Category: Miscellaneous


Food for Thought…

Abstinence is only 100% effective if you reject the immaculate conception.

Also, I doubt Joseph found it all that immaculate.

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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

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.

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.

Dinner

Going soon to have dinner with some friends, most of whom don’t know I’m sick. I’ll put on a happy face, tell my stories, my jokes. I’ve always been the funny guy. That has made it easy to hide what’s going on. To them I just walk funny.

I wonder if every one of my friends is hiding the way I am. I wonder if everybody is.

I gave an exam yesterday. Now I have around 480 pages of grading to do.  The students certainly don’t appreciate being told that they can’t duplicate what I’ve done for them several times. So why even do it?

It makes me wonder if there isn’t something else I should be doing with my time.