Tanya Khovanova's Math Blog

Here is a fun math activity I use with my students, after I teach them to play the game of set. To other teachers — feel free to clone this idea.

First, I ask the students if they know what a magic square is. They usually do know that a magic square is a three-by-three square of distinct digits, so that every row, column and diagonal has the same sum. Then I ask them what a magic set square might be. Often they guess correctly that it is a three-by-three square made of set cards, so that every row, column and diagonal form a set. Once that’s established, I have them build magic set squares.

While they’re building them, I ask a lot of questions, from how many cards there should be in the deck to how many different sets there are.

Once the squares are built, I ask them what a magic set cube might be. Their next task is to use their magic set squares as the bottom layer in building magic set cubes. In order to see all the cards in the cube, I instruct them to arrange the layers (bottom, middle and top) side-by-side.

As they’re working on their cubes, I continue quizzing them. How many main diagonals does a cube have? Once they confirm that the answer is four, I ask them to show me those diagonals in their magic set cubes and check that they are sets. I might also ask them how many different magic set squares should be inside a magic cube. This is a theoretical math question they need to answer before finding them in their own model. Next they need to identify the different sets that form lines inside their cubes.

At this point, some students guess my next request: to construct a magic set hypercube.

After students build their hypercubes, they never want to destroy them. They like comparing the different hypercubes and often take photos of them. If there’s still time left, I can continue in several directions. For example, they can count the main diagonals of the hypercube and find them in their models. Alternatively, they can find a “no set” — the largest possible set of cards inside a magic set hypercube that doesn’t contain a set.

Math is usually about thinking, but this is one activity the students can do with their hands. And that adds another layer of magic.

By Tanya Khovanova and Richard Stanley

This essay is written especially for high school and undergrad math lovers who enjoy problem solving and who plan to major in mathematics. One of the authors, Tanya, often received this advice when she was an undergraduate in Russia: “Problem solving is child’s play. You’ll have to change your attitude if you plan to succeed in research.”

Perhaps that’s why some famous problem solvers, even those who won gold medals at IMO, became not-so-famous mathematicians. To help you avoid that fate, we’ll discuss the ways in which research is unlike problem solving.

Is research different from problem solving?

Yes and no. There are many mathematicians who continue problem solving as their form of research. Remember Paul Erdos who used to suggest a lot of problems and even offered money rewards for solutions. Many mathematicians solve problems posed by other people. You might consider Andrew Wiles as the ultimate math problem solver: he proved Fermat’s last theorem, which had been open for 400 years. Though he could not have done it without the many theories that had already been generated in the search to find the elusive proof.

You can become a mathematician and continue to look around you for problems to solve. Even though this is still problem solving, the problems will be very different from competition problems, and you will still need to adjust to this type of research.

Problems you solve during research

So, what is the difference between problems that mathematicians solve during competition and the problems they tackle for their research?

Expected answer. In competition problem solving you know there is a solution. Often you know the answer, but you just need to prove it. In research there is no guarantee. You do not know which way it will go. For this reason finding counter-examples and proving that some ideas are wrong is a positive contribution, for it can eliminate some possibilities. So one adjustment is that you might start valuing negative answers.

Difficulty level. Competition problems are designed to be solved in one hour, so you are expected to generate an idea in just minutes. In research the problem might drag on for years, because it is far more difficult. If you get used to the instant gratification of competition problem solving, you might find the lengthy work of research frustrating. It’s very important to adjust your expectations so that you won’t drop a problem prematurely. You need to measure progress in small intermediate steps and learn to appreciate this different rhythm.

Motivation. Although you miss the euphoria of finding quick solutions, you get a different kind of reward with research. Because no one knows the answer in advance, when you solve the problem, you are the first to do so. You have opened up a new truth.

Time limits. In competitions you have a time limit for every problem. In research you set your time limits yourself. That allows you to put a problem aside and come back later if necessary. In a sense you can think about several problems at the same time.

Your passion. You can choose your problems yourself. Research is much more rewarding if you follow your heart. In competitions you have to spend time on problems you might not like. Here you have an option to choose and pick only the problems that appeal to you. Thus, you become more motivated and as a result more successful.

Finding a problem

After solving problems posed by other people, the next step is to pose math problems yourself. As we mentioned before, in research you do not always have a strictly-defined problem. It is a significant adjustment to move from solving already-defined problems to posing the problems yourself.

Generalizations. Often you can generalize from an existing problem to more general cases. For example, if you see a problem for n=3, you can wonder what happens for any n, or for any prime n.

Being on the lookout. Sometimes a situation puzzles you, but you can’t formulate a specific problem around that situation. For example, why do most of the terms in the sequence end in 9? Is there a reason for that? Or, you might find that a formula from your integrable systems seminar is similar to a formula from your representation theory class. This might lead you to the essential research question: “What is going on?” You always need to be on the lookout for the right questions.

Value. When you create your own research problems it is crucial to always ask yourself: Is the problem I am creating important? What is the value of this problem? There is no a good reason to create random generalizations of random problems. If the problem you found interests you very much, that is the first sign that it might interest other people; nonetheless, you should still ask yourself how this problem will help advance mathematics.

Mathematics is not only problem solving

There are other things to do than solve problems. There are many mathematicians who work differently, who don’t solve problems or don’t only solve problems. Here are some of the many options mathematicians have:

Building structures. You may not be interested in calculating the answer to a question, but rather in building a new structure or a new theory.

Advancing the language. When you invent new definitions and new notations, you will help to simplify a math language so that the new language will allow you to prove your results and other peoples’ results faster and clearer.

Unification. Sometimes you notice two results in two different areas of mathematics with some kind of similarity. Explaining why these results are the same might create a new understanding of things. It is great to unify two different areas of mathematics.

Explaining. Very often proofs are not enough. Why is something true? What’s the reason and what’s the explanation? It is good to ask yourself a “why” question from time to time, such as, “Why is this proof working?” When you find an answer, it might become easier to understand what to do next and how to generalize your proof.

Directions. Many mathematicians are valued not for the problems they solve or suggest, but for ideas and directions they propose. Finding a new direction for research can generate unexpected opportunities and create tons of math problems on the way. It can be valuable to come up with good conjectures, even if you have no hope of solving them yourself. Two example of this are the Weil conjectures (eventually proved by Deligne) and the Langlands program, which is still incomplete but which has generated a huge amount of important research.

Vision. What is the most general thing that can be proved by this technique? What kinds of improvements and refinements are there? It is good to step back from the problem you solved and meta-think about it.

As you can see, problem solving is just the beginning of all that mathematics can offer you. Mathematicians find these other options very rewarding, so it’s worth your while to try these varied aspects of mathematical work to see if you have a taste for other things. If you don’t venture beyond problem solving you might miss the full beauty of mathematics.

My guest blogger is David Wilson, a fellow fan of sequences. It is a nice exercise to understand how this graph works. When you do, you will discover that you can use this graph to calculate the remainders of numbers modulo 7. Back to David Wilson:

I have attached a picture of a graph.

Write down a number n. Start at the small white node at the bottom of the graph. For each digit d in n, follow d black arrows in a succession, and as you move from one digit to the next, follow 1 white arrow.

For example, if n = 325, follow 3 black arrows, then 1 white arrow, then 2 black arrows, then 1 white arrow, and finally 5 black arrows.

If you end up back at the white node, n is divisible by 7.

Nothing earth-shattering, but I was pleased that the graph was planar.

My son, Sergei Bernstein, got accepted to MIT through early action. Because the financial costs of studying at MIT worried me, I insisted that Sergei also apply to Princeton and Harvard, as I had heard they give generous financial packages. In the end, Sergei was rejected by Princeton and wait-listed and finally rejected by Harvard. Though many people have been rejected by Princeton and Harvard, not too many of them have won places on US teams for two different international competitions — one in mathematics and the other in linguistics. To be fair, Sergei was accepted by these teams after Princeton had already rejected him. Nonetheless, Sergei has an impressive mathematical resume:

• In 2005 he was the National MathCounts Written Test Champion.
• In 2005 he was the National MathCounts Master’s Round Champion.
• In 2007 and 2009 he was a USAMO winner.
• In 2008 he passed Math 55a at Harvard taught by Dennis Gaitsgory, which is considered to be the hardest freshman math course in the country. More than 30 students started it and less than 10 finished. Sergei was one of the finishers, and he was only a high school junior.
• In 2007, 2008 and 2009 he competed at a 12th grade level at the Math Kangaroo, while he actually was in 10th, 11th and 12th grade. He placed first all three times.
• In 2009 he was on the US team at the Romanian Masters in Mathematics competition, which might be a harder competition than the International Mathematical Olympiad. He got a silver medal and was second on the US team.
• In 2009 he placed 5th in the North American Computational Linguistics Olympiad, making it to the Alternate US Team for the International Linguistics Olympiad.

I am trying to analyze why he was rejected and here are my thoughts.

1. His application forms to Harvard and Princeton were different from MIT. Yes, MIT was his first choice and he wrote a customized essay for MIT. For other places he had a common essay. But as he was supposed to be flagged as a top math student, his essay should have been irrelevant, in my opinion.
2. Admissions offices made a mistake. I can imagine that admissions offices never heard of the Romanian Masters in Mathematics competition, because it is a relatively new competition and the USA only joined it in 2009 for the first time. On its own, though, it should have sounded impressive. Also, they might not have known about the Math 55 course at Harvard, as usually high-schoolers do not take it. But that still leaves many other achievements. Many people told me that admissions offices know what they are doing, so I assume that I can disregard this point.
3. Princeton and Harvard knew that he wanted to go to MIT and didn’t want to spoil their admission rate. I do not know if colleges communicate with each other and whether Princeton and Harvard knew that he was admitted early to MIT. Because he had sent them a common application essay, they may have been suspicious that they weren’t his first choice.
4. Harvard and Princeton didn’t want him. I always heard that Harvard and Princeton want to have well-rounded people, whereas MIT likes geeks. I consider Sergei quite well-rounded as he has many other interests and achievements beyond mathematics. Perhaps his other accomplishments aren’t sufficiently impressive, making him less round than I thought he was.
5. Harvard and Princeton are not interested in mathematicians. Many people say that they want future world leaders. I think it is beneficial for a world leader to have a degree in math, but that’s just my personal opinion. And of course, to support their Putnam teams, it is enough to have one exceptional math student a year.
6. Sergei couldn’t pay. Yes, we marked on the application that we need financial help. In the current financial crisis it could be that even though Harvard and Princeton do not have enough money to support students, they do not want to go back and denounce their highly publicized generosity.

Many people told me of surprising decisions by Ivy League schools this year. The surprises were in both directions: students admitted to Ivy League colleges who didn’t feel they had much of a chance and students not admitted that had every right to expect a positive outcome. I should mention that I personally know some very deserving kids who were admitted.

I wonder if there has been a change in the financial demographics of the students Harvard and Princeton have accepted this year. If so, this will be reflected in the data very soon. We will be able to see if the average SAT scores of students go down relative to the population and previous years.

I do not know why Sergei wasn’t accepted; perhaps I’m missing something significant. But if it was because of our finances, it would be ironic: Sergei wasn’t admitted to Princeton and Harvard for the same reason he applied there.

I recently got a new job — to coordinate math students at RSI (Research Science Institute). RSI provides a one-month research experience based at MIT for high school juniors. The program is highly competitive and kids from all over the world apply for it.

Before the program started, I asked around among mathematicians for advice on how to do a great job with these talented kids. I was surprised by the conflicting opinions on the value of the program. I thought you’d be interested in hearing those opinions, although I confess that I do not remember who said what, or anyone’s exact words. I will just repeat the gist of it.

Former participants:

• I went there, it was awesome.
• I went there, it was underwhelming.
• Canada/USA math camp is more fun for sure.
• RSI is an absolutely fantastic experience for students, and I think the adults who take part enjoy it very much as well.

Potential participants:

• Cool, if I get there I’ll try to prove the Riemann Hypothesis.
• Last year Eric Larsen won $100,000 as a result of this program. If twenty math students participate, then the expected return is $5,000 per one month of work — not bad for a high-schooler.
• MIT is my dream school; just to be there will be inspiring.
• I will prove the Riemann Hypothesis.
• Yeah, I can become famous.
• Cool, I want to be a mathematician — I should try this.
• I love Canada/USA math camp and I’d rather go there.

Grad students, former and potential mentors:

• My professor doesn’t have a good problem for me. If he gives a nice problem to a high school student, that will be unfair.
• It’s just a job.
• What if I solve the problem first, do I keep silent? — That doesn’t make any sense.
• What if this high school student is better than me? That would be a bummer.
• This job was a lot of fun; I enjoyed it.
• I used to participate in RSI myself, and that was great. Now I would like to be on the giving side.
• RSI teaches students how to get versed in impressing people. For the Meet-Your-Mentor Night the students showed up in suits. How many real mathematicians do you know that own a suit?

Professors on the program in general:

• Usually students study mathematics for many years. RSI allows them to actually do mathematics.
• I studied for many years before I could start to do research. This RSI experiment is degrading to mathematics and disrespectful to mathematicians.
• Most students are wired towards problem solving, and very often they need only one basic idea and 15 minutes to solve a problem. Research has a completely different pace; it is important that kids try it.
• Some students go to this program because they want to win competitions and get to good colleges. These goals should be secondary. We should accept students because they want to try research.
• One month for research? Is this a joke? Do you like fast food?
• These are the best students from around the country. It feels nice when a potential future Fields medalist looks up to you.
• These students might be better than average undergraduate students at MIT. It might be fun to work with them.
• I think that the number of students who might be a good fit for such a program is very small; the number of professors who might be a good fit is very small too. If this program grows it might become completely useless.
• High school students are being mentored by grad students, who themselves have just started their own research. Grad students do not have enough experience to really guide people through research.
• It is such a great opportunity to get a taste of research while you are in high school.
• People usually choose projects for their research. These kids are given projects: this is not research — it’s slave labor.
• One month is not enough for interesting research. It would be good if students use this month to jump-start some research and then continue it after the program.
• It’s a waste of time to learn mathematics for many years and then discover that you do not like research. This program gives an opportunity for students to decide whether they are interested in research very early in their lives. This is tremendously useful.

I asked some math professors to suggest problems for these students:

• I have some problems I can give, but they require deep knowledge of topology. The students would need to take some courses to understand the second paragraph of the paper I would give them, which they can’t succeed in doing in a month. Can we replace this program with my course?
• It wouldn’t be nice to give them a problem that is too difficult. If the problem is easy, then I usually have an idea how to solve it. Instead of wasting two hours describing an easy problem to students, I can use this time to solve it myself.
• Ask Ira Gessel or Pavel Etingof. I have heard that they generate problems faster than their graduate students solve them.
• I have some leftover problems I can give away. However my concern is this: what if they solve it or mostly solve it, but then go back to school without writing their paper. What do I do? Giving the same problem to someone else or writing a paper myself without mentioning the student would not be kosher. Writing a joint paper for them is a burden. I need to think about a leftover problem I do not care about.
• If I have a good project, I will give it to my graduate students. Why would I invest in a high school student who is here for a month and probably is not ready for this anyway?
• That’s great, the online database of integer sequences contains tons of conjectures. They even have an index pointing towards “Conjectured sequences” and towards “Unsolved problems”. Besides, you can search the database for the words “conjecture”, “apparently” or “appears”. There is also an article by Ralf Stephan describing 100 conjectures from the OEIS.
• I have some things I need to calculate, but I do not know programming. If someone can do this for me that would be good.
• They usually want to submit papers for competitions, which means they do not want me to be a coauthor. I do not have problems I just want to throw away.
• Richard Stanley keeps a list of unsolved problems, ask him.
• There is a list of unsolved problems on wiki, but they are too difficult.
• They can always try to find a different proof for something.

The 2009 RSI has just begun. We have awesome students, great mentors and quite interesting problems to solve. I am positive we’ll prove the negativists wrong.

There is interesting data to show that MIT takes math students more seriously than Harvard and Princeton. By Michael Sipser’s suggestion I looked at the Putnam Competition results. Out of the top 74 scorers of 2007, 21 were from MIT, 9 from Harvard and 7 from Princeton. Keep in mind that the total freshman enrollment at MIT is much lower than at Harvard or Princeton. This story repeated itself in 2008: out of top 79 scorers 23 were from MIT, 11 from Harvard and 11 from Princeton.

Ironically, MIT’s team didn’t win Putnam in those years. MIT’s team won the third place after Harvard and Princeton. If you look at the results more closely, you will notice that had MIT arranged teams differently, MIT would have won.

It appears that MIT put their three top scorers from the previous year on their lead team. MIT shouldn’t assume that those three continue to be their strongest competitors. Instead they should probably test their students right before the Putnam competition, because if you look at MIT’s top individual performers, had they been on a team together, they would have won.

Maybe MIT should rethink its algorithm for creating teams, or maybe we should just wait. As it is obvious that MIT is more serious about math, all top math students may want to go to MIT in coming years. If this happens, the mathematics field will be absolutely dominated by MIT.

I have a tiny book written by Vladimir Arnold Problems for Kids from 5 to 15. A free online version of this book is available in Russian. The book contains 79 problems, and problem Number 6 criticizes American math education. Here is the translation:

(From an American standardized test) A hypotenuse of a right triangle is 10 inches, and the altitude having the hypotenuse as its base is 6 inches. Find the area of the triangle.
American students solved this problem successfully for 10 years, by providing the “correct” answer: 30 inches squared. However, when Russian students from Moscow tried to solve it, none of them “succeeded”. Why?

Arnold has inflated expectations for kids. The book presents the problems according to the increasing order of difficulty, and this suggests that he expects kids under 10 to solve Number 6.

Arnold claimed that every student from Moscow would notice what is wrong with this problem. I can forgive his exaggeration, because I’ve met such kids. Anyways, I doubt that Arnold ever stumbled upon an average Russian student.

My own fundamental interest is in the state of American math education, so I decided to check his claim concerning American students. I asked my students to calculate the area of the triangle in the above puzzle.

Here are the results of my experiment. Most of them said that the answer is 30. Some of them said that it is 24. In case you’re wondering where the 24 is coming from, I can explain. They decided that a right triangle with hypotenuse 10 must have two other legs equal to 8 and 6.

Some of the students got confused, not because they realized that there was a trick, but because they thought the way to calculate the area of the right triangle is to take half the product of its legs. As lengths of legs were not given, they didn’t know what to do.

There was one student. Yes, there was one student, who decided that he could calculate the legs of the triangle from the given information and kept wondering why he was getting a negative number under the square root.

You decide for yourself whether there is hope for American math education. Or, if you are a teacher, try running the same experiment yourself. I hope that one day I will hear from you that one of your students, upon reading the problem, immediately said that such a triangle can’t exist because the altitude of the right triangle with the hypotenuse as the base can never be bigger than half of the hypotenuse.

So many people liked the puzzles I posted in Subtraction Problems, Russian Style, that I decided to present a similar collection of multiplication and division puzzles. These two sets of puzzles have one thing in common: kids who go for speed over thinking make mistakes.

Humans have 10 fingers on their hands. How many fingers are there on 10 hands?

This one is from my friend Yulia Elkhimova:

Three horses were galloping at 27 miles per hour. What was the speed of one horse?

Here is a similar invention of mine:

Ten kids from Belmont High School went on a tour of Italy. During the tour they visited 20 museums. How many museums did each kid go to?

Another classic:

How many people are there in two pairs of twins, twice?

Can you add more puzzles to this collection?

My son Sergei brought back the Flip-Flop game from Canada/USA Mathcamp, and now I teach it to my students. This game trains students in the multiplication table for seven and eight. These are the most difficult digits in multiplication. This game is appropriate for small kids who just learned the multiplication table, but it is also fun for older kids and adults.

This is a turn-based game. In its primitive simplification kids stand in a circle and count in turn. But it is more interesting than that. Here’s what to say and do on your turn, and how the game determines who is next.

First I need to tell you what to say. On your turn, say the next number by default. However, there are exceptions when you have to say something else. And this something else consists of flips and/or flops.

So what are flips? Flip is related to seven. If a number is divisible by seven or has a digit seven, instead of saying this number, we have to say “flip” with multiplicities. For example, instead of 17 we say “flip” because it contains one digit seven. Instead of 14 we say “flip”, because it is divisible by seven once. Instead of 7 we say “flip-flip”, as it is both divisible by seven and has a digit seven. Instead of 49, we say “flip-flip” as 49 is divisible by the square of seven. Instead of 77 we say “flip-flip-flip” as it has two digits seven and is divisible by seven once.

Flop relates to eight the same way as flip relates to seven. Thus, instead of 16 we say “flop” as it is divisible by eight; instead of 18 we say “flop” as it contains the digit eight; and for 48 we say “flop-flop” as it is both divisible by eight and contains the digit eight.

A number can relate to seven and eight at the same time. For example 28 is divisible by seven and contains the digit eight. Instead of 28 we say “flip-flop”. The general rule is that all flips are pronounced before all flops. For example, instead of 788 we will say “flip-flop-flop-flop” as it is divisible by eight and contains the digit seven once and the digit eight twice.

The sequence of natural numbers in the flip-flop version starts as the following: 1, 2, 3, 4, 5, 6, flip-flip, flop-flop, 9, 10, 11, 12, 13, flip, 15, flop, flip, flop, 19, 20, flip, 22, 23, flop, 25, 26, flip, flip-flop, 29, 30, 31, flop, 33, 34, flip, 36, flip, flop, 39, flop, 41, flip, 43, 44, 45, 46, flip, flop-flop, flip-flip, 50, 51, 52, 53, 54, 55, flip-flop, flip, flop, 59, 60, 61, 62, flip, flop-flop, 65, 66, flip, flop, 69, flip-flip, flip, flip-flop, flip, flip, flip, flip, flip-flip-flip, flip-flop, flip, flop-flop, flop, flop, flop, flip-flop, flop, flop, flip-flop, flop-flop-flop, flop, 90, flip, 92, 93, 94, 95, flop, flip, flopflip-flip-flop, 99, 100.

So how does the turn change? Everyone stands in a circle and says their number the way explained above. We start clockwise and move to the next number. For every flip we reverse the direction and for every flop we skip a person. That means that if we have two flips, we don’t change the direction, while for two flops we skip two people. If we have flips and flops together, for example 28 corresponds to “flip-flop”, then first we change the direction and then we skip a person.

On top of that, there is an extra rule for what you do on your turn. If you say something other than a default number, you switch your position from standing to sitting and vice versa. Sometimes I skip this extra feature — not because I am too lazy to exercise, but because I usually conduct this game in a classroom, where all the desks prevent us from fully enjoying such physical activity.

There are two ways to play this game: as a competition or as practice. When we are competing, a person who makes a mistake drops out. If we’re just practicing, no one drops out. Sometimes I am particularly generous and allow my kids one mistake before making them drop out after the second mistake. So far we have played up to 100. I am curious to see if we can ever reach 700 and how long we will be able to continue the game after that.

Testing in the US is dominated by multiple-choice questions. Together with the time limit, this encourages students to stop thinking and go for guessing. I recently wrote an essay AMC, AIME, USAMO Contradiction, in which I complained about the lack of proofs in the first two rounds of math competitions.

Is there a way to improve the situation? I grew up in the USSR, where each round of the math competition had the same format: you were given several hours to write proofs for three or four difficult problems. There are two concerns with organizing a competition in this way. First, the Russian system is much more expensive, whereas the US’s multiple choice tests can be inexpensively checked by a computer. Second, the Russian system is prone to unfairness. You need many math teachers to check all these papers on the highest level. Some of these teachers might not be fully qualified, and it is difficult to ensure uniform checking. This system can’t easily be adopted in the US. I am surprised I haven’t heard of lawsuits challenging USAMO results, but if we were to start having proofs at the AMC level with several hundred thousand participants, we would get into lots of trouble.

An interesting compromise was introduced at the Streamline Olympiad. The problems were multiple choice, but students were also requested to write proofs. Students got two points for a correct multiple choice answer, and if the choice was correct the proof was checked. Students could get up to three points for a correct proof. This idea solves two issues. The writing of proofs is rewarded at an early stage and the work of the judges is not as overwhelming as it would have been, had they needed to check every proof. However, there is one problem that I discussed in previous posts that this method doesn’t solve: with multiple choice, minor mistakes cost you the whole problem, even though you might have been very close to a solution. If we want to reward thinking more than accuracy, the proof system allows us to give credit for partial solutions.

I can suggest another approach. If the Russians require proofs for all problems and the Americans don’t require proofs for any problem, why not compromise and require a proof for one problem out of the set.

But I actually have a bigger idea in mind. I think that current development in artificial intelligence may soon help us to check the proofs with the aid of a computer. Artificial intelligence is still far from ready to validate that a mathematical text a human has produced constitutes a proof. But in this particular case, we have two things working for us. First, we can use humans and computers together. Second, we do not need to check the validity of any random proof; we need to check the validity of a specific proof of a simple problem that we know in advance, thus allowing us to prepare the computers.

Let us assume that we already can convert student handwriting into computer-legible text or that students write directly in LaTeX.

Here is the plan. Suppose for every problem, we create a database of some sample right, wrong and partial solutions with corresponding scores. The computer checks the students’ solutions against the given sample. Hopefully, the computer can recognize small typos and deviations that shouldn’t change the point value. If the computer encounters a solution that is significantly different from the ones in the sample, it sends the solution to human judges. Humans decide how to score the solution and the solution and its score is added to the sample database.

For this system to work, computers should be smart enough not to send too many solutions to humans. So how many is too many? My estimate is based on the idea that we wouldn’t want the budget of AMC to go too much higher than the USAMO budget. Since USAMO has 500 participants, judges check just a few hundred solutions to any particular problem. With several hundred thousand participants in AMC, the computer would have to be able to cluster all the solutions into not more than a few hundred groups. The judges only have to check one solution in each group.

As a bonus, we can create a system where for a given solution that is not in the database, the computer finds the closest solution and highlights the difference, thus simplifying the human’s job.

In order to improve math education, we need to add proofs when teaching math. My idea might also work for SATs and for other tests.

Now that there is more money available for education research, would anyone like to explore this?