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.
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.
Whenever you’re called on to make up your mind, 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.
— Piet Hein
(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.
5:27 PM phenomist: do you use gridderface to make
pretty puzzles?
…
5:52 PM phenomist: actually nvm excel is probably
easier lol
Okay I’m sorry this is a horrible puzzle where the rules don’t make
sense and I didn’t even get it testsolved. I just wanted an image to
concisely demonstrate the capabilities of
gridderface, my
puzzle marking and creation program, for the project homepage, after
somebody expressed interest in using the program to write a puzzle. Then
I got tremendously carried away.
These are the thoughts that sometimes keep me awake at night.
These are things I don’t want to think about. These are things I’ve
spent hours thinking about, never productively. They are worrying, but
unlike typical worries in my life, it is fundamentally impractical to
take steps to resolve or mitigate them, after which I may rest assured
that I’ve done my best. The reason is that they also happen to be either
untestable/unfalsifiable or only testable if one incurs absurd and
irreversible costs, mainly dying.
Sometimes I explain them away to myself successfully and move on.
Sometimes I read what I’ve written and think about these thoughts and do
the cognitive equivalent of looking at them funny, as I’m expecting most
readers to feel if they get that far — why would anybody be bothered, or
afraid, or soul-crushingly panicked about these things? Life is so busy,
there are literally more than sixty-four items on my HabitRPG to-do
list, and besides, there are so many serious global issues humanity is
actually facing right now, and people who are actually deprived of basic
rights and resources and have to struggle to stay alive. How can I
possibly be bothered by these absurd remote thoughts?
But I know that other times I do feel those emotions exactly. And if
I stare just right, I can feel those emotions bubbling beneath
the surface in me. Sometimes I can’t explain the issues away to myself,
and a deep soul-sucking pang grows in my stomach. I’m irrational — I’m
afraid of some of these thoughts — and I have submitted to the fact that
there are some edges of my irrationality that would not be worth the
effort to fix if just not thinking about them is better.
Sometimes these thoughts make me wish I were not so rational.
Sometimes they even make me wish I were religious; it would be easier if
(I believed) consciousness were, somehow, special. I suspect if I tried
really hard, I could make myself believe something like that sincerely.
But I think that’s a betrayal of myself I’m not willing to take. I think
there are better ways to remain happy.
I want to maximize happiness. Thinking about more general moral
principles will help with that, but the remoteness of these particular
thoughts is such that I doubt I’ll ever have to make a choice that would
benefit from me having thought about them. At least, I think the chance
is small enough to not be worth the negative utility spent thinking
about them.
So: “There is nothing to fear but fear itself.”
But I feel frustrated: not thinking about something just doesn’t seem
like a solution. I don’t know how to come to terms with just how
irrational happiness fundamentally is. And I still can’t resist thinking
about them sometimes…
Well, it’s been over a week, which is a long time for blog posts to
be delayed after the event they’re documenting in probably all of the
world except my blog. So.
I guess this post should start with a bit of background. I’ve been
puzzlehunting for… wow, three and a half years now. I was introduced to
puzzlehunts from AoPS, when some fellow members got together a team for
CiSRA
2011, and I think I’ve participated to some degree in every known
internet Australian puzzlehunt since.
But as for my experience with the MIT Mystery Hunt in particular, I
sort of hunted with a decidedly uncompetitive AoPS team in 2012 (I think
we solved one puzzle exactly), but my serious hunting career began when
dzaefn recruited me into the Random team (then Random Thymes)
for the 2013 hunt (and I did
blog obliquely about it). We didn’t win (and I actually didn’t
participate that much because I was traveling with family) but the next
year (as One Fish Two Fish Random Fish Blue Fish (1f-2f-17f-255f (I am
evidently in a parentheses mood today because as you’ve probably
noticed, the amount and depth of parentheses in this sentence are
positively alarming (lol)))) we won.
And I do have a half-written post about that which will never get
posted (and I also didn’t participate that much, because my
family was moving that weekend) but okay, let’s just drop any semblance
of chronological coherence on this blog and dump a short version of the
list of puzzles and parts towards which I contributed solving, as I
wrote them down one year ago:
These things are a thing. They’re like mini-trampolines that you are
supposed to strap onto your feet and jump around on to have fun, if
you’re, like, six years old, maybe?
Anyway, if you happen to run into a pair of these, please be careful,
and please think twice before doing something like, I don’t know,
immediately strapping them on after you see them and jumping as hard as
you possibly can, especially not when you are jumping on a hard
unforgiving surface, or when you are surrounded by a bunch of
impressionable children whom you could be concerned might follow your
example or be swayed by your bad influence or something.
I’m assuming you want to use D largely, but not entirely, for
competitive programming. That’s me right now.
Basics
Syntax is very similar. Function definitions, semicolon-terminated
statements, variable declarations, and so on. You can declare
int main() {...} or void main() {...} or
something with arguments.
Basic types like bool and int and
double are all there. Wonderfully, long is 64
bits. Instead of unsigned whatever, just prefix a
u, e.g. uint.
Arithmetic operators and bit operators are all there too, including
unsigned right shift >>>. Although ^
is still xor, D has exponentiation as ^^. Sadly,
% is still same-sign remainder; there’s no true mod.
import std.stdio;
Casts look like cast(int) x;
Control Flow
if, while, for,
do, and even switch all work as you’d expect,
along with break and continue.
foreach is the nice addition though. Not only can you
iterate over arrays and stuff, but range loops go like:
Well, there are better memory-triggering songs but I think this
pretty much sums up how I feel about blogging right now (possibly
including the very act of choosing that song.) And college apps. And
life. Plus, the music video is silly in its own incredible way.
Anyway. Around this time a year ago, I made
a post talking about how around
a year before that,
I paused my participation in big high-school competitions, for a variety
of reasons.
And then I rambled on life and programming competitions.
If you didn’t get it yet, this post so far has been written to
meaningfully echo the last one. Nothing so abrupt has happened this
year, but I just realized how nice it was to have a paragraph humorously
listing the weird stuff I had gotten myself into over the course of
2013, so I’m going to do so again, even more completely.
