Find and replace

One time I wanted to rename a single-letter variable in my code for a USACO problem so I can present it publicly with less confusing variable names.  But turns out, it replaced every instance of that letter in the entire file, and not just the variable reference, and the result was... chaos.

I don't know why, but we seem to have the tendency to look for very simple patterns, and just call it done.  Even when those simple patterns make absolutely no sense.  For example, today I was trying to calculate the angle between vectors $\mathbf a=\langle2,-1,2\rangle$ and $\mathbf b=\langle2,2,1\rangle$, so I use

$$\cos (\mathbf a, \mathbf b)=\frac{\mathbf a\cdot \mathbf b}{\|\mathbf a\|\|\mathbf b\|}=\frac{4}{3\cdot 3}=\frac 49$$

...but when I write down the answer, since it looked like a perfect square, I thought that was $\cos^2$ and wrote down the $\cos$ as $\frac 23$.

The thing is that this isn't even your typical engineer's induction, like how Borwein Integrals eventually fail; this is just... I don't know what to call it.  Stupidity.  Ignorance.  Perhaps arrogance would be a better label—you think you're so good at math that you start doing things that are just too square to be true.

Now of course, in section where I described my first naive attempt in my previous blog post ("An unusual way of telescoping") you get another example of this—me assuming the lower bounds for all telescoping sums were $1$ and messing things up in contest, but this isn't exclusive to math either.

Years ago, I had to go to a summer camp my mom signed me up for, and I didn't want to go.  Her response was that learning happened everywhere even when I'm supposedly just doing nothing.  Turns out it's not, this is yet another stupid pattern-matching of mine because I recognized some boredom in this situation.

Why, you may ask?  Well, I discovered some kids playing the game Uno, learnt the rules, and joined in.  And look, when I had 1 card left and was about to win for the first time (in forever...) my hilariously stupid brain just happened to decide to randomly run around and disrupt the room and celebrate wildly.  My brain recognized something to celebrate, and did exactly that without even asking me if I should.  And, of course, I didn't say the word Uno.  This happened another time, and I ended up losing.

So why the heck do I assume so much?  I keep seeing everything in the world through a funny little dumb brain of mine, which just constantly fools me into thinking that I do understand things, when I don't.  I want to actually understand things, but I guess my brain wants to think I understand things.