Spoiler alert: I'm gonna start off talking about kung fu and then segue into probably the nerdiest stuff I've ever said.

There is a Bruce Lee quotation: "I fear not the man who has practiced 10,000 kicks once, but I fear the man who has practiced one kick 10,000 times."

I, of course, immediately began thinking about the math involved here.

We could represent this as 1 kick practiced 10,000 times = 1 fear. 10,000 kicks practiced 1 time = 0 fear

or F(Kicks, Repetitions) = fear. Where F(10000, 1) = 0 and F(1, 10000) = 1.

So what could the function be? Let's first look at our edge cases. Presumably, only positive integers are allowed. It doesn't make any sense for somebody to try to intimidate Bruce Lee by bragging about the half a kick they practice negative one million times.

We'll also say, for purposes of keeping our variables sane that the range of F(K,R) has to be between 0 and 1. Ultimately, there is a bottom limit of fear; eventually, Bruce can't consider something less than non-threatening. The upper limit would be the point when the fear is so overwhelming that it kills Bruce. This means that 0 and 1 are asymtopes representing our lower and upper bounds. It's distinctly possible that F(10000,1) does not actually equal 0, and F(1, 10000) does not actually equal 1, but are within our reasonable margin of error. (Using double-precision floating points, that's something in the range of 2^{-2047}).

So what kind of function will give us these results?

Is F(K,R) a polynomial? F(K,R) = a*K^{n} + b*R^{m} + c

The difficulty with this is that it's difficult to come up with value of a that is non-negative, which would result in negative values for F(K,R) at very high values of K

A rational function? F(K, R) = (b*R^{m}+c1)/(a*K^{mn}+c2) + c3 (You know what? I don't want to keep track of these constants. y'all can figure them out)

Presumably, K would be in the denominator. Since increasing R should increase F(), and increasing K could presumably decrease it. I dunno though. You'd think that the guy who's practice 2 kicks 10,000 times each would be scarier than the guy who'd practice one kick 10,000 times. Plus, I think this function can increase without limit.

An exponential function then? F(K, R) = a*n^{K} + b*m^{R}

I'm pretty sure these things all increase without limit pretty quickly.

Frig. The wikipedia list of types of functions is fast getting out of my range. I only got to Calc 2, man!

Wait! I've got it! Trigometry! F(K,R) = a*sin(n*K) + b*sin(m*R)

Yeah! These functions are self-limiting. They naturally cap out between 0 and 1 no matter how ludicrously large your values are! Great! Bruce Lee had a complex cycle of how many kicks he considered intimidating. (Granted, this presumably means that they guy who practices 1 kick 20,000 times gets not scary again, but every system has its flaws.

Q.E.D.

Photo credit: public domain via Wikipedia.