What was something that someone did that made you think they were really smart?  by Anthony Cummins

I had a friend once. Let's call her A.

A had recently enrolled as a pupil in our school as a transfer student from Germany (she was born in the US, but had spent most of her life there).  Amicable though she usually was, A seemed to be distant and apathetic whenever our calculus class rolled around. Her head was often propped up against her forearm, eyes half-lidded and languid, as if merely being around mathematics took so much of her brainpower that she had not enough to remain alert.

A had assumed her characteristic posture one evening, when, in class, our teacher asked about the homework problem from the previous night. Writing it on the board, he inquired how far the class had managed to get. Dejected murmurs and blank stares were the answers he received. (The problem was a particularly nasty one involving double integrals and trigonometric functions, among other things). But while the class seemed to be fairly apprehensive of the castigation which was sure to follow, A had not yet even taken notice of the proceedings.

With a scowl upon his face, the teacher cleared his throat and asked A how she had fared with the assignment. Visibly annoyed at having been disturbed from her semi-conscious state, A looked at the board for several seconds and said a certain number. (I can't remember what it was, but let's call it 0.323)

The rest of the class snickered, knowing that there was no way that silly girl could have gotten anywhere near the right answer. But the teacher's jaw became unhinged and he began to shake visibly as he punched numbers into a calculator (as mathematicians know, we often leave numbers in exact form, rather than a decimal approximation,  but we often have an idea of what number the exact form of the answer is close to). A was actually correct. The class fell quiet and all eyes were upon her.

He called her to the board and told her to explain how she had solved the problem. Marker in hand, A filled about half the whiteboard with the working to the question, concluding with the answer in exact form. Again, let's imagine it's $\sqrt{\cos\left(\frac{7 \pi}{15}\right)}$.

The other half was filled with series expansions of the above answer to convert it to decimal form, in case we were wondering how she had arrived at .323. And A had done all of that in her head in the space of about 5 seconds.

It turned out that A was a child prodigy in math (she was two years younger than the rest of us, but she was very tall, so we hadn't noticed). She later told me that math was like an involuntary reflex action to her. She could easily multiply 7-digit numbers in her head and solve differential equations; numbers manifested themselves as shapes and patterns in her mind. A had struggled with math for years, though, because writing out the working was tedious and primitive to her, similar to us thinking that 1+1=2 is self-evident (yes I know there is a proof of this, but you get the idea). But once she got the hang of it, she was unstoppable.

What was something that someone did that made you think they were really smart?