POTW Beta

As I hinted toward recently, I was thinking about running problems of the week. Well now I have created a beta test of the system to run over winter break. I am calling on you to help me test in the time between all your fun winter break activities. If you are a current member of VMT, go to http://activities.tjhsst.edu/vmt/pages/training/index.php after logging in to the wiki and you should be able to access the problems by clicking on the link that says POTW Beta. If you are not a member of the TJ math team, that link should direct you to a login/registration page where you can register for an account and log in as a guest. This beta will be open until the real POTW starts.

A note about the interface: All of the answers for this beta are positive integers. Therefore, there is no need to use the preview button. That is there so that if a problem has a more complicated answer, you can check to make sure that the system is parsing it correctly, and that the fact that your answer is incorrect is a result of your answer being wrong, rather than that it is formatted incorrectly.

I ask the following from you if you decide to participate in the beta:

• Use your real name and grade and keep IDs appropriate. I will delete accounts that violate either one of these conditions.
• Report any bugs with the system to me! These include broken links, bad formatting, anything that you think could be improved. This also includes feature requests!
  
• Do not look up the problems or cheat on them in any other way. Because this is just a beta, I did not use original problems. However, these are still good practice problems and I think that much of the TJ math team can benefit from actually doing them.
• Tell your friends! I want to get as many people as possible into the POTW system and to do that I need your help. However, make sure that they don't leave between now and the start of the real POTW.
  

Without further ado, I declare the beta for POTW open! The actual POTW will hopefully start shortly after break (maybe with 1-2 weeks in between). Note: This beta is completely voluntary and will not count for anything.

Ohaithar

Snow!

Rounds 3, 4, and 5

The contests for HMMT team selection have now concluded. However, I started writing this blog post before performance contests 4 and 5, so you will see my thoughts progress over a week.

http://activities.tjhsst.edu/vmt/wiki/index.php/2010_Performance_Contest_3

Performance test 3. Halfway home. Something is clearly different. This time the curve is much more spread out. This is pretty much the exact score distribution that I was going for, but I wanted this to happen on performance test 1. So when I heard that the top 15 average was almost 10, I honestly winced: "Was this test too easy?" That question is still going through my mind. I like the score distribution, but I can't help but think that I only got it because the test wasn't at the same level as the last two.

And this is confirmed by others. When people commented on this test, they often said "I think this contest was much better than the last ones, mainly because problem 3 was completely trivial." When I look back over the test, I can't say that I really think it is too easy. Certainly problem 3 was easier than it has been before, but the hard problems were, I think, the same level as the ones before. The difference was that people had more time to work on them.

So no, this test wasn't too easy. What really gets me now is that nobody got a 4. Somehow people felt that problem 3 was easier than problem 2, and in fact more people solved problem 3 than problem 2. I don't know what to make of this. Problem 3 was a very classical problem that many people had seen before, but problem 2 was supposed to be an easy problem regardless of whether you've seen it before or not.

Problem 4 did pretty much exactly what it was meant to do: people who actually finished the problem (proved their answer) got it right, whereas people who sort of tried a few values and hazarded a guess didn't. Problem 5 turned out a bit easier than I anticipated, but that didn't affect the score distribution much at all. Finally, I was hoping more people would get problem 6, but I'm happy that one person got it right, even if he didn't quite finish the computation.

Let's go on to contests 4 and 5, which both happened on December 16, the day where all the seniors would find out about MIT early action decisions.

http://activities.tjhsst.edu/vmt/wiki/index.php/2010_Performance_Contest_4

http://activities.tjhsst.edu/vmt/wiki/index.php/2010_Performance_Contest_5

The graphs really interest me because the score distributions for the later contests were much smoother than those for the earlier contests. Contest 4 is the kind of score distribution that I would like if I were writing a real contest, where I'd want to name a top 3. This is because the top 3 are really distinct from the rest of the crowd. Test 5 is something that would be better for choosing a top 10 (although it looks like it's really only top 8), since there is a hump at 7. This means that the people who did better than that hump are distinguished, and should be named in the top 8. If you were to ask me my interpretation, I would say that contest 4 is better for training for the contests that we will be going to, and contest 5 is better for choosing the team (although the way our system works, getting an 8 on contest 5 doesn't help very much, unfortunately).

So what caused this change in score distribution? I can think of two possibilities: either my contests got easier or the team got used to my contests (hopefully improving in the meantime). I don't know if the contests really got easier. I will say, however, that I did feel that test 5 was definitely easier than test 4. However, what I didn't expect was that a nonzero number of people solved the last problem on test 4, whereas as far as I know only one person was close to solving the last problem on test 5 (and he finished incorrectly). This has a very simple explanation: the last problem on test 4 was a geometry problem, and I am relatively bad at geometry. Therefore, the geometry problems I think are the right difficulty are going to generally be easier than the nongeometry problems I think are the right difficulty. When I noticed that test 5 was easier than test 4, I thought about it a bit, then decided that I really liked test 5's problem 6, so I left it that way to give people time to work.

For those of you reading this, take a look at the tests now, after everything is done. Did the tests really get easier? Or can I safely say that the team is improving? Perhaps both, but I hope the tests didn't really get that much easier. So now I am interested in what people thought of this experiment (since it really was an experiment). Did the performance tests go well? Did you feel that they were better or worse than the other contests for determining teams (consider all three teams - I really didn't write these for the purposes of determining C team, but they had an effect down there)? Finally: did you like them?

My guess is that next year I won't be able to write performance tests for the team, although I had a lot of fun putting them together. However, I'm still not sure what I'll be doing second semester. I might (don't hold me to this) put together some contests and run them problem-of-the-week style. Would the people currently at TJ be interested? (this would also be open to people outside of TJ, so feel free to say that you would be interested if you aren't at TJ as well). If I get enough interest, I'll definitely try to do it.

I have some other blog posts that I want to write now, so this post is a bit short compared to some of my previous ones. For those of you who haven't read the solutions to performance contest 5 yet, I invite you to try this generalization of the last problem:

Brian flips a coin repeatedly. Before each flip, he decides randomly with equal probability to either continue flipping or to stop (so he flips 0 times half of the time). What is the probability that, after he stops, he has flipped exactly k more heads than tails in terms of k?

The Second PUMaC 2009

This story begins long before Saturday, and in fact I'm writing this sentence on Wednesday. But while there's an interesting story that started far earlier, I'm going to start at the Saturday a week before the competition, when the power round was sent out. The power round this year was about lattices, meaning subsets satisfying the following properties:

• 


• If , then


• If and then

Those of you who like thinking of as an abelian group should immediately notice that a lattice is simply a subgroup of . Those of you who are reading my mind and thinking of it as a -module are noticing that it's a submodule (now of course abelian groups and -modules are the same thing, but some of the ideas later are better to think of in terms of modules).

The power round was basically then an excursion into basic results for -modules. It defined isomorphisms and asked for some isomorphism invariants, which were relatively simple. It then asked to show that all these submodules of are finitely generated, and then finally went into the canonical form of the submodules (which is better known for the quotient modules as the Structure Theorem for Finitely Generated Modules over a PID). So essentially I had made a post on this power round more than a month before it was released. Awesome.

So how did the power round go? Well basically I didn't want to do my homework on Sunday, so I did the power round again. By Sunday night, I had done every problem done except 3.5 and 5.6, both of which I knew how to do but it involved proving or citing the above mentioned Structure Theorem, or at least special cases (other teams went the citing route, I would have proven it but I didn't want to go through the entire proof). On Monday I came up with a clean proof of 3.5, but 5.6 still eluded me. Finally I gave up and just wrote up the proof...it started on page 5 and ended on page 10 of section 5 (well at the time it took up fewer pages, but we later changed it from 11 point to 12 point font and while the wording didn't change, the number of pages did).

So now that the power round was written up, I sent an email to the team telling them to check it for readability and correctness. So while Sam and Adam had checked the round fully before the Wednesday mandatory practice, the other five team members were made to read over the round, even if they didn't know linear algebra. By the time I left for school on Friday morning, this was the chart of who checked what (the initials of the team members are at the top):

Yes, I have a whiteboard in my room.

Okay, so every problem was checked by at least half of the team except for 5.7. Looks like we should be good to go for power. After printing the twenty-two pages that can be found on the wiki, I snapped the above photo and then went to school.

At lunch, the car arrangement was a source of some great entertainment. We set everything up, including a car with Grace, Allison, and Divya. Then, as a joke, we switched Divya with Lawrence to see Lawrence's reaction. Allison arrived first, and reacted with "WHAT THE HELL?" But after a few minutes, she said "Whatever, I'll have Grace's laptop with Asian dramas." XD

Then Lawrence arrived, and he also reacted with "WHAT THE HELL?" But instead of just switching himself with Divya (as we expected), he switched himself with Seung In. Somehow that stuck and the car ended up being Allison, Grace, and Seung In. I still have no idea how that actually happened.

My car consisted of Aviv, Sam, Renjie, Akshar, Jenny, and me, so we were a bit cramped in the van (Jenny had to sit in the middle of the back between Akshar and me). On the car ride, we played hearts for a bit and house for a bit. I sat out the first game of hearts, where Akshar got completely destroyed. Second game, Sam and Renjie decided to play as a team and I was in. I got to shoot once, having two runnable suits (I believe my distribution was 2056) and by leading the spades early, so that when I got the lead again I took the last 7 tricks, or something insane like that, which contained all of the points. Shortly afterward, I ended up getting the queen a few times and eventually lost, but that round of shooting was quite fun :).

Upon getting to the hotel, I was disappointed to find out that I had one of the small rooms, but it was fine. I played around with google wave for a few minutes (I had finally gotten my official invite just that morning) and realized that it was actually pretty laggy. I wouldn't want to use it for most communication in its current form. Especially big threads start to lag massively. I'm not sure what it is, but it reminds me of the lag you get when you're X forwarding. But really, if document viewers can display 200+ page pdfs without lag, you'd think that google wave would be able to handle waves with only 100 messages.

After just a few minutes of google wave and finding the latex bot (watexy@appspot.com for those of you who don't know about it yet) I went back to the lobby to wait until dinner time. We ate dinner at Quaker Ridge Mall, which is a horrible idea. There's basically nothing there to eat, except for an applebee's out in the parking lot, which we went to. The food was decent, but then Jenny tried to cheat us out of $1.50. We caught her when we ended up short a bit of money, and then she decided to complain that she didn't have any coins so she had to pay $.50 too much. I tried to give her the rest of my coins ($.03) but she wouldn't take it. This was after she decided to buy $.60 gravy...

Anyway after dinner I took my laptop up to Renjie's room where we were going to play mafia. While playing I read through the google wave API. It looks simple enough, but based on my googling javascript has unfortunately little support for dynamic graphics and I don't want to use flash, so I want to find a good way around that. The game of mafia went extraordinarily well. After the first day, where we killed Seung In for voting for no lynch, Sam and I saved Luke, who was also targetted by the mafia. Second day, we killed a mafia, and then Sam insisted on saving himself. I had no better ideas, so I shrugged and let him do it. But the best part was after the next night, when Sam wanted to save himself again. I was like man this is a pretty silly and pointed to someone else, but Sam eventually won the fight. Next day, Sam said "I think Brian is the mafia, because he and I are the medics and he wouldn't save me last night!" The game ended where, in the last night, I wanted to save Jenny, Sam wanted to save himself, and then after we agreed on Jenny, Sam said (in the middle of the night), "Wait! Do you think it's more likely that they killed one of them or one of us?" Turns out Sin had tried to kill Jenny, and I was right. Hah.
(On the other hand, if Sin believes that Sam and I are the medics, targetting Jenny ensures his loss. However, he says that he didn't think that Sam was telling the truth, so his decision is defensible.)

Anyway, it's time to talk about competition day. I skipped breakfast, as always, and we headed out shortly after our hoped departure time of 730 (Aviv was late, as always). We got to Princeton at some time which I don't know because I don't wear a watch, and then after Mrs. Gabriel registered us we started walking toward McCosh, during which I heard Jenny yell from behind me, "Hey Brian, look behind you!" I turned around and saw Sherry there, so I slowed down a little, but she stayed behind me by a bit. Oh well.

On the way into McCosh 50, I saw Amy Zhou just a few feet in front of me! But she didn't notice me. Anyway, once we got in, I decided I wanted to talk to some people, so I went back outside and went to the registration table where Damien and his team were standing. While I was there, several members of the Exeter team showed up and I talked to them. It was pretty much my first time talking to David Xiao since red mop. We talked about the power round, and I found out that both North Carolina and Exeter had failed to solve problem 5.6. Awesome, we looked to be in good shape. Good lead on all of the other teams. In the middle of the courtyard, Sherry was pacing around waiting for the rest of the AAST team, and I considered going over and talking to her, but her parents were there and I would have felt kinda awkward. So instead I just continued talking to Damien and the Exeter people. It was pretty good to catch up with them. Also that this time I said hi to Amy (she actually saw me this time), who I hadn't seen for over a year since I missed MathCamp '09 :(. Well anyway that was that for the outside talking. We then all went into McCosh 50 and continued talking. There I saw several more people that I knew, but AAST still wasn't there. When finally they walked in, I went up to the door and said "Way to be late, guys." (it was after the scheduled start of the opening ceremony I believe). The opening ceremony started shortly afterward, 5 minutes late I believe. They gave us our proctors and we headed off to our testing room. But when we got there, the door to the building was locked, so we had to go in the other entrance, up to the second floor, and then back down to get to our room. Door unlocking fail. We got to our room and sat down, informed our proctor that Sam and Seung In were switching some subject tests, and then started. Our proctor was Arthur Safira's roommate, and as he put it we "probably got the only non math or science guy" there. In any case, my first test was number theory. Many of the problems were classical, but two of them were pretty decent:

Then again, the second is pretty straightforward if you know continued fractions (and are able to compute that in the time limit, unlike Damien). Some of the problems were a bit too classical. #1 was essentially the same as a Math Prize problem, #2 was find the number of solutions to for positive integers a and b, and #4 is problem 103 in Engel's number theory section, according to Andre.

Next up was combinatorics. Again the test was mostly straightforward, although I made a few mistakes that had to be corrected. And there was the now infamous problem:

As pointed out on AoPS, the fourth largest number of the set is 2, not 4. Many people interpreted it the other way (including the test writers and myself), so the answer 606 was considered correct, rather than 303. This could have been worded much better ("When these numbers are sorted in increasing order, what is the expected value of the fourth number?" or a variety of other ways), but regardless I got the points :). Unfortunately, Sam didn't, so our team suffered a bit because of the unclarity. Darn.

At this point PUMaC's amazing grading system that would announce individual finalists immediately after team round was revealed to us. All the answers were integers, so the proctor called us up one at a time to verify that he typed in our answers correctly to an online submission interface, which presumably handled grading automatically.

So next up was the team round. I forbade the team to discuss the individual round until after the team round, because they might become depressed about how badly they had done (if they had done badly), and also because I didn't want to answer questions about individual during the team round. Team round was quite interesting, with the answer sheet being a crossword puzzle. We got everything (either by solving or guessing) except for one problem, which we had 3 out of 6 digits for, but guessed all the other three wrong. That got us a total of 93.5 points out of 100. Not bad.

After team round, we were told that we had one individual finalist: me. It was a large disappointment that none of the rest of the team made finals, but I was happy that I had qualified. I decided to not bother waiting in line for lunch at that time, and went straight to the finals room where Peter Diao was the proctor. We both kinda partially recognized each other, and then he bothered me about giving him a weird look when he said hi. Then an AAST contingent including Sherry, who thought she had no chance of making finals, walked in. They certainly had more finalists than we did, but by no means did that necessarily mean that they did better on individual, since our scores were all reasonably high, just not high enough on any one test to qualify for finals.

Finals went okay. I looked at the problems and saw how to do problem 2 pretty quickly. It was a simple bounding argument and then a bit of cleanup at the end to eliminate two bad cases. After that problem was taken care of, I looked at the other two. Problem 3 looked obnoxious (47 46-gons? No thanks), so I worked on problem 1, which was a pretty annoying analysis proof that I wasn't completely satisfied with at the end, but I figured it was okay since it looked like 1 and 3 were both hard, so I probably had about as much as anyone else.

After individual finals, Sherry and I lagged behind everyone else, where I found out that Jenny had called her twice and texted her twice during individual finals XD. Way to go, Jenny. She tried calling back, but no answer, so we just went outside and started walking around. There was still free food for the individual finalists, so I decided to take some. I then offerred her some milkis, but she declined for the time being. We found Nassau street and went to the Panera, where I had a turkey artichoke panini and she just had some broccoli and cheese soup because she had eaten the PUMaC lunch before finals. It was a good lunch, though :). Afterward, we walked around Princeton campus for a bit, getting lost because her map from google maps had very few buildings marked on it. Eventually we ran into Divya, Grace, and Allison who were "totally not stalking" us. Of course not.

When I got back to McCosh 50, Greyson said "Congratulations on number theory." Apparently I had gotten all the problems right. I then asked to see the combinatorics answer sheet, and I had gotten all of those right too! With a double perfect score, some of the results were a bit less suspenseful.

The awards ceremony needed another 10 minutes of stalling after minievents to get a powerpoint made with results, so they had the math bowl finals, where AAST had a substantially better showing than last year. And one of the questions was StarCraft! But as Damien pointed out, StarCraft actually came out in 1996, BroodWar came out in 1998. Always recheck your facts. And apparently Damien was actually wrong, so he just fails.

The way they had the divisions split probably contributed a lot to the pumctuality of the awards, though, since we didn't have to listen to all the B division awards. First up was the subject tests. Geometry, Algebra, Number Theory, and finally Combinatorics. Vlad won geometry, so I was really happy off the bat, since he was my roommate at MOP last year. Then Chong won Algebra, who I knew from MOP 2008 (where he will be forever known as stomachache) with a perfect score. Interesting difficulty level, allowing perfect scores. It doesn't really appeal to me that much, since it potentially leaves people having nothing to do at the end of a test, but that's okay I suppose.

They finally got to number theory, and I won (Did anyone not see that coming? If not, you need to read the entire post). When I went up to receive the medal, it wasn't large enough to go over my hair properly, so it got stuck on my hair and glasses, much to the amusement of the audience. Going back to my seat, the rest of TJ said I should just stay there. I decided not to take this piece of advice. So next up was Combinatorics, where I won again (seriously, if you didn't see this coming read the post). And again, the medal got stuck on my hair, again to the amusement of the audience. I now had two medals to show for my trip :) Laura got a picture of this event, which has some hilarious comments.

Then overall results. Here, because of lack of communication about the method for finals, I had no idea if I would place or not. I was especially worried when Vlad placed 8th, since he had gotten problems 2 and 3, while I had only gotten problem 2 and most of 1. But I ended up winning overall as well, completing my sweep of individual events.

Yay two medals an a trophy :)

Next was power results. We sat through places 10 through 2: Beijing STFX1, The Evil Geniuses for a Better Tomorrow, Montgomery Blair High School, AAST Mu B, North Carolina, PEARL, Albany Area Math Circle, Murph and the Magictones, and Lehigh Valley Fire. Now at this point we should probably be suspicious since AAST Mu A and TJ A are both missing, but we didn't think about it too much. As the announcer said "and in first place...", a shout from the audience came: "Brian Hamrick!" (This was from the North Carolina area, so I'm not sure exactly who said it. If anyone knows, I'd be happy to hear). "AAST Mu A"

Okay, at this point, all of us in the back right corner of the auditorium (where TJ always sits) are thinking "WHAT THE ****?" And this wasn't limited to just the TJ team: Peter Diao immediately ran off to the grading room to figure out what went wrong, since we were definitely supposed to place. We were thinking somehow one of our problems got lost, either in printing or by the sponsors, or by PUMaC themselves. We even considered that maybe they mixed up our solutions with another team, but every single page of our power round had TJ A on it, so that was pretty unlikely. Meanwhile, AAST, to our left, was ecstatic because we had apparently massively screwed up the power round.

What?!?

We sat listlessly through the rest of the ceremony, as we placed fifth on the team round and sixth overall. We all knew that we were supposed to place higher, but we didn't know how high. I know I barely could pay attention to the top rankings, as PEARL took third, Beijing STFX1 took second, and Lehigh Valley took first (who is on Lehigh Valley, anyway? They didn't do so well on team and power so they must have had pretty good individuals). After the conclusion of the official awards ceremony, we met Peter againat the front, where he informed us that our power round score had failed to be entered, and that we had actually won the power round and taken third overall. So we missed out on a huge trophy because of a PUMaC screwup. Way to go, guys. Peter said that we got 78/86 on the power round, and I asked him what the second place score was, to which he replied 74. He also said that we had only beaten the teams that were at the actual competition; the German IMO team had gotten a perfect score. (Note: I've now been informed by Peter that he was mistaken and AAST in fact got an 85 on power, so we were actually second on power and third overall. He also says he thinks we lost a substantial number of points on 5.6, which I think is completely correct, so my guess is that the graders simply didn't understand it and took off points, which has happed before, such as to Alex Zhai on USAMO. Also apparently our score was entered, but it was entered as an 11 because someone sucks at reading. Massively.)

Regardless, I was pretty happy with my individual results, and third overall wasn't too bad for the team. When we left, I realized that I still had all of the milkis in my backpack, I had forgotten to give Sherry one after we went to panera :(. I was going back home in my dad's car, rather than in the car I went up with, so I gave them six milkis because we had done pretty well and took the other six in our car. Hopefully Jenny got her milkis this time. She complained after Duke that I had given Sherry milkis and not her. Heh.

The trip back was pretty uneventful. I slept a bit but the way the seat was I had a really sore neck upon waking up. Bleh. Whatever. I got home pretty much satisfied from the trip. While PUMaC certainly still had some kinks, it was run much, much better than past years. I think the contests could have been a bit harder, since the finals cutoffs seemed a bit high (probably around 30) to have 7 and 8 point problems, and a lot of the later problems were actually pretty easy and/or guessable. But besides the screwup on power and the couple of mistakes on the tests, it was a good event, and I got some good stuff out of it.

The spoils

Failure

The second of five performance contests was this Wednesday at math team. This one went a lot worse than the last one, which I think is evident from the score distribution:

First, nobody got a zero! Why? Well, turns out that question 2 was flawed. The triangle with side lengths 5, 15, and 16 does not have an area of 72 and does not have an inradius of 4. This was brought to my attention during the contest and I decided to give everyone credit for the problem. After all, every triangle satisfying the problem statement has a side length of 1, and 2, and 56, and :). (Hint: there are no triangles satisfying the problem statement)

Second, 2 and 4 have about the same number of people! That's pretty easy to explain. The contests are set up so that there will be three tiers. The top tier consists of A teamers who are capable of solving all of these problems. They are expected to always get 1 and 2, usually get 3 and 4, and then sometimes get 5 or 6. The second tier consists of B teamers who are capable of doing well, but either don't have the speed or the knowledge to complete the entire contest in the time allotted. They are expected to always get 1 and 2 and then get some of 3 and 4. Finally, there are the lower teams who are expected to be working on 1 and 2 the entire time.

So what does this mean? This essentially means that A team has a 6 question contest, B team has a 4 question contest, and the rest of the team has a 2 question contest. When #2 turns out bad, it turns the A team's contest into 5 questions, the B team's contest into 3 questions, and the rest of the team's contest into 1 question. The impact on the A and B contests is negligible, especially since they are supposed to always solve the first two problems. However, the contest for the majority of the team was reduced from two problems to one, which resulted in the massive clump at 2 and 4.

Let's look at the test in more detail (you can see it on the wiki page linked to in the first paragraph).

The first problem is straightforward, provided that you can list the primes up to 43 and count them correctly.

The second problem is screwed up, as I said, but you can see what I had intended on the solutions page. (Maybe I should've solved for the true side lengths that would make it work so that you can also solve it with e.g. Heron's formula)

The third problem has an amusing story. It originated from a dinner at IOI when we were eating with the Canadians. One of them proposed a few problems to us. The first was a geometry problem which I can't remember, but Travis eliminated it rather quickly. Then he said "Here's a problem that took me a while to solve. See if you can do it. Find if ." To this, I responded, "Wait, isn't that utterly trivial? It's just the difference of two geometric series..." Their response? "Damn! Why didn't I think of that?"

The fourth problem has a somewhat different history. I first saw this kind of problem at MathCamp 2006 in a class titled "Calculus Without Calculus". It was a nice problem, but I basically never saw it again until last year, when it appeared on an ARML practice. However, the lines in the ARML problem were given to be perpendicular (BAD was right) and so most of the people who solved the problem simply calculus bashed it. After the calculus solution was presented, I went up to the board, drew the circle, and explained that not only could the problem be solved with just power of a point, it was independent of the fact that BAD was a right angle.

The next week, I wrote a TJML and included a similar problem, but changed the angle to something obnoxious: 82.5 degrees. The result was that one person solved the problem and nobody else, and many people didn't even realize that I had presented the exact same solution at the previous week's ARML practice.

Fast forward to the summer when I was writing problems for performance contests. I remembered this failing in the team and decided it was about time that they learned to listen to and remember solutions, so I put the problem in my database and aggressively classified it as a medium level problem.

Problem five is a classic use of Hensel Lifting. Many people commented after the contest "I have never heard of Hensel's Lemma." That was intended, but you can solve the problem without knowing Hensel's Lemma if you think of first solving the equation mod 5, then "lifting" it to mod 25, and then mod 125. This was a problem that was simply meant as a "here is a useful technique that you should remember if you want to do well" problem.

Finally, we get to problem six. This one was thrust into the performance contests because of some team contest last year (I can't remember which contest it was) in which there was a similar problem and none of the rest of the members of my team knew how to do it (meaning none of the top members of last year's math team). Again, this problem was meant as a wake-up call, so that people would learn how to approach problems like this, since they have appeared on various contests in the past (I think there was one at Duke last year). In another sense, it was a problem that said "This is something that is useful to know if you want to win." Essentially, some of my problems are problems that nobody will get, and if you know how to do them you have a huge advantage. An obvious example of this is HMMT 2009 Calculus #10. TJ A essentially got 24 free points because we knew complex analysis and nobody else did (it should have been 40 free points, but Kee Young and I were pretty silly during the test).

So how would I have liked this to turn out? Well, barring the screwup on number 2, I'd expect from our current team:

All of the top 15 to solve #1
All of the top 15 to solve #2
Most of the top 15 to solve #3
Some of the top 15 to solve #4
Few of the top 15 to solve #5
Few of the top 15 to solve #6

and I'd want:

All of the top 15 to solve #1
All of the top 15 to solve #2
Most of the top 15 to solve #3
Most of the top 15 to solve #4
Some of the top 15 to solve #5
Few of the top 15 to solve #6

What I wanted actually pretty much happened for problems 1, 2, 5, and 6. Problem 5 is supposed to get a few more solvers than #6, but still not many The problem is the midrange problems. However, as bad as the distribution is, I strongly disagree with making the middle section of the contest easier. The fact is, our A team is not up to par. I'm actually not sure why this is the case. I said in an earlier post that this might be because the class of 2010 made a block that just rose to the surface as a single chunk as the pieces of the team above it disappeared, so as long as nobody else in that block was working, anyone in the block would see their ranking go up for nothing, so why would they work?

Other people (namely Dan) have suggested that the problem is inherent in the ranking system itself. If rankings were kept private, as schools such as Exeter do, then unless one is vastly superior to the rest of the team, there is no magic website to go to that will tell one whether or not he will be on the A team or the B team or on any team at all. TJ has such a magic webpage, and given the fact that these are TJ people, it's inevitable that somebody will notice that no matter what happens, they will be on A team even if they skip the last practice. And then some people might go further and actually skip the practice.

I'm not going to say that Dan's theory doesn't hold water. It very well could. In fact, there are some people who I think would be likely to fall into such mental traps. However, I do have issues with keeping rankings private. The first is somewhat obvious: TJ kids will figure it out anyway. Even if we radically change how scores are calculated, unofficial results will become commonplace. This has been seen in the world of informatics. USACO releases all scores on a month-to-month basis. While their specific parameters are not releasd, our senior computer team keeps its own rankings page which does a simple average and generally correlates well with the camp selections. Additionally, during the International Olympiad in Informatics, results from day 1 appear on Russian sites even before day 2 starts. The International Mathematics Olympiad does it differently, since all grading is done after both days occur and everyone's score except for one problem is made entirely public officially as coordination proceeds. That is much more similar to what we do at math team. The difference is that when 5 of the 6 IMO problem scores are posted, you might know that you definitely made the gold cutoff, but you still have already done problem 6. When 13 of the 14 math team practices are posted, you might know that you definitely made A team, but you're still able to not go to the last practice, and so some people might just do that.

So maybe Dan's idea has some merit, but I think all of this is fundamentally a problem of the idea that results are what matter. Many people take this so far as to believe in the "big fish in a small pond" theory, which says that being valedictorian at a small, no-name school is better than being simply an above average student at a large, prestigious school. By being at TJ, I think most of you have realized that there are some things more important than being valedictorian. But have you realized that there are more important things than being on our A team?

My worry with the block of 2010ers was that they might think to turn TJ into one of the lesser schools so that they could in turn become higher ranked in terms of their own school and put something more impressive on their college application, which would make the admissions officers think that they are better than they actually are, since the TJ name carries a large reputation, and changes in that reputation won't propagate very fast. And with the large block of lazy 2010ers, this was actually possible.

At this point, I think I've broken the block. Our B team contains almost no seniors and the seniors on A team definitely deserve it, since most of them have performed well on at least one of my performance contests. Looking down at the lower spectrum of the rankings, I can see that many of the seniors that I felt were being too lazy to do as well as they were doing are in fact dropping like flies. But most importantly, the seniors are no longer a huge block floating to the surface. And while there might be a huge group of seniors at the top, in all honesty the gap between the A team and the B team is the smallest it's been in years. If the B team improves just a little bit, I am completely confident that they can continue the TJ legacy in the coming years.

However, some people, who will remain nameless here but probably aren't hiding it very much, believe the big fish in a small pond theory to an unacceptable extent. They think that winning the B division of ARML is better than being a random team in A division because they get more prizes. That's a problem.

Some people, when they read this post, probably thought that when I said our A team isn't up to par, I meant that they can't keep up with the other teams. That's not at all what I meant. Actually, I think that perhaps other than AAST and Exeter, we have one of the strongest A teams in the nation. And with the gap between A and B as small as it is, that also goes for the ARML A team. However, we're still not up to par. We're lacking on the mathematical thought process and how to attack a problem that has never been seen before.

I always wondered: how was it possible that some people made our ARML A team and then performed substantially worse at the competition than members of our B team? Obviously something was wrong with the selection process, but what? Well, I believe the answer was that our contests were too formulaic; they could be mastered by just doing math team for a few years and then recognizing the problems. But what happens then when ARML comes up with a new problem type? Those who are good at problem solving but not as fast because they don't have the problem types memorized are on our B team and get the problem, while those who were fast from just problem recognition flounder. That was the fundamental problem of our selection process.

In fact, 99% of all contest problems are very similar to a previous contest problem. So this formulaic method works to a very large extent, and China is able to use it to great success at IMO. And the results of this method are extremely clear: China dominated problems 1-5, while Japan dominated problem 6. Quite simply, had Japan gotten a perfect score on problem 4 and even a single more point on problem 3, China would not have won the IMO. Why did Japan dominate problem 6? Because nobody had ever seen that kind of problem before. Why did China dominate problems 1-5? Because they had all seen those same types of problems before.

I've had several arguments with Richard Peng, who coached the USA IOI team the past few years, about what the correct training method is. He insists that the Chinese method is the correct one because it brings in the most gold medals, while I say that the correct method is to pretend that the Chinese method doesn't work, and instead work on how to solve new problems.

TJ's method tries to emulate the Chinese method of training by providing a thorough overview of all of the types of contest math problems. The first error is that it only covers an overview of the types of problems that we do during eighth period. Those problems that are written by ARML, HMMT, and PUMaC are completely out there, and our eighth period practices don't cover the correct material. The second error is that there simply isn't enough time to provide a thorough covering of all of contest math. We are in school for about 40 weeks per year, of which about 30 are used for math team. If you do 12 problems every week in eighth period, that is only 360 problems. Can 360 problems cover all of contest math? Not at all.

So back to my performance tests. They are my way of pretending that the Chinese method of training doesn't exist. Instead of having the problems be archetypical problems that are likely to appear on a contest, my problems are the kinds of problems that are challenging and "out there". They're meant to have people exercise the thinking process that they use when they don't know how to do a problem, so when they invariably are stuck on a problem at PUMaC or HMMT, they have had practice with dealing with that situation before. In fact, since Arvind is reviewing the performance contests, I'll say that it's not unlikely that HMMT will specifically dodge the types of questions that I have given you. That should not detract from their usefulness if I have done my job right.

Will it be enough to win? In truth, it doesn't matter. Winning is great fun when it happens, but it shouldn't go to your head and you should be focusing on learning first, winning second (if even that). I have made the math team wiki public for that reason: I would rather have our competitors know much of what we know and give us a good contest than have a trophy on my shelf.

AAST

is dense.

Sucky Contests

Sometimes I wonder if math team is worth it. How does one get onto the teams that we send to Duke Math Meet, the Harvard-MIT Mathematics Tournament, the Princeton University Mathematics Competition, and the American Regions Mathematics League? Well you have to do well on contests given during eighth period. Fine, so what contests do we do? Well the main one that counts for these away contests (except for ARML) is the NYCIML: the New York City Interscholastic Mathematics League (Mandelbrot also counts, but occurs much less often). When I first joined math team, this was fine. Now I'm not so sure.

So what is wrong with NYCIML? For a large portion of the math team, absolutely nothing. However, for the people who actually have the potential to make our A team, there is a lot wrong with NYCIML. The biggest is that it is completely unrepresentative of the contests that they will have to face if they get selected for the team. For example, freshman year I qualified to be one of the members of TJ A at PUMaC. I got a 0 on number theory, what I considered at the time to be my best subject. That's right. I didn't get a single number theory question correct.

What went wrong? Well, it was a combination of PUMaC being crazy hard and a severe lack of preparation for that kind of math. To put it simply and bluntly, the contests that we do simply do not prepare people for contests, and they furthermore test for something completely different than what PUMaC and HMMT test for.

After three years of experience at PUMaC and HMMT, I think I have a good idea of what characteristics makes one successful at each of them. PUMaC rewards knowledge and an ability to work through standard computations, regardless of how ugly they get. Many PUMaC problems are not theoretically difficult, but finding the actual numerical answer is the bulk of the work. HMMT, on the other hand, focuses much more on problem solving. Most of their problems require little prior knowledge; almost everythign necessary can be derived on the spot. Some exceptions exist: the fact that has been used, for example. But by and large, HMMT is a contest based around how well you can handle things you've never seen before.

So what is NYCIML? NYCIML is just a hard math test. Practically everything can be seen instantly. There is no creativity involved. In fact, the last NYCIML this year essentially repeated two of the problems from the previous one. Contests that do that are generally bad. They become formulaic; anybody who has seen enough problems will do well, regardless of how good they are at math. As a result, for the people competing for the top spots on our math team, NYCIML is not about solving problems. It's about eliminating stupid mistakes. While I am all for stupid mistakes being considered in rankings, it should not be the determining factor as it is here.

This is why I wrote more than 50 problems over the summer. I wanted to create contests that would prepare the team for what they will face at PUMaC and HMMT. So far, the team has not met my expectations, but I believe that they will grow now that they have been faced with the challenge and the knowledge that there are three more to come. The biggest difference between the contest that I wrote and the NYCIMLs is the average. Out of 6 points, the NYCIML top 15 averages have been around 5. Out of 19, the top 15 average on my contest was a little over 6. What does this mean? It means that you have more than one route to victory. The first is to do the same thing as NYCIML: do the problems you know how to do and do them right. The problem is that I ramped the difficulty up quickly, so many of these people probably got a 4, or possibly a 2 because they still missed one of the first two problems (the contest was six problems weighted 2,2,3,3,4,5 respectively). The other method to victory is to be more ambitious, solving more problems, and then maybe you make a mistake or two that drops your score from 11 to 8. You'll still be ahead of most of the people. However, making a single mistake on a NYCIML costs you dearly, even if you know how to do all the problems.

At HMMT, the top score is not 50. If you know how to do all the problems, you will win, even if you make stupid mistakes. I speak from experience: On the 2009 Calculus test, I solved all 10 problems. I made some stupid errors on 9 and 10, dropping my score down to 35. As a result, I tied with Kee Young for first, but lost the tiebreaker. Regardless, the fact that I could make two mistakes and still tie for first demonstrates how different HMMT and NYCIML are. The two mistakes probably cost me the overall first place, but I was second, not eighth. One mistake is much more acceptable at HMMT than on NYCIML.

So why do we choose our team based mostly on a contest where problem solving is minimal? And why do we pass it off as good practice? It's good practice for most of the team, but not A team. But A team is the team that represents us at competitions, so it seems weird that we train the rest of the team more. Hopefully these performance contests will help jolt people into realization that being good at NYCIML does not mean that they will do well at every contest and will continue to work to improve in order to conquer the challenges I have given them.

The performance contest (to be done in 35 minutes):


And as an amusing sidenote:

Information is not Knowledge

In the search for an answer to the question, I have come to the realization that a certain message from various sources (among them is Zuming) is extremely important for people to understand. This is a place where conventional math classes have a huge failing, and before people have realized it, they've assimilated the same failing into all of their studies, both inside and outside of school.

I realized that this is a problem when Dan sent out his latest email. Here is an excerpt from it:

"For a small subset of you, it was your mathematical knowledge that prevented you from making the calculation. To this subset, I assure you that we will attempt to teach you the mathematics. However, at the meeting last Tuesday, I believe that the majority of the people who did not take the relevant derivative, even after repeated requests from me, had the capacity to do so but lacked the motivation."

What is this saying? It's basically saying what I said in my last blog post: that people don't want to work when they know that they can. But I have to question something that Dan said. Do people actually have the capacity to do what they have the information to do? There are some people who "know" the product rule but would almost certainly be unable to do the problem that Dan posed. That is, they have the information but not the knowledge.

I have observed this failing in many places, not just at TJ. Perhaps the most surprising place for some of you would be red MOP. Yes, not everyone at MOP is good at math, as shocking as that may seem. They may be better than you. That doesn't make them good.

I was walking through the blue room (the main lounge) one day and I overheard some red moppers talking about abstract algebra. Some of the things I heard were along the lines of "Yeah, rings are really cool...except I don't really know what they are." This was when I realized the truth of Zuming's statement, but the importance had not yet hit me. What I did know at that point was that informing them of what a ring is would be completely useless. They would surely forget it by the next day, and most of the effect would be lost on them.

Now I think I understand what's going wrong. People love information. When they feel they understand one piece of information, they immediately grab for the "next" piece. Then why they think they're done with that piece, they grab for the next, and the next, and the next. What is the result of this process? It's a ton of information, but no knowledge. When you follow a trail of crumbs, you eat one, then the next, then the next. But did you ever wonder why the second crumb comes after the first, and not before? Sometimes it's obvious - the second explicitly cites the first. But what about when the first cites the second? Why do we learn about real numbers before we learn about Cauchy sequences or Dedekind cuts? Why is calculus the "next" step after algebra and geometry? The people who don't consider these questions are doomed for failure. They will never understand where they are going, and when the line of breadcrumbs dies out, they won't know where they are, unable to find any familiar landscape, hopeless until they find a new trail of bread crumbs that won't help, but rather just lead them further into the wilderness.

So what is the answer? I am not saying that you should be contrary and head off into a completely different section of the woods from where the crumbs lead you. This will probably lead you to just as much failure. After all, there are crumbs there for a reason: this path has worked for other people before, and it will work for you too - if you walk it the right way. You should walk this path, but don't swallow bread crumbs as fast as possible. Savor them. Enjoy the scenery. Maybe even deviate from the path in order to learn the surrounding area. Then come back to the next crumb and do the same thing. Each time, you'll feel more and more comfortable with the area, allowing you to move further and further out into the wild, while still always knowing where you are.

All around me, I see people who have been adversely affected by the bread crumb trail. There was a trail of crumbs. They ate the crumbs; they did their homework. Every once in a while, there was a signpost; they took a test. They got to the end of the trail, marked by a final signpost: the final exam. Where were they? They had no idea. What they did know is that there was another trail of crumbs waiting for them: the next class.

People tend to like hammers, like the Fundamental Theorem of Calculus, or Muirhead's Theorem, or Combinatorial Nullstellensatz. But think: If you have no idea how to use them, how useful are they? Not at all. Information is not knowledge.

Smith Normal Form...Finally

Okay, so I'm writing this post about Smith Normal Form just as I decided that my first point of attack is going to be avoiding it. This is because the fastest algorithm I found to compute Smith Normal Form is general is time, whereas an algorithm that uses geometric properties of spaces can compute homology groups in time. This is a huge gap, although the big issue is going to be how to handle torsion, since the paper that computes homology groups in assumes the space is embedded into , which implies that there is no torsion (so the homology groups are all isomorphic to for some k). Well let's look at why Smith Normal Form is hard to compute. To do that, I should explain what it is.

Consider an m x n matrix A whose entries are elements of a PID (in our case, ). Then we can find invertible square matrices P and Q with entries in the PID such that PAQ is diagonal and if i < j. This matrix need not be square, for varying i. Here is how I think you should think about it:

• A is a matrix that represents a quotient module of . Each column represents one generator, and each row represents one relation.
• P represents combining of the relations. This is like how you can take and to get and by subtracting the second from the first. With P alone you can get into a row reduced form, but not necessarily diagonal.
• Q represents combining the generators. So maybe instead of using the generator set we want to use . Q lets us do that.

These two matrices together allow us to eliminate all the off diagonal elements, and when that's done we have our module as something of the form . However, this may cause the elements on the diagonal to get larger. This is probably not obvious to you, but when I explain the algorithm for computing the normal form it should become more clear. The divisibility condition is also one of the weirder things, but it is essentially just removing all the ambiguities caused by the Chinese Remainder Theorem: . Smith normal form uses the divisibility condition to give the latter, although in the form .

Okay, so how to actually compute this mysterious Smith normal form? Well, let's use a simple example. Suppose we have the following module that we want to simplify:

• The generators are
• The relations are and .

We represent this module by the matrix . We then do row and column operations to eliminate entries: subtracting two times the first row from the second row gives . Then subtracting twice the first column from the second column and one times the first column from the third column, we get , which is our Smith normal form, and we now know that our module can be better written as . The generators corresponding to each of these three components can be obtained from the column operations that we did. In this case, the generating set is , and our relations become and .

There is still a question, though. What do we do when the top left element doesn't divide the row or the column elements? We need to make sure that P and Q have inverses with integer entries, so how do we do this? The answer lies in Bezout's Identity. Suppose we're looking at a portion of our matrix that looks like . To make it simpler, let's pretend it just looks like . It might take a while, but you should be able to see that it doesn't matter what the dimensions of the stretched version is, as long as we multiply by square matrices on either side. So now say we want to eliminate c. How can we do that? Well, what we do is we write where g is the greatest common divisor of a and c. Then , so we know that is invertible because the determinant is 1. Furthermore, when we multiply we get , and we have eliminated c, just as we wanted. The problem is: this might have undone our elimination of some of the columns! Well, never fear! We can alternate between eliminating the rows and eliminating the columns, leading to a sequence of top left elements where each divides the previous. Therefore at each step the top left element gets smaller, meaning it terminates eventually. (For general PIDs, we create an increasing sequence of ideals and then use the fact that our ring is a PID to show that it is Noetherian, and therefore the sequence becomes constant eventually). So once both the first row and first column are eliminated, we ignore the element that's now in the top left and proceed to diagonalize the rest.

Each of these row or column operations can be represented by an invertible matrix that can be absorbed into either P or Q (P for row operations, Q for column operations), giving us the construction we wanted before. That the divisibility condition can be satisfied with row and column operations is a simple exercise.

So how does this relate to homology? Well it's relatively simple. We can represent our bundary homomorphisms easily with matrices. The kernel of a homomorphism is just the null space of the corresponding matrix and the image by its column space. So to compute the homology group, we just reduce the matrix into Smith normal form, and then read off the module. The one thing that we need to watch is that we don't use the components that correspond to rows that don't exist, because our module is generated by elements of the null space, not just any columns. That means that the rank of our module can't be more than m, as it could before.

As I said, making progress here seems difficult. The methods to compute Smith normal form are pretty complex and it would be very difficult to surpass them. However, the large gap between and gives me hope for finding at least something for orientable manifolds in dimensions higher than 3 that runs faster than just computing the boundary matrices and their smith normal forms.