How Not To Play Dice

The question is as follows: Can two loaded dice be constructed such that, rolling the pair gives the numbers 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 with equal probability?

Now, a couple of warnings going in. First and foremost, I don’t really know how you go about loading dice. I’m a mathematician, not a dicetition, so this takes a very abstract approach to loading dice. For each die, we’re basically choosing the probability with which each number occurs in a roll.

Second, as little as I know about loading dice, I’m even less sure about my abilities with probability. Honestly, I avoid doing probability as often as I possibly can. Well, I don’t go out of my way to find probability to avoid, but it does make me very uncomfortable. So if I make any mistakes, feel free to comment. The only reason I’m doing this problem at all is that it involves a neat algebraic jump.

So here we go.

What Is Math? (Baby Don’t Hurt Me, No More)

Today’s XKCD

Now, I generally avoid philosophy at all costs. All costs. For many reasons. Any good thought has likely been thought and better expressed before. It’s hard to separate philosophy from the pretensiousness of many who like to discuss it. Many reasons. But I am a mathematician, and it’s hard not to think about what it is I’m actually doing.

So, the comic expresses the traditional stacking of sciences. Biological, Chemical, Physical, Mathematical. And somewhere, burning in some deep, dark pit of hell, Computational Linguistics. It’s an easy stack to make. As in the quantum physics course I just took, it’s often difficult to tell which came first, the physics of quantum mechanics, or the math of it. Indeed, it often appears as though quantization in physics, is a result of a necessity of the math. Relativity, I have argued, is a logical necessity – from the math. Sometimes, it’s even more difficult to separate the math from ‘reality’. So it begs the question, I think so at least, what is math?

To which I would like to make a proposal.

News Of Interest, Stuff to Think About

I feel so neglectful. Here’s what my day was like today.

Fox sits down at computer, piece cake in hand. Paw, rather. “Ah, today’s XKCD provides an excellent platform on which to launch a discussion of the nature of Mathematics. … But, if I move already filtered strings to a seperate list, this will prevent them from being filtered multiple times and thus decrease the net time the program spends on filtering…” Fox wanders off in thought, coding madly in his head before sitting down and spending the next few hours banging out yet another version of the project, cake and philosophy forgotten, one of which is promptly eaten by BrotherFox. Damnit.

Anyway, before I get back to serious math writing, some points of interest.

Carl Zimmer is one of my favourite science writers, and here he discusses one of the more interesting experiments I’ve seen in a while. Researchers grew/cultured/observed a single colony of e. coli over 20 years, periodically freezing samples, so they could observe changes over time. And recently, their e. coli developed the ability, through random mutation and natural selection, to digest a completely new type of food. Now, e. coli have been observed in nature with this digestive ability, but it has always been due to plasmids, sections of genetic code that the organism snitched from others. But this represents an actual genetic change to the e. coli itself. And because of the way they froze samples over time, the scientists were able to look back, and see exactly how and when the bacteria developed this ability. It’s very exciting stuff. Carl Zimmer expresses all this in much more detail and far better than I ever could. Go read. Go, go!

Following the sort of ‘biology’ trend, next we have from the BBC a video of a live whale birth. I’m not sure if it’s awesome, or terrifying.

Then, also from the BBC, something more physicsy. This is old stuff, I think, but interesting and worth a mention. Scientists (I love using the general ’scientists’ – it’s as though there’s just a building full of scientists somewhere, just thinking stuff up and doing it) think they can predict earthquakes up to two weeks in advance. The geologic activity that leads up to the actual quake, so the theory goes, results in this massive release of electrons that make their way to the surface, resulting in a detectable change in infrared radiation over the area about to be hit. This is very exciting because current earthquake detection systems give you minutes warning. So I hear.

And then, still on a physics bent, but incredibly more mindbending, scientists have evidently shown that reality doesn’t exist when you’re not looking at it. Of course, the fact of the matter is far more complex than that. In short, they had two competing theories about measuring the polarization of light, one based on quantum mechanics, the other based on the idea that light had some fixed polarization before it was measured. And as it turns out, the results were several orders of magnitude away from what was predicted assuming a pre-measurement polarization. In shorter short, the polarization seems to have not existed before they attempted to measure it. It’s all very philosophical. And stuff ^^

Stuff to think about!