When solving these problems, learning by doing is much better than learning by reading. I encourage to you read only as far in the solution as you need, then trying to solve the problem. If you get stuck, try reading a little further. And of course, let me know if you find a better solution!
Tuesday, February 11, 2014
Triangular manhole cover -- can it fall in?
Can a triangular manhole cover fall into a triangular shape pipe that it covers? Assume that the triangle covers it perfectly (e.g., not bigger than the pipe)
Well, this is an interesting problem. We could imagine a triangular shaped pipe, and now we want to see if a triangle can fall into it.
As an example, we can try the easiest case, which is an equilateral triangle (all sides the same) on the end of an equilateral pipe. In this case, we can imagine putting a pin through the middle of the triangle and turning it.
Well, it won't fall in here, and some basic math around edges being longer than the width of the triangle can convince of that.
But, that only proves that the simple solution does not work.
Now, let's try some other options, and some math.
Each edge of an equilateral triangle is of length x, however the height of the triangle is (3^.5/2)*x which is less than x. Therefore when oriented vertically with along one edge, it can be rotated so as to fall in.
This is also true of a regular pentagon, and hexagon for which you can also work out a proof. I believe that it is true for any regular polygon -- and only circles do not exhibit this. I haven't proved it but I think that it's true for any convex polygon (I'll leave that to the mathematicians) But, this is a good reason to make manhole covers round!
Here is a general test:
Imagine that you place a stick through the middle of the lid. Next pick it up and look at it from the edge so all you see is a line. Now as you look at if from the edge, spin it on the stick through the central axis. If the projected line length changes over a rotation, it can fall in.
Thank you to Wayne L. for help with this problem!!