theoretical and applied randomness by betaveros
Darn, I only lasted four days. That’s pretty bad.
And I broke like 90% of my HabitRPG streaks too. I was busy running Monte-Carlo calculations to estimate the number of domino logic puzzles, and forgot about midnight. Okay, before that I spent twice as much time on Flight Rising for whatever reason. Bad life decisions.
I guess that means today I have to post one now, before going to sleep, and one later. Eh, time to harvest really weird mini-posts from nowhere.
(Faux-philosophical blog content, posted as part of a daily posting streak I have openly committed to; standard disclaimers apply)
This is a hard essay to write because (1) it’s very irrational and I should (and I do) know better — death by car accident is much more likely than death by an airplane crash, but the latter is scarier because it’s more vivid and we have less control over it, and (2) people don’t like talking about it. When I tried writing it, though, I realized I already burned through most of the down-to-earth worries in the posts I made between April and August of 2010. They still coherently and accurately sum up my current thoughts surprisingly well. And most of the irrational, overly philosophical fears appeared in Thoughts at Midnight. So there used to be a lot of fluff here like this, which was inducing procrastination because I don’t know what to include and what to cut, but now that I have a daily deadline, I cut most of it. Here’s what’s left.
One, xkcd:
Two, bonus quote: As really-long-term readers know, I have had a reason to think that I might actually die in the past few years, a real reason that has stayed with me and gotten me thinking now and then about what my meaning of life is, instead of a short-lived fuzzy philosophical feeling obtained from reading Tuesdays with Morrie (which is not to say that Tuesdays with Morrie isn’t a good book; I just suspect no book can convey everything a personal experience can.) Anyway, it’s over in all likelihood, but the point is that in the middle, I wrote an essay for class in ninth grade, which I find equally coherent and equally representative of my views. The conclusion runs thus:
(Frivolous blog content, posted as part of a daily posting streak I have openly committed to; standard disclaimers apply)
Out of boredom and curiosity, I graphed how many emails colleges sent me, excluding the colleges I actually applied to. I am being extremely polite and just calling them emails. I’ve wanted to make this for a long time, but it wasn’t until I saw this post about an email experiment on waxy.org/links that I understood which tools I could use to quantify my emails. (And then I actually made it and procrastinated posting it here for two months. If you look at my GitHub page or activity you might have seen it already, though. Oops.)
I don’t think the results were expected. Other than saying that, I leave the interpretation up to the reader because I’m on a tight blogging schedule. Cool? Cool.
Step-by-step instructions:
(Frivolous blog content, posted as part of a daily posting streak I have openly committed to; standard disclaimers apply)
It is quite interesting that Wikipedia’s article on Ninety-nine (addition card game), plus many of the following search results (ignoring the identically-named trick-taking game that is guaranteed to show up), have the same basic idea but wildly differing assignment of special cards from the one I’m familiar with, which everybody I can recall having played with agrees on. (Admittedly I’ve only ever played this among Taiwanese friends.) The only special-card assignment method that came close was a certain person’s “stuff from my old blog” dumping post I bumped into very accidentally. (His 5 is our 4; our 5 skips to an arbitrary player. The post also clarifies that negative totals bounce back to zero, and includes a clause whereby players must state the running total after playing and lose if they’re wrong. Interesting.)
Anyway, yes, I am documenting the rules to a card game on this blog. I think this deserves to exist online.
These rules are not completely rigorous because I don’t know them completely rigorously. You can use common sense to reach a consensus in corner cases.
Use a normal deck of playing cards, or two or more identical decks if you want. Deal five cards to each player and set the rest aside to form a draw pile. Cards are played into a discard pile in the center. Players sit in an approximate circle and take turns along the circle, playing one card and then, usually, drawing one replacement card from the draw pile, so in normal 99, hands stay at five cards. When the draw pile runs out, shuffle the discard pile to become the new draw pile.
Obligatory life update: I have graduated [from] high school.
But that’s not what this post is about. I contemplated setting up a schedule for my blogging three long years ago, and decided against it, because I didn’t think writing was a high enough priority for me. Well, I am setting up a schedule now: I am going to post something on this blog every day until I have to leave the country (which is happening once before college, so it’s not for as long as you think; but I might decide to continue the schedule anyway after I get back. We’ll see when the time comes.)
Note on notation: I’m going to use \(\text{Stab}(x)\) instead of \(G_x\) for the stabilizer subgroup and \(\text{Cl}(x)\) instead of \(^Gx\) for the conjugacy classes. For the orbit of \(x\) I’ll stick with the norm and use \(Gx\), although it’s only used in confusing summation notation that I’ll explain with words too.
We keep using this silly counting argument which I thought was something like Burnside’s lemma but actually is a lot simpler, just partitioning the set into orbits and slapping the orbit-stabilizer theorem on.
If \(G\) is the group and \(S\) is the set then
Note on notation: to be maximally clear, I have bolded all my vectors and put tiny arrows on them. Normal letters are usually reals, uppercase letters are usually bigger matrices. Also, \(\cdot^T\) denotes the transpose of a matrix.
Let \(\vec{\textbf{v}}_1, \ldots, \vec{\textbf{v}}_m\) be elements of \(\mathbb{R}^n\) where \(m \leq n\), i.e. column vectors with \(n\) real elements. Let \(V = [\vec{\textbf{v}}_1, \ldots, \vec{\textbf{v}}_m]\). This means pasting the column vectors together to make an \(n \times m\) matrix (\(n\) rows \(m\) columns).
Consider the thing \(\vec{\textbf{v}}_1 \wedge \vec{\textbf{v}}_2 \wedge \cdots \wedge \vec{\textbf{v}}_m\), which can be visualized as the hyperparallelogram \(\left\{\sum_{i=1}^{m} t_i\vec{\textbf{v}}_i \,\middle\vert\, t_i \in [0,1], i = 1, 2, \ldots, m \right\}\) but is apparently a different thing in a different vector space of things. We wonder how to compute the hyperarea of this hyperparallelogram.
A PSYCHOLOGICAL TIP
Whenever you’re called on to make up your mind,
— Piet Hein
and you’re hampered by not having any,
the best way to solve the dilemma, you’ll find,
is simply by spinning a penny.
No — not so that chance shall decide the affair
while you’re passively standing there moping;
but the moment the penny is up in the air,
you suddenly know what you’re hoping.
(By the way, apparently spinning a penny is a terrible randomization process; studies have shown they come up tails 80% of the time. Tossing or flipping is better but there’s still a faintly biased 51% chance it lands with the same face it started with (PDF link). Entirely irrelevantly, is the meter amphibrachic? Nice. I’m sorry, but the impenetrable English names they give to metrical feet just sound so cool.)
As May 1 has been coming up, I’ve been half-seriously giving this advice to others who still haven’t decided. But I knew this wouldn’t work for me. I knew where I intuitively wanted to go all along.
The reasons holding me back were more… reasonable. Mostly the money. Call it an id-superego conflict.
I don’t know if the difference between my choices would mean I’d have to take out loans, or work a lot during college, or both. I don’t think either of those things would be difficult. I think tech internships over the summer could just cover the parts assigned to parental contribution (which I’m not going to let my parents pay, unless they start earning a lot more money than expected) and I think I have the skills to get those internships. But of course that’s a tradeoff. Maybe there will be something more self-actualizing or more helpful to my future career that I could do during the summer. I’m not so sure that I’ll find the same drive to program for a job instead of for a personal project I really want to use myself, or for putting off something more boring. I don’t know yet.
(Get it? Drive? Program? Um, never mind, I guess that’s a hardware problem.)
CLICKBAIT PERSONALITY TEST THAT YOU CAN DO WITHOUT SOLVING THE PUZZLE: What do you see in the puzzle image below? I have my own thoughts but I won’t bias you by posting them yet. Sound out your thoughts in the comments below! (I don’t expect this to work but I’d love to be proven wrong)
Okay so apparently how puzzles work is I go nearly a year without posting one and then when I post a terrible one, I feel guilty and obligated to post a legitimate one soon after. Testsolved by chaotic_iak.
This is a Fillomino (write a number in every empty cell so that every group of cells with the same number that is connected through its edges has that number of cells) where each tetromino has had their 4s replaced by one of L, I, T, or S describing their shape, and they obey the rules of LITS — they can touch if they are not congruent, they must all be connected, and their squares cannot form a 2×2 block. In addition, cells separated by a thick border may not contain the same number or letter.