These are simple, obvious concepts and constructions that we live with each and every day. The first is observable with blocks and introduced reasonably early on in education, and the last two are directly observable.

Unpacking thehowof both concepts comes significantly later, if at all.

Consider our right-triangle friend on a Cartesian grid.Yes, we all learned this in school and filed it away somewhere. Along with the Quadratic Equation, except that the Pythagorean Theorem can help you figure out if a window frame is square or not (e.g. not tweaked), as you can decompose the shape into a rectangle diagonally-divided--itself two triangles. If these are found to be right triangles (e.g. the Pythagorean Theorem holds), then you've just made your window frame true and can start nailing it into its surrounding framing members.The sum of the squares of the sides are equal to the square of the hypotenuse.

It's useful for this.

But what is really going on here? It wasn't until my first year at university that I saw the diagram from Euclid'sElementsposted on a faculty member's door, and thought it a brilliant illustration:

(Clicky-link to image here. I believe that Moo is looking to do away with IMG tags, anyway.)

Aha! Certainly different than the recitation of the theorem and scribbling down of its elements.

The other two questions are so simple that even a child can ask these without prior knowledge. Answering them, too, is seen as simple--but it's not. The first response requires knowledge of optics, and the second requires knowledge of chemistry.

And it is here that the thread opens:what other questions seem terribly simple, are posed (or their answers used) routinely, and are not thought through?I'm curious to know these things, and this curiosity once had me scribble out 3 pages of equations in a high school physics exam. I only memorized a scant few equations, you see, and derived the rest. My answer went through kinematics, optics, E&M, and circled back in on itself--and in the end, I'd proven that, yes,p=mv. If I still have that exam result, I'll dig for it to post here--it was a blast to think through and write*.

Oh. And the Fibonacci series. Specifically, the equation that generates the series. It'scrazyto think that thing generates integers.

*I received a zero on that exam question. Oh well.