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I’ve just opened the research thread for the mini-polymath4 project over at the polymath blog to collaboratively solve one of the six questions from this year’s IMO.  This year I have selected Q3, which is a somewhat intricate game-theoretic question.  (The full list of questions this year may be found here.)

This post will serve as the discussion thread of the project, intended to focus all the non-research aspects of the project such as organisational matters or commentary on the progress of the project.    The third component of the project is the wiki page, which is intended to summarise the progress made so far on the problem.

As with the previous mini-polymath projects, I myself will be serving primarily as a moderator, and hope other participants will take the lead in the research and in keeping the wiki up-to-date.

Just a reminder that the mini-polymath4 project will begin in three hours at Thu July 12 2012 UTC 22:00.

Two quick updates with regards to polymath projects.  Firstly, given the poll on starting the mini-polymath4 project, I will start the project at Thu July 12 2012 UTC 22:00.  As usual, the main research thread on this project will be held at the polymath blog, with the discussion thread hosted separately on this blog.

Second, the Polymath7 project, which seeks to establish the “hot spots conjecture” for acute-angled triangles, has made a fair amount of progress so far; for instance, the first part of the conjecture (asserting that the second Neumann eigenfunction of an acute non-equilateral triangle is simple) is now solved, and the second part (asserting that the “hot spots” (i.e. extrema) of that second eigenfunction lie on the boundary of the triangle) has been solved in a number of special cases (such as the isosceles case).  It’s been quite an active discussion in the last week or so, with almost 200 comments across two threads (and a third thread freshly opened up just now).  While the problem is still not completely solved, I feel optimistic that it should fall within the next few weeks (if nothing else, it seems that the problem is now at least amenable to a brute force numerical attack, though personally I would prefer to see a more conceptual solution).

Two polymath related items for this post. Firstly, there is a new polymath proposal over at the polymath blog, proposing to attack the “hot spots conjecture” (concerning a maximum principle for a heat equation) in the case when the domain is an acute-angled triangle (the case of the right and obtuse-angled triangles already being solved). Please feel free to comment on the proposal blog post if you are interested in participating.

Secondly, it is once again time to set up the annual “mini-polymath” project to collaboratively solve one of this year’s International Mathematical Olympiad problems. This year, the Olympiad is being held in Argentina, with the problems given out on July 10-11. As usual, there will be a wiki page, discussion thread, and research thread for the project. As in previous years, the first thing to resolve is the starting date and time, so I am setting up a poll here to fix a time (and also to get a preliminary indication of interest in the project).  (I am using 24-hour Coordinated Universal Time (UTC) for these times.  Here is a link that converts the first time given in the poll (Thu Jul 12 2012 UTC 6:00) into other time zones.) Given that the last three mini-polymaths were reasonably successful, I am not planning any changes to the format, but of course if there are any suggestions for changes, I’d be happy to hear them in the comments.

High school algebra marks a key transition point in one’s early mathematical education, and is a common point at which students feel that mathematics becomes really difficult. One of the reasons for this is that the problem solving process for a high school algebra question is significantly more free-form than the mechanical algorithms one is taught for elementary arithmetic, and a certain amount of planning and strategy now comes into play. For instance, if one wants to, say, write ${\frac{1,572,342}{4,124}}$ as a mixed fraction, there is a clear (albeit lengthy) algorithm to do this: one simply sets up the long division problem, extracts the quotient and remainder, and organises these numbers into the desired mixed fraction. After a suitable amount of drill, this is a task that can be accomplished by a large fraction of students at the middle school level. But if, for instance, one has to solve a system of equations such as

$\displaystyle a^2 + bc = 7$

$\displaystyle 2b - c = 1$

$\displaystyle a + 2 = c$

for ${a,b,c}$, there is no similarly mechanical procedure that can be easily taught to a high school student in order to solve such a problem “mindlessly”. (I doubt, for instance, that any attempt to teach Buchberger’s algorithm to such students will be all that successful.) Instead, one is taught the basic “moves” (e.g. multiplying both sides of an equation by some factor, subtracting one equation from another, substituting an expression for a variable into another equation, and so forth), and some basic principles (e.g. simplifying an expression whenever possible, for instance by gathering terms, or solving for one variable in terms of others in order to eliminate it from the system). It is then up to the student to find a suitable combination of moves that isolates each of the variables in turn, to reveal their identities.

Once one is sufficiently expert in algebraic manipulation, this is not all that difficult to do, but when one is just starting to learn algebra, this type of problem can be quite daunting, in part because of an “embarrasment of riches”; there are several possible “moves” one could try to apply to the equations given, and to the novice it is not always clear in advance which moves will simplify the problem and which ones will make it more complicated. Often, such a student may simply try moves at random, which can lead to a dishearteningly large amount of effort expended without getting any closer to a solution. What is worse, each move introduces the possibility of an arithmetic error (such as a sign error), the effect of which is usually not apparent until the student finally arrives at his or her solution and either checks it against the original equation, or submits the answer to be graded. (My own son can get quite frustrated after performing a lengthy series of computations to solve an algebra problem, only to be told that the answer was wrong due to an arithmetic error; I am sure this experience is common to many other schoolchildren.)

It occurred to me recently, though, that the set of problem-solving skills needed to solve algebra problems (and, to some extent, calculus problems also) is somewhat similar to the set of skills needed to solve puzzle-type computer games, in which a certain limited set of moves must be applied in a certain order to achieve a desired result. (There are countless games of this general type; a typical example is “Factory balls“.) Given that the computer game format is already quite familiar to many schoolchildren, one could then try to teach the strategy component of algebraic problem-solving via such a game, which could automate mechanical tasks such as gathering terms and performing arithmetic in order to reduce some of the more frustrating aspects of algebra. (Note that this is distinct from the type of maths games one often sees on educational web sites, which are usually based on asking the player to correctly answer some maths problem in order to advance within the game, making the game essentially a disguised version of a maths quiz. Here, the focus is not so much on being able to supply the correct answer, but on being able to select an effective problem-solving strategy.)

It is difficult to explain in words exactly what type of game I have in mind, so I decided to create a quick mockup of what a sample “level” would look like here (note: requires Java). I didn’t want to spend too much time to make this mockup, so I wrote it in Scratch, which is a somewhat limited programming language intended for use by children, but which has the benefit of being able to churn out simple but functional apps very quickly (the mockup took less than half an hour to write). (I would certainly not attempt to write a full game in this language, though.) In this mockup level, the objective is to solve a single linear equation in one variable, such as ${2x+7=11}$, with only two “moves” available: the ability to subtract ${1}$ from both sides of the equation, and the ability to divide both sides of the equation by ${2}$, which one performs by clicking on an appropriate icon.

It seems to me that one could actually teach a fair amount of algebra through a game such as this, with a progressively difficult sequence of levels that gradually introduce more and more types of “moves” that can handle increasingly difficult problems (e.g. simultaneous linear equations in several unknowns, quadratic equations in one or more variables, inequalities, etc.). Even within a single class of problem, such as solving linear equations, one could create additional challenge by placing some restriction on the available moves, for instance by limiting the number of available moves (as was done in the mockup), or requiring that each move cost some amount of game currency (which might possibly be waived if one is willing to perform the move “by hand”, i.e. by entering the transformed equation manually). And of course one could also make the graphics, sound, and gameplay fancier (e.g. by allowing for various competitive gameplay modes). One could also imagine some other types of high-school and lower-division undergraduate mathematics being amenable to this sort of gamification (calculus and ODE comes to mind, and maybe propositional logic), though I doubt that one could use it effectively for advanced undergraduate or graduate topics. (Though I have sometimes wished for an “integrate by parts” or “use Sobolev embedding” button when trying to control solutions to a PDE…)

This would however be a fair amount of work to actually implement, and is not something I could do by myself with the time I have available these days. But perhaps it may be possible to develop such a game (or platform for such a game) collaboratively, somewhat in the spirit of the polymath projects? I have almost no experience in modern software development (other than a summer programming job I had as a teenager, which hardly counts), so I would be curious to know how projects such as this actually get started in practice.

I’ve just opened the research thread for the mini-polymath3 project over at the polymath blog.  I decided to use Q2 of this year’s IMO, in part to see how the polymath format copes with a geometric problem.  (The full list of questions for this year is available here.)

This post will serve as the discussion thread of the project, intended to focus all the non-research aspects of the project such as organisational matters or commentary on the progress of the project.    The third component of the project is the wiki page, which is intended to summarise the progress made so far on the problem.

As with mini-polymath1 and mini-polymath2, I myself will be serving primarily as a moderator, and hope other participants will take the lead in the research and in keeping the wiki up-to-date.

Just a reminder that the mini-polymath3 project begins in 24 hours, on July 19, 8pm UTC.

Following the results from the recent poll on this blog, the mini-polymath3 project (which will focus on one of the problems from the 2011 IMO) will start at July 19 8pm UTC, and be run concurrently on this blog, on the polymath wiki, and on the polymath blog.

In the last two years, I ran a “mini-polymath” project to solve one of the problems of that year’s International Mathematical Olympiad (IMO).  This year, the IMO is being held in the Netherlands, with the problems being released on July 18 and 19, and I am planning to once again select a question (most likely the last question Q6, but I’ll exercise my discretion on which problem to select once I see all of them).

There are some kinks with our format that still need to be worked out, unfortunately; the two main ones that keep recurring in previous feedback are (a) there is no way to edit or preview comments without the intervention of one of the blog maintainers, and (b) even with comment threading, it is difficult to keep track of all the multiple discussions going on at once.  It is conceivable that we could use a different forum than the WordPress-based blogs we have been using for previous projects for this mini-polymath to experiment with other software that may help ameliorate (a) and (b) (though any alternative site should definitely have the ability to support some sort of TeX, and should be easily accessible by polymath participants, without the need for a cumbersome registration process); if there are any suggestions for such alternatives, I would be happy to hear about them in the comments to this post.  (Of course, any other comments germane to the polymath or mini-polymath projects would also be appropriate for the comment thread.)

The other thing to do at this early stage is set up a poll for the start time for the project (and also to gauge interest in participation).  For ease of comparison I am going to use the same four-hour time slots as for the 2010 poll.  All times are in Coordinated Universal Time (UTC), which is essentially the same as GMT; conversions between UTC and local time zones can for instance be found on this web site.   For instance, the Netherlands are at UTC+2, and so July 19 4m UTC (say) would be July 19 6pm in Netherlands local time.  (I myself will be at UTC-7.)

Gil Kalai has officially started the Polymath3 project (Polynomial Hirsch conjecture) with a research thread at his blog.

The original aim of this project is to prove the polynomial Hirsch conjecture, which is a conjecture in the combinatorial geometry of polytopes.  However, there is a reduction due to Eisenbrand, Hahnle, Razborov, and Rothvoss that would deduce this conjecture from a purely combinatorial conjecture, which can be stated as follows.

Combinatorial polynomial Hirsch conjecture. Let ${\mathcal F}_1,\ldots,{\mathcal F}_t$ be non-empty collections of subsets of $\{1,\ldots,n\}$ with the following properties:

1. (Disjointness) ${\mathcal F}_i \cap {\mathcal F}_j = \emptyset$ for every $i \neq j$.
2. (Connectedness)  If $1 \leq i < j < k \leq t$ and $A \in {\mathcal F}_i, B \in {\mathcal F}_ k$, there exists $C \in {\mathcal F}_j$ such that $A \cap B \subset C$.

Then t is of polynomial size in n (i.e. $t = O(n^{O(1)})$).

For instance, when n=3, one can obtain such a family with t=6, e.g.

${\mathcal F}_1 = \{ \emptyset\}, {\mathcal F}_2 = \{ \{1\}\}, {\mathcal F}_3 = \{\{1,2\}\}, {\mathcal F}_4 = \{\{1,2,3\}\},$

${\mathcal F}_5 = \{\{2,3\}\}, {\mathcal F}_6 = \{\{3\}\};$

one can show that this is the best possible value of t for this choice of n.  The best possible value of t for n=4 is still not worked out; it is between 8 and 11.

One appealing thing about this problem is that there is a simple elementary argument that gives the bound $t \leq n^{\log_2 n + 1}$ for all $n \geq 2$; and so in some sense one is “only a logarithm away” from proving the conjecture.  Anyway, the project is just starting, and does not require any particularly specialised background, so anyone who may be interested in this problem one may want to take a look at the research thread.