#math trivia

What’s your favorite math

hmmm.... i'm a pascal's triangle enjoyer. lots of fun identities, and i dont come across binomial expansions that often but i always use pascal's triangle for that. if you highlight all the odd numbers, you get sierpinski's triangle. the diagonals give you the triangular numbers, the tetrahedral numbers, and so on. the rows sum to powers of 2. the hockeystick lemma. probably some other stuff that i dont know.

this is a hard question because the word "favorite" makes me think about picking the absolute best one and having a passionate explanation for why i like it so much. pascal's triangle is not groundbreakingly cool or advanced, but it's kind of fun.

other math party tricks that i like:

the dropping sticks approximation of pi (buffon's needle)

the Long Line

the well ordering theorem

axiom of choice proofs

So how big does the inner circle need to be in order to put those four points on one straight line?

It turns out to be about 38%. But this is math. We want to know exactly how much and we want to know how to find the answer!

Let's consider the bigger circle a unit circle, so R = 1. Then consider this triangle.

Ten equal sectors define the entire 360°, so they're 36° each. The center-touching angle of this triangle encompasses 2 of those sectors. Therefore, 72°.

Unfortunately, I have to use cosines now. Those who fear trigonometry may want to look away.

Cosine equals adjacent over hypotenuse. We defined the hypotenuse to be 1, because it's a unit circle. So the cosine of 72° is the height of that triangle divided by one. Or just the height of that triangle. h = cos 72°

Now lets look at this slightly different triangle.

That's only 1 out of 10 equal sectors that make up the entire circle. That's a 36° angle. And the cosine of that one is the height of the triangle (the same as the other triangle) divided by the radius of the inner circle — exactly we're trying to find! cos 36° = h / r But we know what h is. cos 36° = cos 72° / r Now we solve for r. r⋅ cos 36° = cos 72° r = cos 72° / cos 36°

That's the exact answer: cos 72° / cos 36°

If you throw it into a calculator, you'll get something like ≈ 0.3819660112510519363...

Like I said, about 38%. Or 38.197%, if you want to be pedantic. Which I hope you do.

How to do square roots on paper.

I also made an animation with pause and play buttons here. Reload that page to see it calculate the square root of a different pseudorandom number.

if you were to draw a plot of f(x) = n/x where n is an arbitrary constant, f(x) approaches infinity as x approaches zero, and infinity is not a number

the simple explanation is that dividing any number by zero is impossible

the more complicated explanation is that for any given n, n/0 gives the same result no matter what value n is and that's not particularly useful

incidentally this is similar to why quantum superposition strikes us as weird

the wave function gives more than one valid result and there's nothing in the wave function that distinguishes any valid result from any other

the cat is not both alive and dead in the famous thought experiment

it's just the math we use can't tell us which state the cat is in and you have to open the box and look if you want to find out

the same way you can't calculate which side a flipped coin will be face up when it comes to rest

The Monty Hall Problem

The Monty Hall problem is an interesting exercise in conditional probability. It focuses on a 1970’s American television show called Let’s Make a Deal hosted by television personality Monty Hall.

The game would end with a contestant being shown 3 doors. Behind one of those doors, there was a prize. Behind the other 2, a goat. Monty Hall would ask the contestant to pick a door. He would then open…