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, 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.