Who wants a mask?
Webpages that may or may not be relevant:
Friday, February 25, 2011
Wednesday, February 23, 2011
A Paradigm Shift
This semester, I have picked up a habit that for years I believed disgusting, vile, and absurd - I do math in pen.
Webpages that may or may not be relevant:
Math from grade 1 to about freshman year of college is all about calculating. It's pretty boring and you tend to make a lot of silly mistakes, requiring lots of erasing. It's also not particularly difficult (at least conceptually). Because of my experience erasing lines and lines of work when forgetting to carry an exponent around, and because I have never had to go back and refer to previous work, I had done all math in pencil.
"Real" math is different. Homework problems are sentences; solutions are paragraphs. Instead of having to crunch numbers, you have to grapple with an idea. This semester, Csaba (the animated Number Theory professor) said something the first day that really struck me:
Never use pencil when you're doing math, only pen. If you make a mistake, don't scribble it out so you can never read it again. Just cross it out with a single line. That way, if you try a new method and realize that your first approach was right, it is still there on the page to remind you. If the idea was sound, but it didn't work for that problem, it might work for another problem. If it turns out that it was nonsense, you just won't look at it ever again.It struck me as an interesting philosophy, so I tried it out the first day. I haven't looked back.
Webpages that may or may not be relevant:
Tuesday, February 22, 2011
When in Wien, do as the Wieners do.
Yesterday, I got back from a 3 day trip to Vienna.
Thursday - 2000
I decide I'm dropping Conjecture and Proof (leaving me with 5 math classes) despite it being very interesting. Being (essentially) my only Monday class, I decide to extend my stay in Vienna to Monday afternoon.
Friday - 1700
I meet up with 4 of my friends on the train in Budapest, ready for the 3 hour ride. It's amazing how short of a ride it is to the cities surrounding Budapest, especially when the train is very comfortable and you are learning to play Bridge. Despite being a game for the grandmothers, it is a complex, mentally demanding card game that I very much enjoyed.
We arrive at our hostel at around 21:00 and go to a pub in the center of town to get some good Austrian beer (though I think I like the Hungarian stuff better).
Saturday - 830
The first of three mornings fighting against sleep to get to breakfast before it disappears at 900. This involved getting out of bed, putting on sweatpants and going to 9 other beds and waking everybody up. I learn what Brioche is.
We decide to spend the day walking around the Ring, the center of Vienna, which includes St. Stephen's Cathedral and the Hofburg Palace complex among other things. The cathedral was much more Gothic than other cathedral's I've visited in the past. The South Tower (343 stairs in a spiral staircase) provided the highest viewpoint in all of Vienna. It also provided a view of the roof of the cathedral below:
We then traveled to the Hofburg Palace. Standing in the middle of the complex, you are surrounded by breathtaking buildings, from cathedrals to City Hall. Oh yeah, a castle too.
As we head to dinner, we split into 2 groups (did I mention there were 10 of us?). I made the mistake of choosing the group with Daniel, the food snob. After hunting for a restaurant, I decide to walk into the nearest one and ask for a table. As the menu was in German, Daniel has the waiter translate every dish for him into English (not just the entrees, but the sauces, sides, and garnish on every plate).
Daniel: I'll have the steak with some white wine. Medium-rare.
The steak arrives.
Daniel: [Cutting into the steak] This is medium well. I'm sending it back.
Me: No your not. [Takes steak and devours it].
This whole story is only significant because the reason I was so eager to get out of the restaurant was that our next destination was ice skating:
Sunday
I left with a few others at about 1000 and headed to Schoenbrunn Palace - the summer palace of the seat of the Habsburgs. I've never been inside of a palace before. I'm having a very hard time expressing my thoughts about it and the surrounding gardens (and zoo, hedge maze, columbary, view of Vienna, and artwork), but I'll describe 2 cool things I saw while touring:
There was a table clock, covered with gold, that had two clock faces - one in front and one in back. The clock was situated on a table, in front of a mirror. The face (and mechanism) on the back was designed to be read in the mirror!
Back at the hostel, I learned to play Spades, which is like a casual version of Bridge.
Monday
We head to the Museum of Natural History where, besides an impressive collection of everything related to natural history (not to mention some beautiful art and architecture), they were featuring a Body Worlds-like exhibit. Did you know that the digestive tract of a blue whale is about 0.4 miles? What about the fact that the blood vessels of an ostrich could circle the world twice?
On the way home, I learn to play Cribbage.
Webpages that may or may not be relevant:
Thursday - 2000
I decide I'm dropping Conjecture and Proof (leaving me with 5 math classes) despite it being very interesting. Being (essentially) my only Monday class, I decide to extend my stay in Vienna to Monday afternoon.
Friday - 1700
I meet up with 4 of my friends on the train in Budapest, ready for the 3 hour ride. It's amazing how short of a ride it is to the cities surrounding Budapest, especially when the train is very comfortable and you are learning to play Bridge. Despite being a game for the grandmothers, it is a complex, mentally demanding card game that I very much enjoyed.
We arrive at our hostel at around 21:00 and go to a pub in the center of town to get some good Austrian beer (though I think I like the Hungarian stuff better).
Saturday - 830
The first of three mornings fighting against sleep to get to breakfast before it disappears at 900. This involved getting out of bed, putting on sweatpants and going to 9 other beds and waking everybody up. I learn what Brioche is.
We decide to spend the day walking around the Ring, the center of Vienna, which includes St. Stephen's Cathedral and the Hofburg Palace complex among other things. The cathedral was much more Gothic than other cathedral's I've visited in the past. The South Tower (343 stairs in a spiral staircase) provided the highest viewpoint in all of Vienna. It also provided a view of the roof of the cathedral below:
We then traveled to the Hofburg Palace. Standing in the middle of the complex, you are surrounded by breathtaking buildings, from cathedrals to City Hall. Oh yeah, a castle too.
As we head to dinner, we split into 2 groups (did I mention there were 10 of us?). I made the mistake of choosing the group with Daniel, the food snob. After hunting for a restaurant, I decide to walk into the nearest one and ask for a table. As the menu was in German, Daniel has the waiter translate every dish for him into English (not just the entrees, but the sauces, sides, and garnish on every plate).
Daniel: I'll have the steak with some white wine. Medium-rare.
The steak arrives.
Daniel: [Cutting into the steak] This is medium well. I'm sending it back.
Me: No your not. [Takes steak and devours it].
This whole story is only significant because the reason I was so eager to get out of the restaurant was that our next destination was ice skating:
This rink was in front of City Hall, lit up with beautiful lights, and had music playing (including such classics as "Funky Town"). There are paths that branch off of the main rink and weave through the park. It started snowing about 10 minutes in. It was amazing.
Sunday
I left with a few others at about 1000 and headed to Schoenbrunn Palace - the summer palace of the seat of the Habsburgs. I've never been inside of a palace before. I'm having a very hard time expressing my thoughts about it and the surrounding gardens (and zoo, hedge maze, columbary, view of Vienna, and artwork), but I'll describe 2 cool things I saw while touring:
There was a table clock, covered with gold, that had two clock faces - one in front and one in back. The clock was situated on a table, in front of a mirror. The face (and mechanism) on the back was designed to be read in the mirror!
There was also two bronze Hercules statues (they had a thing for Hercules), one of him killing the Nemean Lion, the other...I can't remember. In both, our boy Herc was pulling the upper and lower jaws of the beast apart, leaving a gaping hole. These hollow statues used to house furnaces, with heat radiating from the jaws of the beast.
Back at the hostel, I learned to play Spades, which is like a casual version of Bridge.
Monday
We head to the Museum of Natural History where, besides an impressive collection of everything related to natural history (not to mention some beautiful art and architecture), they were featuring a Body Worlds-like exhibit. Did you know that the digestive tract of a blue whale is about 0.4 miles? What about the fact that the blood vessels of an ostrich could circle the world twice?
On the way home, I learn to play Cribbage.
Webpages that may or may not be relevant:
Tuesday, February 15, 2011
That's Some Pig
Sorry for the relative lack of updates. There is much to write about, and I will try to catch up before heading to Vienna on Friday. In the mean time, click "fullscreen" to see some T double-E double-R double-R double-I double-F double-I double-C-C-C pigs in their full glory.
Webpages that may or may not be relevant:
Webpages that may or may not be relevant:
Thursday, February 10, 2011
First Class First Class Moments
I'm not going to go into depth about everything. I'll just mention that there are a whole lot of good classes here and I can't take them all. I will share two moments from this first week that really struck me.
1. Number Theory is taught by a short man named Csaba. He's rude, makes off-color jokes, loud, throws chalk at people and at the board, and waves his hands a lot. Essentially, he's Mr. Kinberg but for math. Anyway, we were talking about the idea of primes and thinking about primes existing if we restrict our world to just the even numbers. I don't want to get technical, but the short of it is that we concluded, after a very detailed explanation, that 6 is prime in the evens. Then, he turns around and says, "So where's my mistake?" We stare blankly. "Any proof I give may very well be wrong on purpose," he goes on to explain.
Now, for years I've been tutoring math. I do this all the time - I make a big fuss about it. Never has it happened to me. It was awesome and humbling - it didn't help that none of us could find his mistake.
2. Classical Algebra is a 3-week refresher on things like complex numbers, polynomials, etc - stuff that Hungarian students learn when their in womb. A strong tradition in math competitions and education, along with a rigorous curriculum and some number theory means that Hungarian high schoolers can whoop American math majors on the Putnam. This is something that every one of my teachers likes to point out. Back to the story. We're talking about sums of binomial coefficients:
The first expression sums to 2.
Then 5.
11
22
"Can anybody guess the next one?"
"45?"
"Almost, 43. How about the next one? I'll give you three guesses."
Before you peek, take a guess. Actually, take three.
Have you guessed? It's 85.
Now, I don't know about you, but when the professor said three guesses, about 30 students all guessed 85, 86, and 87 in unison. Do you see it now?
If the 5th term is x, then the 6th term will be 2x-1, 2x, or 2x+1. This is true for any kth and (k+1)th terms.
Now, this in itself isn't that interesting. What is (I think) amazing is that we (EDIT: by "we" I mean humans, not that particular group of math students) are so good at finding patterns that, seeing 2, 5 (2x2 + 1), 11 (2x5 +1), 22 (2x11), and 43 (2x22 - 1), we were all able to guess the pattern. It's hardly a pattern at all!
Okay, maybe it's not that exciting. Maybe the pattern is pretty straight forward. Granted, we're math kids - our brain our trained to look for fun patterns. Even so, this got me thinking about how awesome our (EDIT: again, the collective "we" of the species) minds are.
Webpages that may or may not be relevant:
1. Number Theory is taught by a short man named Csaba. He's rude, makes off-color jokes, loud, throws chalk at people and at the board, and waves his hands a lot. Essentially, he's Mr. Kinberg but for math. Anyway, we were talking about the idea of primes and thinking about primes existing if we restrict our world to just the even numbers. I don't want to get technical, but the short of it is that we concluded, after a very detailed explanation, that 6 is prime in the evens. Then, he turns around and says, "So where's my mistake?" We stare blankly. "Any proof I give may very well be wrong on purpose," he goes on to explain.
Now, for years I've been tutoring math. I do this all the time - I make a big fuss about it. Never has it happened to me. It was awesome and humbling - it didn't help that none of us could find his mistake.
2. Classical Algebra is a 3-week refresher on things like complex numbers, polynomials, etc - stuff that Hungarian students learn when their in womb. A strong tradition in math competitions and education, along with a rigorous curriculum and some number theory means that Hungarian high schoolers can whoop American math majors on the Putnam. This is something that every one of my teachers likes to point out. Back to the story. We're talking about sums of binomial coefficients:
The first expression sums to 2.
Then 5.
11
22
"Can anybody guess the next one?"
"45?"
"Almost, 43. How about the next one? I'll give you three guesses."
Before you peek, take a guess. Actually, take three.
Have you guessed? It's 85.
Now, I don't know about you, but when the professor said three guesses, about 30 students all guessed 85, 86, and 87 in unison. Do you see it now?
If the 5th term is x, then the 6th term will be 2x-1, 2x, or 2x+1. This is true for any kth and (k+1)th terms.
Now, this in itself isn't that interesting. What is (I think) amazing is that we (EDIT: by "we" I mean humans, not that particular group of math students) are so good at finding patterns that, seeing 2, 5 (2x2 + 1), 11 (2x5 +1), 22 (2x11), and 43 (2x22 - 1), we were all able to guess the pattern. It's hardly a pattern at all!
Okay, maybe it's not that exciting. Maybe the pattern is pretty straight forward. Granted, we're math kids - our brain our trained to look for fun patterns. Even so, this got me thinking about how awesome our (EDIT: again, the collective "we" of the species) minds are.
Webpages that may or may not be relevant:
Táncház - Hungarian Folk Dancing
Instead of going Israeli dancing tonight, Ben proposed that we go Hungarian dancing. We got about 8 people together and went to Godor (no, not Gondor) at 8:00pm. This is an underground establishment, but, unlike any other nightlife hub in Budapest, this one was really nice: no smoke, nice decor, and dressed-up folks. It had some seating area, a bar, a main dance floor, and an empty room in the back.
We spot a band tuning on the main floor and head over, expecting to start the festivities. The music begins, and it's a slow, plaintive Hungarian ballad. As I've alluded to earlier, these Hungarian folk songs go on and on.
"They're probably just starting slow because once the dancing starts, it'll be very upbeat."
Another song - beautiful, but slow.
"So where's the dancing? Ooh look - that 2 year old is dancing!"
...
"I think I hear something over there"
*Walk to not-so-empty room in the back*
There's another band, wailing away at their instruments (upright bass, a drum, violins, recorders, and ud, along with 60 or so people in a circle doing an absurdly simple circle dance (3 steps in, 3 steps out, all the time moving counter-clockwise).
When that song ends (I would estimate that every song was 7-10 min), this guy with a fedora gets in the middle, starts jabbering in Hungarian, and begins to slowly demonstrate some steps. As the whole crowd joins in, the band begins to play - slowly at first, but always accelerating. 2 minutes later, with a loud whistle and some hooting, the fedora'd man is running around in the center of 3 concentric circles totaling over 100 people.
As the night progressed, more and more people began to join. The room began to fill up, and a classic instance of the Prisoner's Dilemma (rather, the Tragedy of the Commons) unfolded. Everybody wanted to be in the center circle. Going into the center gives one a nice benefit of more fun dancing, costing everybody else very little (the center circle is just that much bigger). As people enter the center, the inner circle pushes outwards, limiting the space of those on the outside and making the inner circle not so inner any longer. Those on the outside begin to suffer and move in more rapidly. Now, the majority is in 1 big inner circle (when we started with 2 circles of maybe 3:1, and everybody suffers from having no room. This reminded me of 'Yalla' on a Friday night at camp. People eventually figured it out.
Also there was a girl there whose outfit reminded me of Chun-Li from Street Fighter.
The dancing was very fast paced, almost exclusively circle dancing. Most of the dances consisted of 2 phrases of ~8 counts that were repeated ad infinitum. There were 2 partner dances: one involved facing your partner and spinning for about 10 minutes; the other was about half running around in a big circle with everybody and half spinning with your partner.
One of the dances was particularly difficult. I think the sequence of steps was 10 beats, meaning that it was realigning itself with respect to the 4/4 music.
The room was underground, poorly ventilated, and filled with 100+ people running around. My undershirt (I had by now taken off my real shirt) was see through mid-way through the dancing. During the one dance I sat out for, I went outside (it's about 0 Celsius at this point), took off my shirt, and sat down for about 5 minutes. When I put my shirt back on, the sweat on it hadn't dried, just cooled down to near freezing. I promptly ran back inside (where the same song hadn't yet finished).
I think this was the most fun I've had in Budapest so far. I will be going back.
Webpages that may or may not be relevant:
Saturday, February 5, 2011
Falafel for Five Hundred Forints: A Feature on Fast Food
The fast majority of local eateries fall into one of three species: Gyros, Bakeries, and Pubs
Gyros:
Of the three predominant species in Budapest, this invasive species, originally from the region around the Eastern Mediterranean Sea has established itself in the local fast food niche. On my way to the subway station, I can turn left or right: if I turn left, I pass "Pizza Non-Stop," a 24-hr pizza and gyros place; if I turn right, I pass 2 more 24-hr gyros places.
Bakeries:
This city is filled with bakeries. From the language school, I can get to 3 bakeries without crossing a street (this is not due to the presence of some fiendishly clever system of sidewalks). These bakeries have kilogram loaves of bread for 200 HUF, about $1. Even better than the bread is the unbelievable abundance of pastries: burekas, strudel, rugalach, chocolate snails, cream-filled croissants, and many more that I couldn't name in English or Hungarian. My particular favorite bakery also has a microwave too heat up your pastry - chocolaty-goo heaven.
Pubs:
On the same walk to the subway (see "Gyros"), I pass 1 pub before reaching the fork, 2 more when I go left, and at least 4 more when I go right. Keep in mind the walk is less than four blocks. Everybody has the Hungarian Dreher on tap (I've discovered I like dark beer) and hot mulled wine - perfect for when you've been walking all day in -2 °C. It also turns out that it's very difficult to find gulyas (that's how they spell it) in restaurants; on the other hand, every pub has a huge pot brewing in the back.
On a related not, I went out to dinner with some friends for my birthday. We went to a restaurant (that was also a pub). I couldn't decide between french onion soup (which apparently means something different here) and gulyas as a starter, so I got both and passed on the entree. It was the tastiest decision all week.
Webpages that may or may not be related:
Friday, February 4, 2011
Some Light Math with Tom and Hans
Today we had the BSM orientation where we listened to the director of the program talk about BSM, the facility, Budapest, and the courses. We had a reception with some of the professors and all of the students on the program, allowing us to meet new students and talk to the professors.
I spoke with the Combinatorics 2A professor who was handing out problems for the first extra credit assignment:
Webpages that may or may not be relevant:
http://wikitravel.org/en/Szentendre
http://www.wimp.com/glasstransparent/
http://www.wimp.com/hawaiianukulele/
I spoke with the Combinatorics 2A professor who was handing out problems for the first extra credit assignment:
1. In a group of 70 students, for every choice of distinct students A, B, student A knows a language which student B does not know. At least how many languages do they know together.
2. I am in a 36-story building. I have with me two glass balls. I know that if I throw the ball out of the window, it won't ever break if the floor number is less than X, and it will always break if the floor number is equal to or greater than X. Assuming that I can reuse the balls which don't break, find X in the minimum number of throws. (Copied without permission from http://www.tanyakhovanova.com/Puzzles/).
Looking at these, I wondered if he had heard the one with the chessboard and the mathematicians (if you haven't heard this story, suffice it to say that it was one of my triumphs of fall semester). He hadn't:
3. The Devil kidnaps 2 mathematicians: A and B. He tells them both that he has a chessboard in another room; on each square is a coin that is either heads or tails independently of all the other coins. He will take A into the room and designate a specific square on the board. Mathematician A will then have to flip exactly 1 coin on the board (it can be any coin on any square) and then leave the room. Mathematician B must then enter the room and indicate which square the Devil originally designated, knowing nothing except the configuration of the board after A flipped one coin (which one is a mystery). The two have time to discuss a strategy beforehand. How do they succeed?When I got home, my roommates and I discussed the classes we're thinking about taking. I'm considering Conjecture & Proof, Topics in Geometry, Combinatorics 2A, Theory of Computing, and Topics in Analysis (the full list of classes can be found here). We then decided to solve problems 1 and 2, which we did, at which point Tom related another problem he had heard from a professor:
4. You're on a boat 1 mile from a perfectly straight shoreline. It's too cold for you too go above deck, so you cant actually see anything, and you don't know which way land is. What is the best path to take to minimize the upper-bound on the time it takes you to find the shore.We think we have the answer to this too. Feel free to comment your thoughts on the puzzles. If these have been too easy, I'm still stuck on one Erez gave me.
Webpages that may or may not be relevant:
http://wikitravel.org/en/Szentendre
http://www.wimp.com/glasstransparent/
http://www.wimp.com/hawaiianukulele/
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