Jan 27, 2024

What a day! So much to cover!

After breakfast this morning, students were treated to a presentation by Professor Pat Devlin of Swarthmore College on“Interactive Proofs.” The lecture started with the intriguing assertion that Professor Devlin is psychic and can predict the roll of a die. He predicted he would roll a 6 and then did so, repeatedly! Of course he then admitted that the die was weighted in such a way that a six was overwhelmingly likely. The situation begged the question, “How can someone prove to you that they are a paychic, or how can you disprove that they are? Student suggestions included doing a higher number of trials, providing their own die, etc. The basic point seemed to be that the challenger should control the challenge and repeat to eliminate the luck factor.

In a parallel example, Professor Devlin showed three cards that seemed to look the same on their backs (specifically the 2 of hearts, 2 of diamonds, and a joker) and claimed that he could find the joker just by looking at the backs of the cards. He asked the students to devise a challenge where his chances of getting lucky would be less than one in a million. One student suggested adding more cards, but that wouldn’t help because the claim was limited to those three cards. Other students said the challenger should shuffle rather than let the person making the claim do it. Given that each time the Professor identified the joker he had a 1/3 chance of getting lucky, how many trials would be convincing? Professor Devlin had all the students stand up and make predictions on a series of attempts as to where the joker would be, with students sitting down when they missed a guess. As ot happened virtually nobody was left standing after 4 trials. (Eventually Professor Devlin showed the students that the joker was a marked card, proving he that he could in fact tell the difference, but at the cost of revealing the mechanism underlying the trick.

Then came a version of Where’s Waldo called“find the face” - on a card with many faces drawn on it, assuming cash prizes for fastest, second-fastest, etc., how can the first person claiming to find the desired face prove that they actually found the face without ruining the game for everyone else? Just by saying they found it? The difficulty is that we can’t let them point it out or it ruins the possibility of a good battle for second place, etc., so how can they prove they found it without revealing it?

One way is to have the guesser cut a hole in a blank card to cover everything but the face. But if we show that cutout card it still reveals where the face is. And what if nobody can be trusted to know the correct answer to assess the claims? It gets very complicated, but it was a good example of this type of security regimen might work in the real world.

Our next presenter up was…me! Traditionally when I present to the students I try to pick one specific style of logic puzzle to explain, and then give them ample time to work out several example exercises. Last year I explained “skyscraper puzzles,” which are fun challenges. This year we covered nonograms aka nonogrids. These are logic puzzles on a grid in which a picture is increasingly revealed as students evaluate how darkened boxes in each row or column are placed. They also happen to be my favorite kind of logic puzzle, and apparently Professor Devlin’s as well! If you’ve never done them before they are a very relaxing pursuit. Here’s my favorite place to find them, though a simple google search will find other resources: nonograms.org. Just to keep my presentation interesting I awarded prizes for the most puzzles completed before lunch by a team (congrats to team 3 students!) and for most puzzles completed before lunch by an individual (congrats to student 8-10!). Some of the students seemed very engrossed in the puzzles; many were seen continuing to work on them hours later, even after there were no more prizes to be won.

After a break for lunch, students returned to another lecture from Paul Ellis, who they might have recognized since he spoke to them about knot theory two days ago. Today he led them through some strategy exercises inspired by the game NIM, one of the oldest known games in the world. It is basically a removal game, with players removing stones from piles until one person removes the final stone.

Students started by figuring out the optimal playing strategy for 2 pile NIM - the strategy is to keep the piles even - and then were asked to think about 3 pile NIM, which is significantly more complex. Professor Ellis revealed to the students that the first player to move definitely can win every game, but left them to figure out: what should their first move be?

That left just one final lecture to round out the afternoon, and this one was by perennial Math League Summer favorite Steve Miller. His topic of the day was “from the quadratic equation to differentiation, part I,” and he wasted no time before polling the students to see how much math they had been exposed to in school. He then set a goal to teach most of the A/B calculus in 60-90 minutes!

Steve started by asking, “what does it take to solve an equation?” He then ran through a series of equations to solve: first came 3x + 6 = 0, then the general case of ax + b = 0 (specific and general linear equations). Next were equations of degree 2: x^2 = 0; then x^2 + c = 0; then ax^2 + c = 0; then a(x-h)^2 + c = 0. Professor Miller then showed that for cubic equations there is a ridiculously complex formula that will find all 3 roots; then for quartic equations there is an even more ridiculous formula to find just one root; then for 5th degree equations we really don’t even have a formula!

Professor Miller then led the class through an exercise in finding the approximate value of sqrt 3 by the“divide and conquer” approach. First he established that its value is between 1 and 2; then he tested 1.5, and found that our desired answer is between 1.5 and 2…after that he challenged the tables of students to get as many digits as possible in 4 minutes. From there It was only a matter of a surprisingly small amount of time before Professor Miller really did start explaining differentials to the students! More on that tomorrow.

Then came dinner and, of course, the big event of the evening - the awards ceremony! We were thrilled that Dan Flegler, senior director of Math League, was able to make it to the show and participate fully despite his recent difficulties with mobility. This year he even treated us to a riveting account of the historical matchup between two of the greatest racehorses of all time, in one of the most exciting race finishes in history. I, on the other hand, was moved to retell the least successful joke I had told during the talent show, and perhaps not surprisingly it was just as unsuccessful in front of our awards ceremony audience. You win that round, Dan!!

Seriously, though, It is always such a satisfying experience to see how excited the students get as we reveal the winners of first the individual and then the team events. They are also always so supportive of one another! It really is a night filled with special moments. It was sometimes hard to tell who was most excited - the students, the parents, or the counselors! I can tell you that our counselors work very hard to make our students comfortable and successful, and by the time of the awards ceremony they are extremely proud and excited when the students on their teams win awards! I think that was pretty obvious to anyone who was paying attention tonight.