The other day our 9-year old said to me, “A is 1 inch from B, and B is 1 inch from C. How far apart are A and C?” It turns out this was a joke, and the answer was supposed to be 1 inch (I think he got it from Calvin and Hobbes, though I don’t remember seeing it there), but of course I couldn’t help mentioning that the answer could be anywhere from 0 to 2 inches.
It got me thinking — what would the “average” distance be? And I think the average would be the square root of 2. My logic goes something like this: I think of the line segment AB being fixed, and then BC rotating around B in a circle. There is symmetry between the angle ABC being acute and being obtuse, so I think the “average” distance AC would occur when ABC is a right angle. In that case, the distance AC would be √2.
So here’s my question:
1) Are there any other reasonable ways of coming up with an answer (whether or not it is the same)?
2) I want the answer, √2, to somehow be an average of the two extremes 0 and 2. But it’s not the arithmetic mean (1), the geometric mean (0), or the harmonic mean (0). That bothers me. Is it some other kind of average? (Yes, I’m looking for a process that gives me the answer I want, which is kind of backwards but there it is.)
Number photo from gokoroko.