Pants, rifle scopes, towels, and TeX
Jul. 2nd, 2007 07:31 pmPANTS.
Pants pants pants pants pants pants pants pants pants pants pants.
Pants pants pants, pants pants.
Pants.
RIFLE SCOPES.
Rifl- OK, that's enough of that.
Now, where was I... oh yes. Pants. Er, rifle scopes. I mean, Duel of Ages. Saturday (or was it Friday?) Grant tells me, he was speaking to Kyle, and he wants to play Duel of Ages again. And so Saturday night Kyle drives over from wherever he lives and we set to killing each other once again, this time, 2½ hours, 4 characters apiece - yes, using our makeshift FFA rules again. (Although we changed the scoring rules a little.) I had Milena Arrebato, Tex, Pvt. Sanchez, and the Martians. Kyle had Frostdancer, Phillip Redlegs, Sir Gawain, and Ardin Glynn. Grant flipped over Kraator, Annie Oakley, Sterling Jack, and... Agent 911. Oh boy.
The board was not your usual parallelogram board; labyrinths were far apart enough that you often had to go for the closest one, +1 bonus be damned, or, well, *not* go for a labyrinth - but if you wanted to stay back and snipe, you'd have to do it in the area; getting from one side of the board to the other was pretty hard. A dismissal roll of 6 could easily screw you.
Kyle got an early lead by killing Kraator[0] and taking his stuff. Meanwhile, here I am with two characters that get a +1 bonus with a long rifle, and the Sniper draws not a rifle but a rifle scope, which gives him an additional +1... as soon as he gets a long rifle. Kyle had one but I had nothing to trade for it. Meanwhile, my Martians had picked up Storm Sprites - the perfect weapon for them, it's based on wits! - and go hide in the woods. "C'mon, I can't seriously be a threat to you here... I'm deep in the woods, I'd get massive hit penalties if I tried to attack you." Not with Storm Sprites! After gleefully taking potshots at everyone around - and killing Annie in melee, taking her Obsidian Tomahawk - an unlucky roll breaks (kills?) the sprites and the Martians were forced to retreat. Kyle was picking up more and more items while Grant was hardly getting any - he got a lucky amaze roll with Agent 911 and got 3 cards for him, but only one of the weapons he picked up (the Colt Peacemaker) was even usable by the Bumbling Idiot, and even that he couldn't hit with. Sterling Jack was in trouble of his own and couldn't trade. Meanwhile Tex finally picks up some weapons, only to respawn right in the middle of a massive shootout. He runs to a building for cover, trying to meet up with the Sniper so they can trade. Unfortunately, they are followed by Kyle's beatup squad who kill them both and take their weapons - and, of course, the rifle scope. Kyle now leads in characters, 4-2-2, and is dominating the board with well-positioned snipers and a ridiculous amount of weapons. He isn't doing quite so well in the labyrinths, though, and Grant and I catch up a bit (Grant less so, as he has to use Agent 911 for this) to end the game 2-1½-1½, Kyle the victor. (We counted ties in an area as half a point for each person). Ouch.
Towels. I walk into the bathroom one day and find the towels missing. Huh? I leave it for the time - I figured maybe someone had gone to wash them, but when Sunday comes, I want to take a shower, and hey, still no towels. I ask Scott - he has no idea. I'm kind of afraid to ask Satory, but if I didn't move them and Scott didn't, well, we need those towels. I bang on his door. "Do you know what happened to the towels?" "Towels?" "The towels. In the bathroom." "Oh. I threw those out." "YOU THREW THEM OUT?!" "They weren't ours, they were left here by the people before us. You never said you wanted to use them - you should have written your name on them."[10] "What... what.. no! No! You write your name on something to say other people can't use it, not so you can use it yourself! By definition, you *want* to use things! That's what they're for! Throwing them out is simply wasteful!"
I'll be honest: I don't get it. Now Satory, as I said, keeps entirely to himself so I really don't have any good idea of him. My first impression was simply, oh, he just likes to keep to himself. My next impression was hearing him and Scott arguing over something and thinking, oh, he sounds like he's going to be a pain to work with. Then there was the bread incident - and the noticing that he wasn't eating any of our food and assumed it went without saying that we wouldn't eat "his" - giving me the snippet, oh, he cares only for himself. And finally Sunday - oh, he's... wasteful? What? Maybe it's just me, but "selfish" and "wasteful" aren't two words I would normally put together, except where kids are involved. Usually your archetypical image of the selfish person is the the maximizer, the efficiency nut, trying to get every last benefit out of whatever he has. But I think that's not it - it's oh, he assumes everyone else is like him, i.e., that they don't give a shit about and won't use things that aren't "their" property - obviously we weren't using those towels since they weren't "ours", and so it was OK to get rid of them. Gah. (Why getting rid of them would be a *good* thing remains a mystery.) He's sure going to have some trouble in life. How did he ever, you know, live in a family? (I'd say "or in a dorm", but, well, different dorms are different...)
Perhaps I'm being a bit unfair to him, but I'll consider that possibility when he actually does something to contradict my image of him. ("Think of it as evolution in action." :P )
Meanwhile, he tells me there are towels in the pantry. Those aren't towels, those are washchloths. Not quite the same thing. As it happens, later I hear him talking to Scott, and he says, he didn't throw them out - he put them in the closet. Well, there are no towels in the closet. "I didn't go in the closet," I say. "I didn't go in the closet," Scott continues. So... what the hell happened? My own towel is lost so I had to borrow Scott's. Of course, I can't keep doing that, but where am I going to buy towels around here?
Anyway, now we come to the last and longest part of this entry (this is what happens when I don't write anything for a while): the diagnostic exam. I mentioned last week that we had to come up with a diagnostic exam to split our kids into groups. We started on it that Monday, of course, drawing up a preliminary version. Now there was to be a multiple choice section and a free response section. Problem: It seems all the people who wanted lots of easy problems ended up on multiple choice, while the people who wanted more hard problems worked on free response. In particular, what happened in multiple choice was that someone suggested that what we really want to test is what these kids have already seen, and that therefore we should boil down the problems to their most basic form. I liked this idea and immediately started on the boiling, until before long it seemed all the actual difficulty in the multiple choice section had, well, gone up in steam.
Meanwhile, free response was headed by resident supervillain Sam Ratskin[3], who came up with the following diabolical problem: We present them with an arbitrary K7 embedded in R³, and ask them to find a knotted cycle. We ourselves do not have to take time to specially construct our K7, or check for a knotted cycle beforehand, because it is a theorem that one always exists! We need only check their answers when they turn them in, and see whether or not they are knotted. (This, of course, raises the question of how difficult it would be to actually check, but I doubt Sam was thinking that far ahead. It was ultimately Jonathan who ended up grading that problem, but I get ahead of myself.)
It was only towards the end of our meeting that I realized what I was doing to the multiple choice section. I was acting these were, like, the 7th and 8th graders, but these were the 11-12ers we were dealing with! A lot of these kids are probably going to be math kids already - going to math team, doing AMCs... these problems are too easy! I mean, yes, we need some easy problems, even some super-easy problems ("divide these complex numbers") because it's a diagnostic test so you want to hit all difficulty ranges, but these problems were *all* easy! I didn't know what was going on with free response at the time, but I figured, you know, we should still have a good spread with multiple choice alone. I quickly tried to replace some problems with harder ones, but not many made it through - it was hard finding problems to replace and harder finding good problems to replace them with.
That night found me working more on this, but on free response - I had made quite a mistake of a suggestion that day. You see, the kids were going to be going to lectures on two things: Knot theory (hence the K7 problem, as well as quite a few other knot problems we put on), and computability theory. We had nothing computabilitical (to make up a ridiculous-sounding word), so I suggested a "robot" problem. You have a robot moving around on a square grid, his program is such-and-such, his environment is such-and-such, what does he end up doing? OK, so this was really just testing their ability to follow directions... but hey, we were already doing things like including basic logic on the multiple choice (which at least one person got wrong). I didn't actually begin writing the problem until that night, when I quickly realized that as the person writing the problem, a complex program and lots of different things for the robot to interact with was the last thing I wanted. I spent much of the night trying to come up with a program that was sufficiently simple, without being too obvious in its effects, that would allow me to send the robot into an infinite loop without the board actually returning to its initial state. It took me a while to realize that all my early programs made this impossible for the simple reason that they were all injective - if you knew where the robot was now, you could deduce where he had been. I eventually managed to get it to work, and sent it off to Corrin[4] with the note that (of course) I think this problem is too easy. The problem was eventually cut, unsurprisingly, for being too much trouble to TeX up.
The next day... if I'm not mistaken, nothing happened on Tuesday. David was supposed to come in with the TeXed up exam, but this never happened. So skip ahead to Wednesday, noting that a draft of the exam was supposed to be in by Wednesday, the final by Thursday (though it didn't actually have to be - we just didn't want to be making all our photocopies Monday morning right before the exam itself). David comes in with his TeXed up version of the multiple choice. Probably free response as well, but I wasn't paying attention to free response at that point. Now we hadn't actually decided on all the problems yesterday - just an outline, what sort of problems they were. Some were specified, but only one had a specified set of distractors. And so David had to make up a lot of the problems. Some were good, some were too easy (or could be reverse-engineered by plugging in the choices), and some were not, in fact, the specified problem. So Corrin and I went over it with him and made a bunch of suggestions. That night I emailed the group and made some more suggestions as to what should be taken out, asking for suggestions for what to replace them with. I put in some of my own, but I got no reply. So when the next day I found myself writing real sets of distractors, I just made the changes I had suggested - Corrin approved them, so I figured that meant they were OK - but I still needed something to replace problem 18, and I had come up with nothing. Eventually, my train of thought - "We need to make this harder. But it's multiple choice. How do you make multiple choice hard? It's inherently easier than the free response... wait, what the hell am I saying? Take a look at the AMC! *That's* multiple choice! And for 11-12ers, it *is* hard! Hm, AMC..." - combined with my wanting to add at least one number theory question (not at all present on the exam at that point) - led me to copying an old AIME problem from memory as the new problem 18 (though of course I had to write distractors). Of course, a lot of problems on the test were copied from the diagnostic test from older years anyway - specifically the tests from two and four years ago, because the program alternates between geometry years and... uh... whatever the other years are. Number theory years, is it? More on that later.
Note that this means it's now Thursday and we're still working on it. Corrin sends me the TeX file and I apply my modifications to the multiple choice, also reading over the free response for clarity and correctness and making corrections where necessary.
That night I send out an email pointing out a potential problem I just realized. (What, us, finished on time? Hah!) I had looked over the free response finally - for clarity, correctness, and proper TeXing only, really, but now it hit me: The result of our lack of coordination was that two of the problems were too similar. We had two roots of unity problems, one in multiple choice, one in free response. One of them, I suggested, had to go. Do you guys agree? Disagree? Suggestions for replacements? No response.
OK, so our lack of coordination had resulted in another problem as well - a huge difficulty gap. It looked like multiple choice had easy covered, and free response had hard covered, but that all-important middle had managed to evade us. And that was what we would spend Friday trying to fix, as best we could. Or perhaps would have, had we not had lots of other things to fix.
And so came Friday. Well, Corrin (or someone) had looked over the exam and decided - the roots of unity problems were OK, but the two basic modular arithmetic problems - one in multiple choice, the other in free response - really were too similar. That's one problem cut from free response, replacement needed. Also, the "crazy mathematician" problem has to go - it was on the test last year. See, copying from two years ago is OK, because then any 12th graders coming here would have been in 10th grade, and would have seen a different test entirely. But copying from last year? Not so good. That's two problems cut from free response, replacements needed. (We did, actually, end up deliberately keeping one problem from the year before - Calvin's catch-all problem, meant to be far too hard for any of them to answer. Indeed, none of them did.) And, oh, we need to make another change to multiple choice as well.
You see, one of the multiple choice problems was "Are these two knots equivalent?". That has the problem that there are only two possible answers: Yes, and no. David, when he TeXed it up initially, had the choices "Yes", "No", "Cannot be determined", and "42". Naturally, that had to go. My solution was to add a third knot to the problem, turning the question into "Which of these three knots are equivalent?"[5]. Of course, this called for a new diagram.
The diagrams were a can of worms all by themselves. When David TeXed it up initially, he didn't have all the diagrams. Neither did I. In fact, not all of them existed. One enormous time sink that day was scanning in the existing ones and making nonexistent ones - either drawing them and scanning them in, or constructing them in the GIMP, which none of us knew how to use, or Xfig, which, until the end, none of us thought to use. Remember that K7? That was indeed drawn at the last minute as an arbitrary K7... exactly as Sam's evil plan had called for.
Getting back to the knot problem, the answer to the original question was "yes", so this led to the question of whether the third knot should be equivalent to the first two. I didn't like the idea of making all three equivalent, but on the other hand, we couldn't possibly ask them to prove that two knots were inequivalent; they didn't know anything about knot invariants! (This was a big problem on free response, as well - one of the problems asked, "Is this knot equivalent to the unknot? Prove your answer." As I did not have the diagram at that point, I commented in the file, "The answer had better be yes...". Sam assured me that it was.) We eventually made the third one inequivalent with the justification that we weren't asking them to prove it. We, of course, made sure that the third one was actually inequivalent - the first two were not tricolorable, so for the third we used the trefoil. This would actually become the most missed question (seemingly, anyway; I don't have the actual data), with the most common answer being "I and III".
Meanwhile, we still had two problems in need of replacement. (Actually, if we wanted to turn the difficulty gradient into something sensible rather than the oh-this-is-easy-HOLY-CRAP we had right then, we would have had a lot more work to do, but we kind of didn't have time to do that anymore. Remember, we were already overdue.) Modular arithmetic got replaced with proving power of a point (my suggestion, and, I think, actually a very good one). Largely the reason that ended up getting used was because we managed to pull the diagram (with some editing) from a math website rather than making our own. "Crazy mathematician" got replaced with one of the puzzle problems Babai had given us - OK, probably pushing up the difficulty too far there, but we needed to put /something/ in[8].
Finally the test was ready in its final form, and we went to class and then when that was over, was the weekend. I went to Grant's, saw Nadja, played Smash in the Reg, brought it home with me, and played DoA against Grant and Kyle, as has been previously recorded. Then came today, Monday, when we finally met the kids.
Funny story - one kid, a sixth-year - Mark was his name - wasn't on the list. An administrative screw-up? No; he hadn't actually applied at all. He had just sort of assumed he could come back. Since Paul Sally seems to like him, and, well, Paul Sally tends to get what he wants, it's far from impossible that he'll actually be staying. Also, I just have to point out, we have in our class Alex Zorn, great-grandson of Zorn of the Lemma.
So we took attendance and we explained the instructions for the exam. I saw some kids looking at copies of The Contest Problem Book V, and I mumbled to Corrin, "We definitely made the exam too easy." 25 minutes later, grading it, I have to say I was pleasantly surprised. The multiple choice, as stupid as the problems may have looked, apparently wasn't too easy at all as we got a very good spread, from 3/20 ranging pretty linearly up to 19/20[11]. The free response, on the other hand, was too hard. Calvin's catch-all wasn't the only problem to go entirely unanswered - well, OK, the one other was Babai's puzzle problem, which I think we knew was too hard, but problem 9 only got a single answer (IIRC) - and, as it turned out, problem 9 was also copied from last year, though none of us but Sam (who didn't know we weren't supposed to do that, and who was responsible for both that and the crazy mathematician problem) knew this. And the scores, as well as being much lower (unlike on the multiple choice, there were actual 0s), were also a lot more uniform (IIRC). Remember "Is this knot equivalent to the unknot?" It was, as Sam had assured me, but only 3 or 4 people recognized that it was. (Hell, I sure didn't when I saw it, though I didn't really get a chance to work on it.) None gave the full proof we were looking for. And, unsurprisingly, not many people actually found a knotted cycle in K7 - after all, it wasn't made to have a knotted cycle easily visible.
And so the making and taking of the exams both are now over.
In other news:
John Wood was apparently over at Grant's the other day, but I still have no idea where he's living and have not been able to get in contact with him. Anyone?
Assist Trophies. If you haven't seen them, take a look.
And finally, Grant has finally gotten a TV, so after the next Smash in the Reg - yes, this is going to be a regular event now - I'm going to have to give it back to him. Aw.
Now I need to eat something.
-Harry
[0]I may have Kraator and Annie confused here.
[3]For the easily confused, yes, his name is actually "Sam Raskin", and you've all seen him on this LJ before, but today, when we actually first met our kids, Sam - I don't remember the circumstances - actually suggested this nickname for itself. And of course I just had to use it there. Ooh, or perhaps "Sam Ratkin"... "Sam, Rat's Kin"... yeah, I think that's as far as I can go with that one.
[4]Corrin Clarkson of Mathews. We (the 11-12 group) all answer to Corrin; she answers to Paul Sally and Diane Herrmann.
[5]Ironically enough, the problem 18 I had insisted we pull earlier was also "Which of these three knots are equivalent?". The knots in that one were much easier, though.
[8]I actually had not been able to do this problem, because I forgot one very important rule of mathematics: If you want to prove that you cannot get from A to B, look for an invariant. If you don't, you're doing it wrong. Once you remember that, the problem gets a lot easier. (I also couldn't do problem 9 that we put on the test, but that was because I somehow forgot that the area of a triangle is half the base times the height. I haven't done geometry in forever.)
For the record, Babai's puzzle problem (actually due to Tom Hayes) is:
Say we take the upper-right quadrant of the lattice, and put a chip on (0,0). We can take a chip on (x,y) and split it in two, removing it from the board and putting a chip on (x,y+1) and (x+1,y) - but only if both those spaces are empty. Prove that you cannot clear the following areas entirely of chips: (a) The spaces [nonstrictly] within Manhattan distance 3 of the origin. (b) The spaces within Manhattan distance 2 of the origin.
The problem number 9 that only person got was, say you have a finite set of points in the plane such that the triangle defined by any 3 has area ≤1. Then all the points lie within a triangle of area ≤4.
Calvin's catch-all problem, which, I should note, not even any of us aside from Calvin can do, is, take n,k∈N, and consider an nkn×nkn square grid, colored with k colors. Prove that there must exist an n×n submatrix that is all one color. Now you see this, you say, oh, pigeonhole principle, right? And of course it begins there, but it requires a lot more. Calvin didn't give us a proof, he just showed us how to reduce it to an inequality that itself is not easy to prove, and doing that was crazy enough. He used Jensen's inequality on x choose k![9] Which I suppose sounds kind of silly to be surprised about once you say it - I mean, it *is* a convex function for x≥k-1 - but it was very surprising to us at the time, let me tell you.
[9]That's k, not k factorial, not that it matters.
[10]See last week's entry.
[11]The one person getting this score missed, of course, the 3-knots problem.
Pants pants pants pants pants pants pants pants pants pants pants.
Pants pants pants, pants pants.
Pants.
RIFLE SCOPES.
Rifl- OK, that's enough of that.
Now, where was I... oh yes. Pants. Er, rifle scopes. I mean, Duel of Ages. Saturday (or was it Friday?) Grant tells me, he was speaking to Kyle, and he wants to play Duel of Ages again. And so Saturday night Kyle drives over from wherever he lives and we set to killing each other once again, this time, 2½ hours, 4 characters apiece - yes, using our makeshift FFA rules again. (Although we changed the scoring rules a little.) I had Milena Arrebato, Tex, Pvt. Sanchez, and the Martians. Kyle had Frostdancer, Phillip Redlegs, Sir Gawain, and Ardin Glynn. Grant flipped over Kraator, Annie Oakley, Sterling Jack, and... Agent 911. Oh boy.
The board was not your usual parallelogram board; labyrinths were far apart enough that you often had to go for the closest one, +1 bonus be damned, or, well, *not* go for a labyrinth - but if you wanted to stay back and snipe, you'd have to do it in the area; getting from one side of the board to the other was pretty hard. A dismissal roll of 6 could easily screw you.
Kyle got an early lead by killing Kraator[0] and taking his stuff. Meanwhile, here I am with two characters that get a +1 bonus with a long rifle, and the Sniper draws not a rifle but a rifle scope, which gives him an additional +1... as soon as he gets a long rifle. Kyle had one but I had nothing to trade for it. Meanwhile, my Martians had picked up Storm Sprites - the perfect weapon for them, it's based on wits! - and go hide in the woods. "C'mon, I can't seriously be a threat to you here... I'm deep in the woods, I'd get massive hit penalties if I tried to attack you." Not with Storm Sprites! After gleefully taking potshots at everyone around - and killing Annie in melee, taking her Obsidian Tomahawk - an unlucky roll breaks (kills?) the sprites and the Martians were forced to retreat. Kyle was picking up more and more items while Grant was hardly getting any - he got a lucky amaze roll with Agent 911 and got 3 cards for him, but only one of the weapons he picked up (the Colt Peacemaker) was even usable by the Bumbling Idiot, and even that he couldn't hit with. Sterling Jack was in trouble of his own and couldn't trade. Meanwhile Tex finally picks up some weapons, only to respawn right in the middle of a massive shootout. He runs to a building for cover, trying to meet up with the Sniper so they can trade. Unfortunately, they are followed by Kyle's beatup squad who kill them both and take their weapons - and, of course, the rifle scope. Kyle now leads in characters, 4-2-2, and is dominating the board with well-positioned snipers and a ridiculous amount of weapons. He isn't doing quite so well in the labyrinths, though, and Grant and I catch up a bit (Grant less so, as he has to use Agent 911 for this) to end the game 2-1½-1½, Kyle the victor. (We counted ties in an area as half a point for each person). Ouch.
Towels. I walk into the bathroom one day and find the towels missing. Huh? I leave it for the time - I figured maybe someone had gone to wash them, but when Sunday comes, I want to take a shower, and hey, still no towels. I ask Scott - he has no idea. I'm kind of afraid to ask Satory, but if I didn't move them and Scott didn't, well, we need those towels. I bang on his door. "Do you know what happened to the towels?" "Towels?" "The towels. In the bathroom." "Oh. I threw those out." "YOU THREW THEM OUT?!" "They weren't ours, they were left here by the people before us. You never said you wanted to use them - you should have written your name on them."[10] "What... what.. no! No! You write your name on something to say other people can't use it, not so you can use it yourself! By definition, you *want* to use things! That's what they're for! Throwing them out is simply wasteful!"
I'll be honest: I don't get it. Now Satory, as I said, keeps entirely to himself so I really don't have any good idea of him. My first impression was simply, oh, he just likes to keep to himself. My next impression was hearing him and Scott arguing over something and thinking, oh, he sounds like he's going to be a pain to work with. Then there was the bread incident - and the noticing that he wasn't eating any of our food and assumed it went without saying that we wouldn't eat "his" - giving me the snippet, oh, he cares only for himself. And finally Sunday - oh, he's... wasteful? What? Maybe it's just me, but "selfish" and "wasteful" aren't two words I would normally put together, except where kids are involved. Usually your archetypical image of the selfish person is the the maximizer, the efficiency nut, trying to get every last benefit out of whatever he has. But I think that's not it - it's oh, he assumes everyone else is like him, i.e., that they don't give a shit about and won't use things that aren't "their" property - obviously we weren't using those towels since they weren't "ours", and so it was OK to get rid of them. Gah. (Why getting rid of them would be a *good* thing remains a mystery.) He's sure going to have some trouble in life. How did he ever, you know, live in a family? (I'd say "or in a dorm", but, well, different dorms are different...)
Perhaps I'm being a bit unfair to him, but I'll consider that possibility when he actually does something to contradict my image of him. ("Think of it as evolution in action." :P )
Meanwhile, he tells me there are towels in the pantry. Those aren't towels, those are washchloths. Not quite the same thing. As it happens, later I hear him talking to Scott, and he says, he didn't throw them out - he put them in the closet. Well, there are no towels in the closet. "I didn't go in the closet," I say. "I didn't go in the closet," Scott continues. So... what the hell happened? My own towel is lost so I had to borrow Scott's. Of course, I can't keep doing that, but where am I going to buy towels around here?
Anyway, now we come to the last and longest part of this entry (this is what happens when I don't write anything for a while): the diagnostic exam. I mentioned last week that we had to come up with a diagnostic exam to split our kids into groups. We started on it that Monday, of course, drawing up a preliminary version. Now there was to be a multiple choice section and a free response section. Problem: It seems all the people who wanted lots of easy problems ended up on multiple choice, while the people who wanted more hard problems worked on free response. In particular, what happened in multiple choice was that someone suggested that what we really want to test is what these kids have already seen, and that therefore we should boil down the problems to their most basic form. I liked this idea and immediately started on the boiling, until before long it seemed all the actual difficulty in the multiple choice section had, well, gone up in steam.
Meanwhile, free response was headed by resident supervillain Sam Ratskin[3], who came up with the following diabolical problem: We present them with an arbitrary K7 embedded in R³, and ask them to find a knotted cycle. We ourselves do not have to take time to specially construct our K7, or check for a knotted cycle beforehand, because it is a theorem that one always exists! We need only check their answers when they turn them in, and see whether or not they are knotted. (This, of course, raises the question of how difficult it would be to actually check, but I doubt Sam was thinking that far ahead. It was ultimately Jonathan who ended up grading that problem, but I get ahead of myself.)
It was only towards the end of our meeting that I realized what I was doing to the multiple choice section. I was acting these were, like, the 7th and 8th graders, but these were the 11-12ers we were dealing with! A lot of these kids are probably going to be math kids already - going to math team, doing AMCs... these problems are too easy! I mean, yes, we need some easy problems, even some super-easy problems ("divide these complex numbers") because it's a diagnostic test so you want to hit all difficulty ranges, but these problems were *all* easy! I didn't know what was going on with free response at the time, but I figured, you know, we should still have a good spread with multiple choice alone. I quickly tried to replace some problems with harder ones, but not many made it through - it was hard finding problems to replace and harder finding good problems to replace them with.
That night found me working more on this, but on free response - I had made quite a mistake of a suggestion that day. You see, the kids were going to be going to lectures on two things: Knot theory (hence the K7 problem, as well as quite a few other knot problems we put on), and computability theory. We had nothing computabilitical (to make up a ridiculous-sounding word), so I suggested a "robot" problem. You have a robot moving around on a square grid, his program is such-and-such, his environment is such-and-such, what does he end up doing? OK, so this was really just testing their ability to follow directions... but hey, we were already doing things like including basic logic on the multiple choice (which at least one person got wrong). I didn't actually begin writing the problem until that night, when I quickly realized that as the person writing the problem, a complex program and lots of different things for the robot to interact with was the last thing I wanted. I spent much of the night trying to come up with a program that was sufficiently simple, without being too obvious in its effects, that would allow me to send the robot into an infinite loop without the board actually returning to its initial state. It took me a while to realize that all my early programs made this impossible for the simple reason that they were all injective - if you knew where the robot was now, you could deduce where he had been. I eventually managed to get it to work, and sent it off to Corrin[4] with the note that (of course) I think this problem is too easy. The problem was eventually cut, unsurprisingly, for being too much trouble to TeX up.
The next day... if I'm not mistaken, nothing happened on Tuesday. David was supposed to come in with the TeXed up exam, but this never happened. So skip ahead to Wednesday, noting that a draft of the exam was supposed to be in by Wednesday, the final by Thursday (though it didn't actually have to be - we just didn't want to be making all our photocopies Monday morning right before the exam itself). David comes in with his TeXed up version of the multiple choice. Probably free response as well, but I wasn't paying attention to free response at that point. Now we hadn't actually decided on all the problems yesterday - just an outline, what sort of problems they were. Some were specified, but only one had a specified set of distractors. And so David had to make up a lot of the problems. Some were good, some were too easy (or could be reverse-engineered by plugging in the choices), and some were not, in fact, the specified problem. So Corrin and I went over it with him and made a bunch of suggestions. That night I emailed the group and made some more suggestions as to what should be taken out, asking for suggestions for what to replace them with. I put in some of my own, but I got no reply. So when the next day I found myself writing real sets of distractors, I just made the changes I had suggested - Corrin approved them, so I figured that meant they were OK - but I still needed something to replace problem 18, and I had come up with nothing. Eventually, my train of thought - "We need to make this harder. But it's multiple choice. How do you make multiple choice hard? It's inherently easier than the free response... wait, what the hell am I saying? Take a look at the AMC! *That's* multiple choice! And for 11-12ers, it *is* hard! Hm, AMC..." - combined with my wanting to add at least one number theory question (not at all present on the exam at that point) - led me to copying an old AIME problem from memory as the new problem 18 (though of course I had to write distractors). Of course, a lot of problems on the test were copied from the diagnostic test from older years anyway - specifically the tests from two and four years ago, because the program alternates between geometry years and... uh... whatever the other years are. Number theory years, is it? More on that later.
Note that this means it's now Thursday and we're still working on it. Corrin sends me the TeX file and I apply my modifications to the multiple choice, also reading over the free response for clarity and correctness and making corrections where necessary.
That night I send out an email pointing out a potential problem I just realized. (What, us, finished on time? Hah!) I had looked over the free response finally - for clarity, correctness, and proper TeXing only, really, but now it hit me: The result of our lack of coordination was that two of the problems were too similar. We had two roots of unity problems, one in multiple choice, one in free response. One of them, I suggested, had to go. Do you guys agree? Disagree? Suggestions for replacements? No response.
OK, so our lack of coordination had resulted in another problem as well - a huge difficulty gap. It looked like multiple choice had easy covered, and free response had hard covered, but that all-important middle had managed to evade us. And that was what we would spend Friday trying to fix, as best we could. Or perhaps would have, had we not had lots of other things to fix.
And so came Friday. Well, Corrin (or someone) had looked over the exam and decided - the roots of unity problems were OK, but the two basic modular arithmetic problems - one in multiple choice, the other in free response - really were too similar. That's one problem cut from free response, replacement needed. Also, the "crazy mathematician" problem has to go - it was on the test last year. See, copying from two years ago is OK, because then any 12th graders coming here would have been in 10th grade, and would have seen a different test entirely. But copying from last year? Not so good. That's two problems cut from free response, replacements needed. (We did, actually, end up deliberately keeping one problem from the year before - Calvin's catch-all problem, meant to be far too hard for any of them to answer. Indeed, none of them did.) And, oh, we need to make another change to multiple choice as well.
You see, one of the multiple choice problems was "Are these two knots equivalent?". That has the problem that there are only two possible answers: Yes, and no. David, when he TeXed it up initially, had the choices "Yes", "No", "Cannot be determined", and "42". Naturally, that had to go. My solution was to add a third knot to the problem, turning the question into "Which of these three knots are equivalent?"[5]. Of course, this called for a new diagram.
The diagrams were a can of worms all by themselves. When David TeXed it up initially, he didn't have all the diagrams. Neither did I. In fact, not all of them existed. One enormous time sink that day was scanning in the existing ones and making nonexistent ones - either drawing them and scanning them in, or constructing them in the GIMP, which none of us knew how to use, or Xfig, which, until the end, none of us thought to use. Remember that K7? That was indeed drawn at the last minute as an arbitrary K7... exactly as Sam's evil plan had called for.
Getting back to the knot problem, the answer to the original question was "yes", so this led to the question of whether the third knot should be equivalent to the first two. I didn't like the idea of making all three equivalent, but on the other hand, we couldn't possibly ask them to prove that two knots were inequivalent; they didn't know anything about knot invariants! (This was a big problem on free response, as well - one of the problems asked, "Is this knot equivalent to the unknot? Prove your answer." As I did not have the diagram at that point, I commented in the file, "The answer had better be yes...". Sam assured me that it was.) We eventually made the third one inequivalent with the justification that we weren't asking them to prove it. We, of course, made sure that the third one was actually inequivalent - the first two were not tricolorable, so for the third we used the trefoil. This would actually become the most missed question (seemingly, anyway; I don't have the actual data), with the most common answer being "I and III".
Meanwhile, we still had two problems in need of replacement. (Actually, if we wanted to turn the difficulty gradient into something sensible rather than the oh-this-is-easy-HOLY-CRAP we had right then, we would have had a lot more work to do, but we kind of didn't have time to do that anymore. Remember, we were already overdue.) Modular arithmetic got replaced with proving power of a point (my suggestion, and, I think, actually a very good one). Largely the reason that ended up getting used was because we managed to pull the diagram (with some editing) from a math website rather than making our own. "Crazy mathematician" got replaced with one of the puzzle problems Babai had given us - OK, probably pushing up the difficulty too far there, but we needed to put /something/ in[8].
Finally the test was ready in its final form, and we went to class and then when that was over, was the weekend. I went to Grant's, saw Nadja, played Smash in the Reg, brought it home with me, and played DoA against Grant and Kyle, as has been previously recorded. Then came today, Monday, when we finally met the kids.
Funny story - one kid, a sixth-year - Mark was his name - wasn't on the list. An administrative screw-up? No; he hadn't actually applied at all. He had just sort of assumed he could come back. Since Paul Sally seems to like him, and, well, Paul Sally tends to get what he wants, it's far from impossible that he'll actually be staying. Also, I just have to point out, we have in our class Alex Zorn, great-grandson of Zorn of the Lemma.
So we took attendance and we explained the instructions for the exam. I saw some kids looking at copies of The Contest Problem Book V, and I mumbled to Corrin, "We definitely made the exam too easy." 25 minutes later, grading it, I have to say I was pleasantly surprised. The multiple choice, as stupid as the problems may have looked, apparently wasn't too easy at all as we got a very good spread, from 3/20 ranging pretty linearly up to 19/20[11]. The free response, on the other hand, was too hard. Calvin's catch-all wasn't the only problem to go entirely unanswered - well, OK, the one other was Babai's puzzle problem, which I think we knew was too hard, but problem 9 only got a single answer (IIRC) - and, as it turned out, problem 9 was also copied from last year, though none of us but Sam (who didn't know we weren't supposed to do that, and who was responsible for both that and the crazy mathematician problem) knew this. And the scores, as well as being much lower (unlike on the multiple choice, there were actual 0s), were also a lot more uniform (IIRC). Remember "Is this knot equivalent to the unknot?" It was, as Sam had assured me, but only 3 or 4 people recognized that it was. (Hell, I sure didn't when I saw it, though I didn't really get a chance to work on it.) None gave the full proof we were looking for. And, unsurprisingly, not many people actually found a knotted cycle in K7 - after all, it wasn't made to have a knotted cycle easily visible.
And so the making and taking of the exams both are now over.
In other news:
John Wood was apparently over at Grant's the other day, but I still have no idea where he's living and have not been able to get in contact with him. Anyone?
Assist Trophies. If you haven't seen them, take a look.
And finally, Grant has finally gotten a TV, so after the next Smash in the Reg - yes, this is going to be a regular event now - I'm going to have to give it back to him. Aw.
Now I need to eat something.
-Harry
[0]I may have Kraator and Annie confused here.
[3]For the easily confused, yes, his name is actually "Sam Raskin", and you've all seen him on this LJ before, but today, when we actually first met our kids, Sam - I don't remember the circumstances - actually suggested this nickname for itself. And of course I just had to use it there. Ooh, or perhaps "Sam Ratkin"... "Sam, Rat's Kin"... yeah, I think that's as far as I can go with that one.
[4]Corrin Clarkson of Mathews. We (the 11-12 group) all answer to Corrin; she answers to Paul Sally and Diane Herrmann.
[5]Ironically enough, the problem 18 I had insisted we pull earlier was also "Which of these three knots are equivalent?". The knots in that one were much easier, though.
[8]I actually had not been able to do this problem, because I forgot one very important rule of mathematics: If you want to prove that you cannot get from A to B, look for an invariant. If you don't, you're doing it wrong. Once you remember that, the problem gets a lot easier. (I also couldn't do problem 9 that we put on the test, but that was because I somehow forgot that the area of a triangle is half the base times the height. I haven't done geometry in forever.)
For the record, Babai's puzzle problem (actually due to Tom Hayes) is:
Say we take the upper-right quadrant of the lattice, and put a chip on (0,0). We can take a chip on (x,y) and split it in two, removing it from the board and putting a chip on (x,y+1) and (x+1,y) - but only if both those spaces are empty. Prove that you cannot clear the following areas entirely of chips: (a) The spaces [nonstrictly] within Manhattan distance 3 of the origin. (b) The spaces within Manhattan distance 2 of the origin.
The problem number 9 that only person got was, say you have a finite set of points in the plane such that the triangle defined by any 3 has area ≤1. Then all the points lie within a triangle of area ≤4.
Calvin's catch-all problem, which, I should note, not even any of us aside from Calvin can do, is, take n,k∈N, and consider an nkn×nkn square grid, colored with k colors. Prove that there must exist an n×n submatrix that is all one color. Now you see this, you say, oh, pigeonhole principle, right? And of course it begins there, but it requires a lot more. Calvin didn't give us a proof, he just showed us how to reduce it to an inequality that itself is not easy to prove, and doing that was crazy enough. He used Jensen's inequality on x choose k![9] Which I suppose sounds kind of silly to be surprised about once you say it - I mean, it *is* a convex function for x≥k-1 - but it was very surprising to us at the time, let me tell you.
[9]That's k, not k factorial, not that it matters.
[10]See last week's entry.
[11]The one person getting this score missed, of course, the 3-knots problem.
