because your pencil would have to travel infinitely fast. The open interval from (0,0) up to but not including (1,0) is also not a curve, because you can't approach a point, closer and closer and closer, without actually touching it. How about the entire square, as x runs from 0 to 1 and y runs from 0 to 1? I'm not talking about just the four sides, but the interior as well. Every point on and inside the square must be covered. Remember that the tip of your pencil is infinitely thin. Surely that is not a curve! Actually it is, and don't call me Shirley. It is possible, at least in theory, to wiggle your pencil around so rapidly, at a microscopic level, that you manage to cover the entire square in one minute. This is extremely counterintuitive, and beyond the scope of this article. You can read about space-filling curves here.
Now that we have defined a curve, what is a line? A line is a curve that fits within a line in the plane. You can draw back and forth, back and forth, but the pencil always stays within a given line. The earlier example, traveling from the origin to (1,0) and then back to the origin, is a line. With this in mind, find a solution to the 9 dots problem comprising a path of 4 lines lying entirely inside the box.
Spoiler. Start at the center and draw a horizontal line, back and forth, covering the middle row, and stop once again at the center. Draw a vertical line, up and down, covering the middle column, and stop at the center. Draw the two diagonal lines and you're done.
Recent statements of the 9 dots problem, including the one in wikipedia, have the additional constraint that you cannot retrace your path. The pencil cannot back up over a line it has already drawn. That rules out the mathematically correct solution described above, and forces you to once again think outside the box.
I will close with an ad campaign that I thought was rather clever. In 1979, Taco Bell encouraged us to "Think outside the bun." If you're hungry for some fast food, it doesn't have to be a hamburger, or even a chicken or fish sandwich, all living inside a bun. Break out of your routine and try some mexican food. It was quite clever, and ran for 35 years, but is being replaced with "Live mas". That slogan doesn't do much for me, but hey, I'm not a marketing expert.