Tanya Khovanova's Math Blog

By Tanya Khovanova and Gregory Bomash

What object is missing?

Should you try to guess an answer to a multiple-choice problem during a test? How many problems should you try to guess? I will talk about the art of boosting your guessing accuracy in a later essay. Now I would like to discuss whether it makes sense to pick a random answer for a problem at AMC 10.

Let me remind you that each of the 25 problems on the AMC 10 test provides five choices. A correct answer brings you 6 points, a wrong answer 0 points and not answering at all gives you 1.5 points. So guessing makes the expected average per problem to be 1.2 points. That is, on average you lose 0.3 points when guessing. However, if you are lucky, guessing will gain you 4.5 points per problem, and if you are unlucky, it will lose you 1.5 points per problem.

So we see that on average guessing is unprofitable. But there are situations in which you have nothing to lose if you get a smaller score and a lot to gain if you get a better score. Usually the goal of a competitor at AMC 10 is to get to AIME. For that to happen, you need to get 120 points or be in the highest one percentile of all competitors. This rule complicates my calculations. So I decided to simplify it and say that your goal is to get 120 points and then see what mathematical results I can get out of that simplification.

First, suppose you are so accurate that you never make mistakes. If you have solved 20 problems, then your score without guessing is 127.5. If you start guessing and guess wrongly for all of the last five questions, you still have your desired 120 points. In this instance it doesn’t matter whether you guess or not.

Suppose on the other hand that you are still accurate, but less powerful. You have only solved 15 problems, so your score without guessing is 105. Now you must be strategic. Your only chance to get to your goal of 120 points is to guess. Suppose you randomly guess the answers for the 10 problems you didn’t solve. To make it to 120, you need to guess correctly at least five out of the ten remaining problems. The probability of doing so is 3%. Here is a table of your probability of making 120 points if you solve correctly n problems and guess the other problems.

We can see that if you solved a small number of problems, then the probability of getting 120 points is minuscule; but as the number of problems you solved increases, so does the probability of getting 120 points by guessing.

The interesting part is that if you have solved 19 problems, you are guaranteed to get to AIME without guessing. On the other hand, if you start guessing and all your guesses are wrong, you will not pass the 120 mark. The probability of having all six problems wrong is a not insignificant 26%. In conclusion, if you are an accurate solver and want to have 120 points, it is beneficial to guess the remaining problems if you solved fewer than 19 problems. It doesn’t matter much if you solved fewer than 10 problems or more than 19. But you shouldn’t guess if you solved exactly 19.

If you are not 100% accurate things get more complicated and more interesting. To decide about guessing, it is crucial to have a good estimate of how many mistakes you usually make. Let’s say that you usually have two problems wrong per AMC test. Suppose you gave answers to 20 problems at AMC. What’s next? Let us estimate your score. Out of your 20 answers you are expected to get 18*6 points for them plus 5*1.5 points for the problems you didn’t answer. Your expected score is 115.5. You are almost there. You definitely should guess. But does it matter how many questions you are trying to guess?

The correct answer to one question increases your score by 4.5 with probability 0.2, and the wrong answer decreases your score by 1.5 with probability 0.8. One increase by 4.5 is enough for you goal of 120 points. So if you guess one question, with probability 0.2, you get 120 points. If you guess two questions, then your outcome is as follows: you increase your score by 9 points with probability 0.04; you increase your score by 3 points with probability 0.32; and you decrease your score by 3 points with probability 0.64. As you need at least a 4.5 increase in points, it is not enough to guess one question out of two. You actually need to guess correctly on both of them. The probability of this happening is 0.04. It is interesting, but you have a much greater chance to get to your goal if you guess just one question than if you guess two. Overall, here is the table of probabilities to get to 120 points where m is the number of questions you are guessing.

Your best chances are to guess all the remaining questions.

By the end of the test you know how many questions you answered, but you don’t know how many errors you made. The table below tells you what you need in order to get 120 points. Here is how you read the table: The number of problems you solved is in the first column. If you are sure that the number of mistakes is not more than the number in the second column, you can relax as you made at least 120 points. The last column gives the score.

If the number of mistakes you made is one more than in the corresponding row of the table, you should start guessing in order to try to get 120 points. Keep in mind that there is a risk: if you are not sure how many problems you solved already and start guessing, you might ruin your achievement of 120 points.

In the next table I show how many questions you can guess without the risk of going below 120 points. The word “all” means that it is safe to guess all the remaining questions.

You can see that if your goal is to get 120 points, your dividing line is answering 22 questions. If you solved 22 questions or more, there is no risk in guessing. Namely, if you have already achieved more than 120 points, guessing will not take you below that. But if you made more errors than are in the table, then guessing might be beneficial. Hence, you should always guess in this case — you have nothing to lose.

Now I would like to show you my calculations for a situation in which you are close to 120 points and need to determine the optimum number of questions to guess. The first column is the number of answered questions. The second column is the number of mistakes. The third column is your expected score without guessing. The fourth column is the optimum number of questions you should guess. And the last column lists your chances to get 120 points if you guess the number of questions in the fourth column.

You can see that almost always if you are behind your goal, you should try to guess all of the remaining questions, with one exception: if you answered 19 questions and one of them is wrong. In this case you should guess exactly two questions — not all that remain.

Keep in mind that all these calculations are very interesting, but don’t necessarily apply directly to AMC 10, because I simplified assumptions about your goals. It may not be directly applicable, but I hope I have expanded your perspective about how you can use math to help you understand how better to succeed at math tests and how to design your strategy.

I plan to teach you how to guess more profitably, and this skill will also advance your perspective.

Lottery is a tax on people bad at math.

In this article I calculate how bad the lottery is as an investment, using Mega Millions as an example. To play the game, a player pays $1.00 and picks five numbers from 1 to 56 (white balls) and one additional number from 1 to 46 (the Mega Ball number, a yellow ball).

During the drawing, five white balls out of 56 are picked randomly, and, likewise, one yellow ball out of 46 is also picked independently at random. The winnings depend on how many numbers out of the ones that a player picks coincide with the numbers on the balls that have been drawn.

So what is your expected gain if you buy a ticket? We know that only half of the money goes to payouts. Can you conclude that your return is 50%?

The answer is no. The mathematical expectation of every game is different. It depends on the jackpot and the number of players. The more players, the bigger is the probability that the jackpot will be split.

Every Mega Millions playslip has odds printed on the back side. The odds of hitting the jackpot are 1 in 175,711,536. This number is easy to calculate: it is (56 choose 5) times 46.

How much is 175,711,536? Let’s try a comparison. The government estimates that in the US we have 1.3 deaths per 100 million vehicle miles. If you drive one mile to buy a ticket and one mile back, your probability to die is 2.6/100,000,000. The probability of dying in a car accident while you drive one mile to buy a lottery ticket is five times higher than the probability of winning the jackpot.

Suppose you buy 100 tickets twice a week. That is, you spend $10,000 a year. You will need to live for 1,000 years in order to make your chances of winning the jackpot be one out of 10. For all practical purposes, the chance of winning the jackpot are zero.

As the probability of winning the jackpot is zero, we do not need to include it in our estimate of the expected return. If you count all other payouts then you are likely to get back 18 cents for every dollar you invest. You are guaranteed to lose 82% of your money. If you spend $1000 a year on lottery tickets, on average you will lose $820 every year.

If you do not buy a lot of tickets your probability of a big win is close to zero. For example, the probability of winning $250,000 (that is guessing all white balls, and not guessing a yellow ball) by buying one ticket is about 1 in 4 million. The probability of winning $10,000 — the next largest win — is close to 1 in 700,000. If we say that you have no chance at these winnings anyway, then your expected return is even less: it is 10 cents per every dollar you invest.

You might ask what happens if we pool our money together. When a lot of tickets are bought then the probability of winning the jackpot stops being zero. I will write about this topic later. For now this is what I would like you to remember. From every dollar ticket:

• 50 cents goes to the state
• 32 cents towards the jackpot
• 18 cents to other winners

I am not at all trying to persuade you not to buy tickets. Lottery tickets have some entertainment value: they allow you to briefly dream about what you would do with those millions of dollars. But I am trying to persuade you not to buy lottery as an investment and not to put more hope into it than it deserves. If you treat lottery tickets as tickets to a movie that is played in your head, you will never buy more than one ticket at a time.

That is it. I advise you not to buy more than one ticket at a time. One ticket will allow you to dream about the expression on your sister’s face when she sees your new $5,000,000 mansion, but will not destroy your finances.

Should you allocate time for checking your answers during important tests? I will use AMC 10/12 as an example, but you can adjust the calculations for any other test.

AMC 10/12 is a math competition that asks 25 multiple-choice questions that you need to solve in 75 minutes. You get 6 points for a correct answer, 1.5 points for an unanswered question and 0 points for a wrong answer.

Whether or not to take time to check your answers depends on you and the situation. If you finished your test and have some time left over, then surely you should use the extra time for checking your answers. If you only have three minutes left and your next problems are too complex to be dealt with in that time, then it is logical to use these moments to check back.

Sometimes, though, it isn’t worth it to check your answers. If you haven’t finished the test, but are a super-accurate person and never make mistakes, then it is better to continue working on the next problem than to waste time checking your correct answers. Also, if you rarely catch your own mistakes anyway, it doesn’t make sense to check.

But things are not usually so clear. By the end of the test, most people need to make a decision: continue working through the problems or use the final moments to check the answers? How can you best decide if you should allocate time for checking and, if so, how much time?

The problems in AMC tests increase in difficulty. I suggest that each time you take the test or practice for AMC, take note of two things. How long did it take you to solve the test’s last problem and what is the level of your accuracy for it. Suppose you know that at the end of the AMC test you can solve a problem in about 10 minutes and it is correct about 90% of the time. That means that investing the last ten minutes in solving the next problem will give you on average 5.4 points. If you remember that a blank answer gives you 1.5 points, you should realize that solving the last problem increases your score by 3.9 points. If you are very accurate, your score can increase more, but not more than 4.5 points.

Try conducting the following experiment. Take an AMC test from a past year. Do it for 65 minutes — the time of the test minus the time you need for that last problem. Then spend the last 10 minutes checking and correcting your answers. Now let us calculate how profitable that would be. Compare the scores you would have gotten without your corrections and with your corrections. If checking increases your score by more than 3.9 points, it is more profitable to check than to solve the next problem. If you do not make errors when you’re trying to make corrections, the rule of thumb is that correcting one mistake is better than solving one problem. Indeed, your score increases by 6 points if you correct a mistake, and by not more than 4.5 points if you solve the next problem.

On all tests that punish wrong answers, correcting an error produces more points than solving a new question.

If you find that checking is profitable, but you can’t check all the problems in ten minutes, you should consider allocating more time. Keep in mind, though, that you should adjust the sample calculation above for the last two problems. Remember, the next to the last problem is generally easier than the last problem. So if it takes you ten minutes to solve the last problem in the test, it most probably will take you less than twenty minutes to solve the last two. Also, since the difficulty increases throughout the test, the accuracy of the second to last problem might be better than the accuracy of the last problem. In addition, the first ten minutes that you check may be more productive than the next ten minutes of checking. So if you wonder if you should forgo the last two problems in order to check your earlier work, you have to redo the experiment anew, measuring both how long it takes you to solve those two problems and the benefits of checking.

This discussion can potentially help you to increase your score. However, there are other strategic considerations to weigh when deciding whether or not to check your work. For example, if the number of mistakes in your tests varies and sometimes you are 100% accurate and you are one problem away from your goal to get to AIME, it is more profitable to go for the last problem and hope for the best. I will discuss the strategic considerations for AMC some other time.

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.

I could no longer resist: I added a section of physics jokes to my math jokes collection:

* * *

A hydrogen atom says to the bartender, “Hey buddy, have you seen an electron around here? I seem to have lost mine.”
“Are you sure you lost it?” the bartender asks.
And the hydrogen atom answers, “I’m positive!”

* * *

Heisenberg gets stopped on the motorway by the police.
Cop: “Do you know how fast you were going sir?”
Heisenberg: “No, but I know exactly where I am.”

* * *

A photon checks into a hotel. The bellhop asks him, “Can I help you with your luggage?”
To which the photon replies, “I don’t have any. I’m traveling light.”

My recent entry, where I asked you to choose the odd one out among these images

was extremely popular. It was republished all around the world and brought my blog as much traffic in one day as I used to get in a month. Not only did I read the many comments I received, I also followed up on other peoples’ blogs who reprinted my puzzle — at least those that were in either Russian or English. I also got private emails and had many conversations in person about it. The diversity of answers surprised me, so I would like to share them with you.

As I’ve said before, I do not think there is a correct answer to this type of question, but I was disappointed by some of the answers. For example, those who simply said, “The green one is the odd one out,” made me feel that either they hadn’t read the question or hadn’t thought about it very much. It’s a shame that these people spent more time sharing their opinion with the world than thinking about the problem in the first place.

I wouldn’t mind someone arguing that the green one is the odd one out, but in this case an explanation is in order. Many people did offer explanations. Some told me that we perceive the color difference stronger than all other parameters I used, and the green figure pops out of the picture more than anything else. In fact, I personally perceive color difference the strongest among all the parameters, but since there are people who are color blind, I would disregard my feelings for color as being subjective.

You can create a whole research project out of this puzzle. For example, you can run an experiment: Ask the question, but flash the images above very fast, so there is no time for analysis — only time to guess. This allows us to check which figure is the first one that people perceive as different. Or you can vary the width of the frame and see how the perception changes.

Color was not the only parameter among those I chose — shape, color, size and the existence of a frame — that people thought was more prominent. My readers weighed these parameters unequally, so each argued the primary importance of the parameter they most emphasized. For example, one of my friends argued that:

The second figure should be the odd one out as, first, it is the only one without a frame, and, second, it is the only one comprised of one color rather than two. So it differs by two features, as others differ only by one feature.

A figure having one color is the consequence of not having a frame, so this particular friend of mine inflated the importance of not having a frame.

However, I can interpret any feature as two features. For example, I can say that the circle is the odd one out because not only is it a different figure, but it also doesn’t have any angles. Similarly, the last one is the smallest one and the border width is in a different proportion to its diameter.

On a lighter side, there were many funny answers to the puzzle:

• The one that says I am special.
• The right one because it is right.
• The fourth one, because four is the only composite index.
• The one that says I am not special.

For the which-is-the-odd-one-out questions, the designer of the question is usually expecting a particular answer. So here’s the answer I expected:

There is only one green figure. Wait a minute, there is only one circle. Hmm, there is only one without a frame and there’s only one small figure. I see! The first one is the only figure that is not the odd one, that doesn’t have a special property, so the first must be the odd one out. This is cool!

And the majority of the answers were exactly as I expected.

Since this is a philosophical problem, some of the responses took it to a different level. One interesting answer went like this:

All right, the last four figures have special features; the first figure is special because it is normal. Hence, every figure is special and there are no odd ones here.

I like this answer as the author of it equated regular features with a meta-feature, and it is a valid choice. This answer prompted me to write another blog entry with a picture where I purposefully tried to not have an odd one out:

Though I wrote that the purpose of this second set of images is to show an example where there is no odd one out, my commentators still argued about which one was the odd one out here.

Finally, I would like to quote Will’s comment to my first set of images:

The prevailing opinion is that the first is least unique and is therefore the oddest. But it is the mean and the others are one deviation from it. Can the mean be the statistical anomaly?

And Cedric replied to Will:

Yes, I think the mean can be a statistical anomaly. The average person has roughly one testicle and one ovary. But a person with these characteristics would certainly be an anomaly.

I started my blog about two year ago. I have written about 200 entries. According to my traffic reports, these have been the top ten most popular entries:

1. The Odd One Out
2. Divisibility by 7 is a Walk on a Graph, by David Wilson
3. The Dirtiest Math Problem Ever (Rated R)
4. My IQ
5. A Miracle Equation
6. Back-to-School Funny Pictures
7. Flexible Zipcar Algorithm
8. A Very Special Ten-Digit Number
9. What Does It Take to Get Accepted by Harvard or Princeton?
10. AMC, AIME, USAMO Contradiction

My most popular category was Math Humor.

Israel Gelfand’s memorial is being held at Rutgers on December 6, 2009. I was invited as Gelfand’s student.

My relationship with Gelfand was complicated: sometimes it was very painful and sometimes it was very rewarding. I was planning to attend the memorial to help me forget the pain and to acknowledge the good parts.

I believe that my relationship with Gelfand was utterly unique. You see, I was married three times, and all three times to students of Gelfand.

Now that I know that I can’t make it to the memorial, I can’t stop wondering how many single male students of Gelfand will be there.

I’ve translated two problems from the 2009 Moscow Math Olympiad. In both of them our characters are genetically engineered octopuses. The ones with an even number of arms always tell the truth; the ones with an odd number of arms always lie. In the first problem (for sixth graders) four octopuses had a chat:

• “I have 8 arms,” the green octopus bragged to the blue one. “You have only 6!”
• “It is I who has 8 arms,” countered the blue octopus. “You have only 7!”
• “The blue one really has 8 arms,” the red octopus said, confirming the blue one’s claim. He went on to boast, “I have 9 arms!”
• “None of you have 8 arms,” interjected the striped octopus. “Only I have 8 arms!”

Who has exactly 8 arms?

Not only do octopuses lie or tell the truth according to the parity of the number of their arms, it turns out that the underwater world is so discriminatory that only octopuses with six, seven or eight arms are allowed to serve Neptune. In the next problem (for seventh graders), four octopuses who worked as guards at Neptune’s palace were conversing:

• The blue one said, “All together we have 28 arms.”
• The green one said, “All together we have 27 arms.”
• The yellow one said, “All together we have 26 arms.”
• The red one said, “All together we have 25 arms.”

How many arms does each of them have?

My students enjoyed the octopuses, so I decided to invent some octopus problems of my own. In the first problem, the guards from the night shift at Neptune’s palace were bored, and they started to argue:

• The magenta one said, “All together we have 31 arms.”
• The cyan one said, “No, we do not.”
• The brown one said, “The beige one has six arms.”
• The beige one said, “You, brown, are lying.”

Who is lying and who is telling the truth?

In the next problem the last shift of guards at the palace has nothing better to do than count their arms:

• The pink one said, “Gray and I have 15 arms together.”
• The gray one said, “Lavender and I have 14 arms together.”
• The lavender one said, “Turquoise and I have 14 arms together.”
• The turquoise one said, “Pink and I have 15 arms together.”

What number of arms does each one have?