Friday, September 14, 2007

Making triangles with Six toothpicks

"Use six toothpicks to make four identical triangles."

This is a problem that was given to my pre-algebra math class in Jr. High School. The problem was straight out of an old math book.  It was presented to us by a substitute teacher whose strategy was to offer free time at the end of class if we accomplishing certain tasks. The math book listed only one solution to this problem. It stated that the only way to form these triangles was to build them into a tetrahedron (a 3D object). In other words, this old text book presented this problem in order to get kids to think in terms beyond the 2D dimensional realm of a flat piece of paper. 

No one in the class figured out the 3D solution. When the teacher presented the 3D solution, he declared that this was the only solution possible. Several people in the class protested by saying that other solutions had to exist. So, he gave us a double or nothing wager. If someone could figure out a 2D solution to the problem by the next day, he would double the amount of free time he offered.

I worked the problem for a couple of hours that night, not because I wanted the free time, but because it really bugged me. I drew many sketches of possible methods, but none produced four identical triangles while the toothpick ends touched. Then it hit me. Nothing in the problem stated that the toothpicks couldn’t overlap. (Nor did the problem suggest that the triangles had to be equilateral.) I drew up my solution: two toothpicks formed an X, and then the other four toothpicks formed a square around that X with overlapping tips. Easy. I read over the problem several times to make sure my solution was in compliance.

The next day, I left my sketched solution on the teacher’s desk and sat down. A minute or two after class started, the teacher discovered my solution on his desk. He was surprised. He showed our class the solution on the chalkboard. Everyone was happy. Our class won our free time. I didn’t really care. I got my rush the night before by solving the problem.

The next day, a deskmate of mine in another class mentioned the incident. She had the same substitute teacher at a different time in the day, who challenged her class with the same problem. So, the teacher presented my solution to her class as well, using my name. She told me that people in her class were upset with her because she sat next to me in another class and didn’t get the solution from me.

Now note, if you look for this problem these days, there are many different versions online. The problem is stated much more specifically, so as to limit the possible answers. These days, this same problem is worded something like this,
“Using six toothpicks, make four identical equilateral triangles and nothing else. (In other words you can’t make six equilateral triangles, or four triangles and a diamond, etc.)”. (backup link)
Of course, being this specific, the only answer is a tetrahedron. Another similar problem I found allows for several 2D solutions (backup link), but of course it also requires equilateral triangles. However, the solutions are similar to my solution to the old problem. 

Now, I’m not saying I was the first person to figure out this solution.  However, I did figure it out on my own in one night; in the days before the Internet. Looking over the Internet these days, I can’t even find this problem improperly stated. Maybe the writers of that old math book just didn’t do their research, tried to dumb down the problem too much, or just didn’t catch the wording error? I can’t imagine that this problem was improperly stated for hundreds of years before it found its way onto my desk.

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