More than Infatuation

Math has always been one of my great loves. Not my only great love, of course, and certainly not that which I love the most, but it ranks pretty high on the list. The way numbers can be juggled and moved, how starting from a very simple base one can build huge impressive structures, and how Fermat didn't have quite enough room left to finish off his proof.

With this love of math, my view of the world is changed. Sometimes I see things as sets, sometimes I find myself counting (sometimes in binary), occasionally I wonder about the number of M&Ms in a jar, and find myself trying to integrate over the jar in different ways to find the best approximation. Then there're the times that I find myself proving the obvious, simply so that I know it's something that, given my axioms, I can know to be true. This also, of course, explains my love for physics.. physics is obviously just math, but less theoretical.

And as my view of the world is changed, I find that I sometimes have trouble understanding other people. I don't know how they can't see what I find to be obvious, I'm not sure where our understandings differ. Perhaps they never had that third-grade teacher that had puzzles and games instead of book exercises. Perhaps they never saw that math could help them win party and carnival games, so opted instead to focus their efforts on poetry and wine.

Perhaps they're simply not me. Perhaps we're all different, we each have our own unique gifts and character flaws. Perhaps I should have taken more psychology classes in college, but they didn't generally have enough math. And perhaps I'll consider mathematical aptitude to be my superpower. It and EMP generation, what more could a guy ask for? Well, those, and the real, true love of his life.

So with those goals accomplished, I can start spending more time wondering... how exactly /do/ you decrypt PGP without the key? And why /does/ a vase just want to keep doing whatever it is it's doing?

Red, Yellow, Green, Red, Blue, Blue, Blue

The first year I went away for school, we got split into groups for a massive "intellectual" competition. Building bridges, flying paper airplanes, coming up with good ways to do fun things. So my group was doing fairly well, and then we get to..the jelly bean competition.

You know, there's a big jar, it's filled with small pieces of candy of some nominally uniform shape and size, often jelly beans or M&Ms. The point is to attempt to guess the number of tiny pieces of candy in the jar, and if guessed correctly you win. Depending on the situation, the prize can differ. For example, in middle school this was a game at a birthday party I attended. Sadly, I was off by 15.. but the next closest was about 200 away, so I got a jar of M&Ms to call my own! Until all the candy was eaten and then I only had a jar until my mother decided to can some more fruit, and then I only had the bittersweet memory left. That and the glory. Can't forget the glory.

This time, though, the prize was more points for your team. Not that the points really mattered either, but it's the whole glory thing again.

So off we went, looked at the jar, smelled it, shook it some, used our x-ray vision and powerful math skills. Anything we could to get the correct number. Compare our answers, and decide which answer to use from them.

I learned one thing from this. Being able to convince someone you're correct is more useful than being correct to start with.

Well, that and you really need more than a day to eat a jar of M&Ms.

A Sixty-to-One Ratio

Being the 4th of July, there were a great number of pyrotechnic displays this evening. I even managed to go see one myself, for the first time in quite a while. I could probably have paid to actually go to the amphitheater and heard the music that the display was set to, but being across the street, behind the hill was good enough for me.

I'm used to much less..impressive displays. A firework, a pause, a firework, a pause, rinse, repeat. For a while that'd continue on until it got late enough that they'd light anything they had left over. But not tonight, no. It was set to /music/. So three times, once for each verse I suppose, it started out with a couple fireworks, and then a few more, and then more than that, until three times we reached the climactic large number of fireworks in the air. Best of all, though, were the smoke clouds that remained after these fireworks. Well, after the one right before the climactic huge number, anyway, the clouds where you could see where each part of the firework flew through the air to its eventual decay and demise.

It was wonderful. It was loud. It lasted about three minutes.

But, hey, I brought a book. So I read a few really good short stories while I waited. Until they turned out the street light, and then I had to wait for half an hour before the fireworks began.

Of course now I'm glad that the fireworks were so close to work, for I'll apparently have to wait another half of an hour before the roads are in a decent shape to drive on. Traffic's horrible, but at least I've got a computer to use at work.

But for the three hours of waiting, it was a good three minutes spent.

How Many Atoms are there in a Dog?

I recently ran across the number 2^24036583 - 1. (No, that's not the answer to the above question.)

Looking at that number, I know that it's huge. There are about 3 decimal places for every 2^10 (2^10 = 1024), so I figure there are slightly more than 24036583/10 * 3 ~= 7.2 million digits in that number. 7.2 _million_. That's one huge number, many times larger than a googol!

Why anyone would want to use such a number is beyond me. Counting things, it's more than anything I could possibly comprehend, I'm sure. Supposedly there are about 10^50 (give or take a couple orders of magnitude) atoms in the earth. So this is much less than a googol, much much much (etc) less than 2^24036583 - 1 (which we'll call p from now on).

~10^60 atoms in the sun (and ~10^60 atoms in our solar system), there's still a lot of fudge room to have a googol atoms in our galaxy, much less p. According to people that know better than I, there are probably less than a googol atoms in the universe, much less than that in just our galaxy.

So what use are the other 7 million-ish digits in p? How can anyone possibly comprehend the /size/ of this number? How can anyone really /care/ about this number (even if everything should be loved, surely there are exceptions!) How can anyone hold enough of it in their mind to start thinking about it.

Why would anyone care to prove that p is prime?

Of course I then realize that while it's larger than tower(5), it's still less than tower(6), so life is again good.

Let's Make A Deal

Recently I was reminded of the (in)famous Monty Hall problem. Most people that have had a probability class have heard the problem and understand the answer, but it still manages to hurt my brain.

So on "Let's Make a Deal", Monty Hall would give the player a choice of three doors, one of which had a good prize, the other two having junk. Well, the player picks one, then Monty shows a bad prize from one of the two remaining doors and asks if the player wants to switch to the last door. (This isn't exactly how the show worked, but makes the problem a _lot_ more interesting).

The question, then, is should you switch?

Now, it seems like there should be a 50/50 chance of getting a winner if you switch, because there are two doors and one winner, right? Except it's actually much more likely that the other door is the one with the prize. Bizarre.

I could go into great detail explaining how and why this works. But I won't. I'll just say that once you've seen a losing door, the door not selected has a 2/3 chance of being a winner, the door picked only a 1/3. Even though it was already known that one of the two doors (at least) was a losing door.

Now, I love math. Thinking about theoretical possibilities, ignoring real world applications, I /enjoy/ spending time that way. But that doesn't mean that Monty doesn't hurt my brain. The really important point of this problem, though, is that Monty does know which door is the winner, and doesn't reveal that one. If he just revealed a random door, it wouldn't make much of a difference. But it's hard to accept that he'd be willing or able to give /any/ help in being a winner.

While some things may seem inconsequential, you already know that one of the doors is a loser, the simple knowledge may make a huge difference.