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Dear Wisconsin

A brief post about the union-smasher bill that I’ve heard just passed in Wisconsin today.

The main issue I have is that, thanks to this, our children will now be dumber. So dumb, in fact, that they may continue to think things like this bill are a good idea.

The five states (before today) without collective bargaining for teachers rank 44th and 47th-50th in standardized test scores. Wisconsin is currently (I believe) 2nd.

The Governor had previously said that this is all about the budget, the budget, the finances. When he realized (that is, was told, but didn’t know, because Wisconsin’s Governor did not graduate from college) that he could pass the non-financial provisions of his bill without a quorum, he stripped it of the elements geared toward the budget and ended collective bargaining anyway. Guess it wasn’t about finances.

As a Wisconsin native and an educator, I can really say only two things. 1) I am very thankful to have come from a state where the education was so good. 2) I am sad that I may be part of the last Wisconsin generation who can say that.

Partition Numbers

This is amazing. Rather than write on it (hey, class in 40 minutes), I’m going to link to the blog of a really cool physicist/mathematician that I read from time to time. Trust me, you’ll want to read up on this:

Finite Formula Found For Partition Numbers.

New Year’s

It’s been a while. I wish I had more time to write. I certainly could, it just isn’t currently prioritized high enough.

2010 was certainly one for the books. Unfortunately, one of the only reasons why is that I found out I’m terminally ill. Other things happened, for sure, but hopefully you understand that this news makes everything else difficult to consider important.

The good news? Every year I live after this will be great, if for no other reason than the fact that I’m still alive.

It’s funny how the biggest things in your life make you appreciate the littlest things.

Ending A Semester

Semester is ending soon. The first semester I’ve completed since my diagnosis.

2 homeworks due Wednesday. Another homework and a paper due next Thursday. Finals to give and grade next Tuesday and Wednesday.

I’ve already started applications for next year. Atop the list is Binghamton (formerly SUNY Binghamton). I will also be applying to Ohio State, and a few other places as well. The applications are unreasonably expensive, but it has to be done.

A semester gone by. I haven’t gotten to the gym but a few times. The one thing I can do to put off my illness as long as possible is be in the best shape I can be. I don’t have time. My other priorities are weighing me down. But, if I take time to go to the gym, I don’t get my work done. What good would it do to be in better shape if I get kicked out of school? I can’t do everything.

I feel like my life is running itself in circles.

Non-Transitive Dice 3

For background (if you care), see this post (which links to the first one as well).

I have proven that balanced, non-transitive n-sided dice exist for every n>2 (and the proof is quite slick, I think). The trick to this is a particular kind of labeling followed by concatenation. I have also proven several nice things (if a set is balanced, all 3 dice add to the same total!) on top of my original thoughts. My task now is attempting to find an upper bound for the victorious probability. My conjecture is 2/3. I also conjecture this is unattainable, that is, it is asymptotic.

I’m going to be writing a lengthy paper on this topic, so it’ll be easier to link to that once it’s done (in April or so).

That’s enough on dice. Back to your regularly scheduled blogramming.

Thanks for reading.

Non-Transitive Dice 2

You’ll want to read the first post to catch up.

So, I’ve decided to call these scenarios “balanced non-transitive dice”. I’ve yet to prove that balanced non-transitive n-sided dice exist for n>2. It should be straightforward, but a simple construction would be nice. At any rate, I’m convinced of this.

The issue beyond is the following: Most solutions so far have a victorious probability of: (floor((n^2)/2)+1)/(n^2). This is fancy language for “as close to half as possible”. The first post mentions this. I have no doubt that this is a sharp bound for n=3. This means that a set of balanced non-transitive 3-sided dice exist with victorious probability 5/9 (it’s written in the first post), but that a set does not exist with victorious probability 6/9.

A professor of mine ran a program for n=4, with interesting results. The only balanced sets he found had victorious probability 9/16. Unless his code has an error, the bound is sharp for n=4.

I then did some diagram chasing by hand, and found, for n=5:

A: 15, 9, 8, 7, 1

B: 14, 13, 6, 5, 2

C: 12, 11, 10, 4, 3

A beats B beats C beats A 14/25 of the time. The bound is broken! Then, just now:

A: 15, 14, 5, 4, 2

B: 13, 12, 11, 3, 1

C: 10, 9, 8, 7, 6

A beats B beats C beats A 15/25 of the time! Now this is cool.

More to come.

Thanks for reading.

Cholera

Over 1100 people have died in Haiti recently because of Cholera. Do you know what the cure for Cholera is? Fluids. WATER. We can send celebrities to Haiti to sing, but can’t send water. Are you going to lose sleep tonight about this?

…You probably should.

My latest fascination: A set of three dice A,B,C is said to be non-transitive if, in a contest between any pair, probabilistically, A will beat B over half the time, B will beat C over half the time, and, oddly enough, C will beat A over half the time.

There are several examples of this using only the numbers 1 through 6, but the dice have things like 4 sides all with 3s, which seems like cheating. So, we are instead going to look at dice where every side of every die has a different number. It is clear (at least to me), that the numbers may be consecutive integers.

An example (using 6-sided dice): in the book Mathematical Fallacies, Flaws, and Flimflam, we find

A: 18, 9, 8, 7, 6, 5

B: 17, 16, 15, 4, 3, 2

C: 14, 13, 12, 11, 10, 1.

In this labeling, we see that A beats B 21/36 of the time. B beats C 21/36 of the time. C beats A 25/36 of the time.

I don’t like this. Why? Well, it isn’t fair! So I set out to find a set of fair, non-transitive dice. After a while:

A: 18, 14, 11, 7, 4, 3

B: 17, 13, 10, 9, 6, 2

C: 16, 15, 12, 8, 5, 1

A beats B beats C beats A with uniform probability 19/36 (as close to being 1/2 as it can be, by the way). This, I like. I then turned myself to the following question. What about dice with other than 6 sides? 5-sided:

A: 15, 11, 7, 4, 3

B: 14, 10, 9, 5, 2

C: 13, 12, 8, 6, 1

A beats B beats C beats A 13/25 of the time (again, barely over 1/2). 3-sided:

A: 9, 4, 2                 9, 5, 1

B: 8, 6, 1     OR      8, 4, 3

C: 7, 5, 3                 7, 6, 2

A beats B beats C beats A 5/9 of the time (again with the 1/2).

These were all done with a particular construction. I won’t go into that here. I will mention that it doesn’t work for 4-sided dice. This bugs me. I even went so far as to conjecture no fair non-transitive triple of tetrahedral dice existed. Oops:

A: 12, 10, 3, 1

B: 9, 8, 7, 2

C: 11, 6, 5, 4

A beats B beats C beats A 9/16 of the time.

This definitely does not work with 2-sided dice (otherwise known as coins), but I will at this point guess that for n>2, three n-sided dice exist which are both non-transitive and fair. My next goal is to explore whether or not we can construct fair non-transitive dice where the probability is farther from 1/2.

Thanks for reading.

History

Does anybody remember the last major hurricane to make landfall and do serious damage before Katrina?

Does anybody remember the last terrorist attack made by a non-American before 9/11?

Does anybody remember what happened the last time we declared war on an idea?

Don’t be surprised when this stuff happens, friends. Those who do not remember history are doomed to repeat it.

More Grading

I’m still grading these exams. I don’t get angry when my students are idiots, which many of them clearly are. If anything, it makes me sad. WE DID THIS.

You can train a monkey to repeat after you. Really. You don’t like math? Fine. You’re not going to need this? Fine. But if you can’t follow instructions and do something you’ve seen a few times, job training is going to be a bitch.