30 March 2016

Back in high school, when I was studying for math competitions, I had a lot of trouble with geometry problems. What made the situation even more difficult is that the advice that I got from the people who were better at geometry problems just seemed to completely fail.

For example, they suggested drawing a large and accurate diagram, to look for any coincidences. The idea is that such a "coincidence" will actually be more likely to be a true statement, and also likely to be a guide toward solving the whole problem. I very clearly remember one session at MOP where I drew such a diagram and stared at it for a long time, still having no idea what to do.

Olympiad-style geometry problems, which are typically synthetic geometry, often have very elegant solutions that involve drawing in just a couple more points and lines than are present in the problem themselves, and then looking at relationships in the resulting diagram. But while these solutions look simple, how do you come up with those points and lines to draw in?

Well, the common answer is that it's intuitive. That somehow, the people who solve these problems just instinctively know which lines are going to be useful, or at least which ones are more likely to be useful. This was completely different from my experience working on such problems.

I distinctly remember two geometry problems that I did successfully solve, and in neither case did I feel that I had any sort of intuition while coming up with the solution.

2009 USAMO Problem 5

Trapezoid \(ABCD\), with \(\overline{AB} \parallel \overline{CD}\), is inscribed in circle \(\omega\) and point \(G\) lies inside triangle \(BCD\). Rays \(AG\) and \(BG\) meet \(\omega\) again at points \(P\) and \(Q\), respectively. Let the line through \(G\) parallel to \(\overline{AB}\) intsersect \(\overline{BD}\) and \(\overline{BC}\) at points \(R\) and \(S\), respectively. Prove that quadrilateral \(PQRS\) is cyclic if and only if \(\overline{BG}\) bisects \(\angle CBD\).

2009 USAMO Day 2 was an interesting experience. In order to qualify for MOP, a good benchmark is solving two problems on each day. That means that if you find yourself weak in a particular area, you want that area to appear in either the first problem on the day, which means the problem could be easy enough to be solved despite the weakness, or the last problem on the day, meaning that it's hard enough that perhaps even people without that weakness will not solve it.

When I was working on that contest day, I solved problem 4 relatively quickly, so I had a choice between working on problem 5 and problem 6. I chose to focus my efforts on problem 6, and to my surprise I actually solved it within a couple hours. This meant that I had a good 1.5 or 2 hours left for problem 5, and really nothing else to work on.

At least for the first hour of that time, I had my typical olympiad geometry experience. I stared at a diagram, and didn't see anything that would really help. I tried adding various things to the diagram and still didn't find any of those intermediate steps that are supposed to help so much. Eventually, I thought, "There are a lot of points on a circle here. What are all of the facts that Pascal's Theorem give?"

Pascal's Theorem is interesting, because it works for absolutely any hexagon inscribed in a circle, even ones where the same vertex is used more than once, or self-intersecting hexagons. That means that when you have six points on a circle, you actually have many choices for the hexagon.

For this particular situation, with somewhere between half an hour and an hour left in the contest, I found an application of Pascal's Theorem where the three collinear points included two points from the original diagram. Once I added the third point, I realized that the new diagram had everything that I needed to solve the problem.

In the end, my solution looked similar to those elegant solutions that I mentioned before. I extended some of the lines of the diagram to meet the circle, applied Pascal's Theorem, and then did a bit of angle chasing to finish it up. But the additions to the diagram were in no way intuitive to me. I didn't feel that solving this problem meant that I was getting better at geometry. It felt like I had just gotten lucky that one of the applications of Pascal's Theorem happened to give me something useful.

2010 USAMO Problem 1

Let \(AXYZB\) be a convex pentagon inscribed in a semicircle of diameter \(AB\). Denote by \(P\), \(Q\), \(R\), \(S\) the feet of the perpendiculars from \(Y\) onto lines \(AX\), \(BX\), \(AZ\), \(BZ\), respectively. Prove that the acute angle formed by the lines \(PQ\) and \(RS\) is half the size of \(\angle XOZ\), where \(O\) is the midpoint of segment \(AB\).

It turns out that this problem was very easy for me, but the reason for that is quite interesting.

At MOP in 2009, a group of us was tasked with collectively solving a set of geometry problems. We had to each solve three problems, but no two of us could submit a solution to the same problem. I had a lot of trouble with solving any of these problems, and mostly needed to get essentially complete solutions explained to me so that I could write them up.

However, there was one problem from that set that I was able to solve completely on my own.

Let \(ABC\) be a triangle, and \(P\) a point on its circumcircle. Let \(X\), \(Y\), \(Z\) be the feet of the perpendiculars from \(P\) to lines \(AB\), \(BC\), and \(CA\), respectively. Prove that \(X\), \(Y\), \(Z\) all lie on a line. This line is called the Simson line of \(P\).

The solution to this problem is a straightforward angle chase, but the fact that I was able to solve a problem on my own in an environment where I was usually hopeless meant that this lemma stuck with me.

So when I saw the USAMO problem, I noticed perpendiculars being dropped from a point on a circle to various chords. Since the Simson line had stuck with me so strongly, I knew that the perpendicular from \(Y\) to \(AB\) was relevant, and it did in fact lead straight to a simple solution.

However, I didn't feel like this was because of intuition. I happened to have had a problem that was very important to me with the same idea as this problem. But there are so many other possibilities, so again I felt like I had gotten lucky.

After these two experiences, I would easily say that I had no intuition for geometry. In fact, I even was willing to go so far as to say that I had no intution at all, and that instead I was working more along the lines of trying various things until one worked. If I got an answer quickly, it wasn't because I had intuition that a method would work, but instead it would be because I tried it in my head and determined that it would work. Thinking back, I think intuition is a lot more nuanced than what I thought then.

Intuition is based on your past experiences.

For most people, studying for olympiad geometry meant doing lots and lots of problems. They didn't have to be particularly difficult. The important part was doing many problems so that over time you'd notice various patterns that show up repeatedly, and build a personal collection of lemmas that have proven to be useful time and time again.

From this perspective, I think it's reasonable to say that my ability to solve 2010 USAMO Problem 1 actually was based on intuition, where my personal collection of lemmas happened to be very small. Maybe I still got somewhat lucky, but that's definitely not the whole story.

This observation is also why I believe that exercises that appear artificial and disconnected from the task at hand are worthwhile. Students often ask "When am I going to use this?" This question has a simple answer, but it's one that I don't expect to satisfy the asker. Exploring various concepts gives you experience, and the opportunity to notice patterns in the structure of things that will give you that intuition that's so valuable in the future.

Incidentally, this is also along the lines of why category theory is so incredibly useful. For almost anyone with experience in abstract algebra, several patterns come up over and over again. For example, one can define the quotient of a ring by an ideal. Similarly, there's a notion of a quotient of a group by a normal subgroup.

It's no coincidence that the word "quotient" is used both times. The structure of the mathematics in both cases are quite similar, and someone familiar with group theory learning about ring theory will probably be drawing analogies in their mind, and vice versa. Category theory takes these similarities and makes them the focus of study.

Category theorists don't talk about functors to sound smart. The idea of a functor is much like the idea of a fruit. Yes, apples and oranges are quite different from one another. But there are similarities, and many of those similarities are described by the fact that they are both fruits. There's also the concept of a citrus fruit, which includes both oranges and lemons, but not apples. Similarly, it is useful to have names for particular types of functors, like monads.

Intuition can be learned.

Since intuition is based on your past experiences, it makes sense that intuition can be acquired by having more experiences. It's surprisingly hard for people to really internalize this idea. When you're just starting out with learning something new, it's easy to look at people that have been doing it for a long time and believe that they just have something that you don't.

My experience with geometry was like that. I didn't work on enough geometry problems to give myself the experience that other people had. From very early on, I found myself not enjoying geometry, and so I would avoid doing geometry problems. That more or less ensured that the situation wouldn't change. I would continue to be unable to solve geometry problems, and trying would only lead to discouragement. I think that if I had done more geometry problems earlier on, I would have ended up with some amount of good intuition, and would have learned to enjoy the subject much more.

That said, teaching intuition is extremely hard. You might try to build up someone's personal collection of lemmas one at a time by giving a lecture on a pattern and giving them a lot of problems that use that lemma. However, in my experience this idea always fails. Much like my experience with the Simson line, people need to find their own personal connection to the lemmas in their toolbox. By trying to tell them what lemmas are important, it instead turns into something that they just try to memorize.

Artificial settings still help with building intuitions.

Don't dismiss training exercises just because they aren't exactly like what you're training for. Board games, like chess and go, are great examples. Puzzles and tsumego are great examples. The exact situation is extremely unlikely to occur in a real game, but the puzzle situations still do contribute to building your intuition.

Intuition can (sometimes) be replaced by fast thinking.

If someone asks me to estimate something that can be calculated, chances are I'm not trying to feel out the solution. Instead, I'll be trying to do the computation, but fuzzing numbers to make it easier to do rapidly.

Game playing engines are another good example. For some extremely tactical games, evaluating a position accurately is not as important as evaluating a lot of positions. In fact, if improving the evaluation involves making that evaluation slower, it can be to the detriment of the overall engine.

It's fair to say that Deep Blue took this approach. The evaluation function used by Deep Blue was not particularly refined, and certainly less accurate than asking a human to evaluate a particular position. However, Deep Blue could evaluate millions of positions per second, and as a result calculate a good evaluation for the positions resulting in possible immediate moves.

However, this approach isn't universal. Modern chess engines search many fewer positions than Deep Blue did, but have a very refined evaluation function that allows them to evaluate positions accurately despite the lack of speed. While that evaluation function isn't really intuition in the usual sense, you can imagine two evenly matched humans where one is calculating faster, but the other has a more refined ability to judge the resulting positions, which demonstrates an analogous tradeoff.

In fact, when you're building up your intuition, it's important to realize that there is an alternative, and to use that. Using board games as an example again, if you're starting to learn go, instead of trying to simply guess the right move, it's much better to try each move and each of your opponent's responses. Over time, you'll learn what types of moves tend to work well and which don't, allowing you to shortcut past some lines of play. Even without that intuition, you can still get to the right answer (though slowly) by simply trying everything.

I hope that more people give thought about where their intuitions come from. Is it really opaque or can you trace back an intuitive solution to some experience you had in the past? How can new people have similar experiences to build similar intuition?