Thinking Outside the Box

Thu, 22 Oct 2009

Quite a few years ago, I took a class, and the instructor gave us the "thinking outside the box" puzzle. This puzzle consists of putting 9 dots in a perfect square on a piece of paper and using a pencil, joining all dots with 4 straight lines or less. Each line starts where the previous line ends. The only thing you can't do is cut or fold the paper (and of course the lines must be perfectly straight).

OK, easy enough. And after a few minutes, most of the class had the answer. Four lines could join all the dots as long as you thought "outside the box" -- that is, you didn't let the dots along the edge that formed the "box" limit you. To solve the puzzle the you had to extend three of the four lines beyond the square formed by the box.

But as I was looking at the paper and the dots I had drawn, and listening to the instructor tell us not to restrict ourselves to the confines of the box, I suddenly realized that this particular geometric shape lent itself to a solution that allowed one to draw one straight line and join all the dots.

I hear you -- not hardly. You have three rows (or three columns) of three dots that are parallel. One continuous straight line can't possibly go through them.

How many of you have ever looked at the tube at the end of a roll of toilet paper or paper towels? The tube is constructed by taking a piece of cardboard approx two inches wide, wrapping it such that the outside edge of one side butts against the inside edge of the other side. With a piece of cardboard long enough, you could keep this tube going forever. And the edges are perfectly straight.

Next time you get a tube, unroll it (you may have to cut along the "line" where it's joined). Stretch it out in front of you going away from you. Notice that the top and bottom are cut at an angle (they are, in fact, perfectly parallel).

Now you have the basic shape visualized. So on your paper, use a magic marker and put 9 dots arranged with 8 in a square and the 9th in the center with the dots each exactly two inches apart side to side and top to bottom. Ensure the square is "square". The top row of dots should be less than 2 inches from the top of the paper.

Imagine the dots are numbered, the top row left to right is 1, 2, and 3 consecutively, the second row 4, 5, and 6, and the bottom row 7, 8, and 9. Now roll the paper such that the top row of dots (dots 1-3) are exactly 2 inches underneath and parallel to the bottom row (dots 7-9). Maintaining the distance, slide the top row to the right so that dots 1 and 2 line up under dots 8 and 9. Assure yourself that dot 1 is exactly 2 inches below dot 8 and dot 2 is also exactly 2 inches below dot 9 and that dots 2, 9, and 6 form a "straight" line.

At this point, if your paper is sufficiently stiff to maintain its shape as a roll, you could hold the roll with one hand and with the other put your pencil on dot 7 and draw one continuous perfectly straight line through dots 4, 1, 8, 5, 2, 9, and 6 consecutively. Unroll the paper, placing it flat on the table, and use a ruler and you can see that the lines are perfectly straight (albeit now 3 perfectly parallel straight lines).

While I've been able to find the four line solution to this puzzle published by everyone who knows of this brain teaser, I've been unable to find my solution. I have a hard time imagining that it's unique, but apparently it's not that common, or I'd have come across it in my searches.

I hope you enjoyed my spatial solution to what most seem to think is only a euclidean geometry problem.


Thinking Outside the Box

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