Archive for the ‘Math in Life’ Category.

Smart Brake Lights

I was driving on Mass Pike, when the cars in front of me stopped abruptly. I hit the brakes and was lucky to escape the situation without a scratch.

Actually, it wasn’t just luck. First of all, I always keep a safe distance from the other cars. Second, if I see the brake lights of the car in front of me, I automatically remove my foot from the gas pedal and hold it over the brake pedal until I know what the situation is.

On a highway, if the car in front of me has its brake lights on, usually that means that the driver is adjusting their speed a little bit. So, most of the time I don’t have to do anything. Seeing that the car in front of me has its brake lights on is not a good predictor of what will happen next. Only after I see that the distance between me and the car in front of me is decreasing rapidly, do I know to hit my brakes. That means that brake lights alone are not enough information. Differentiating between insignificant speed adjustments and serious braking requires time and can cost lives.

I have a suggestion. Why not create smart brake lights. The car’s computer system can recognize the difference in the strength with which the brakes are hit and the lights themselves can reflect that. They can be brighter or a different color or pulsing, depending on the strength of the pressure.

The drivers behind will notice these things before they will notice the decrease in the distance. This idea could save lives.

My Take on Perelman

My American friends often ask me for insights into why Grigory Perelman refused the one million dollar Clay prize for his proof of the Poincaré conjecture. They are right to ask me: my life experience was very similar to Perelman’s.

I went to a high school for children gifted in math. I was extremely successful in competitions. I got my gold medal at IMO and went to college without entrance exams. I received my undergraduate and graduate degrees in one of the best math academic centers in Soviet Russia. Perelman traveled a similar path.

Without ever having met Perelman, I can suggest two explanations of why he might reject the money.

First explanation. To have it publicly known that you have suddenly come into money is very dangerous in Russia. Perelman’s life expectancy would have dropped immediately after accepting the million dollars. Russians that have tons of money either hide their wealth or build steel doors way before they make their first million. In addition to being a life hazard, money attracts a lot of bother. He would have been chased by all types of acquaintances asking for help or suggesting marriage proposals.

Second explanation. We grew up in a communist culture where money was scorned and math was idolized. The goal of research was research. Proving the conjecture was the prize itself. In his mind, receiving the award money might diminish the value of what he did. I understand this way of thinking, but I am personally too practical to follow such feelings and would accept the prize.

My first explanation has a flaw. Though valid, it doesn’t explain why he rejected the Fields medal. So I reached for the book abour Perelman, Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century by Masha Gessen. I like Gessen’s explanation of why he rejected the Fields medal:

His objection to the Fields Medal, though never stated as clearly, seemed to have been twofold: first: he no longer considered himself a mathematician and hence could not accept a price intended for the encouragement of midcareer researchers; and second, he wanted no part of ICM, with all the attendant publicity, speeches, ceremony, and king of Spain.

The reasons are specifically related to the medal, so the Clay prize rejection might not be connected to the medal rejection. This argument slightly rehabilitates my first explanation.

Perfect Rigor

I liked the book. It is a tremendous undertaking — writing about a person who doesn’t want to talk to anyone. After reading it, I have one more possible explanation of his refusal of the prize.

Perelman is a loner. One of the closest people to him was his math Olympiad coach. The coaches tend to understand the solutions on the spot, mostly because they already know them. If in his mind Perelman expected all mathematicians to be like his coach, then he might have expected a parade in his honor the day after he solved the conjecture. Instead, he got silence and attempts to steal the prize from him.

Can you imagine doing the century’s best math work without receiving congratulations for many years? The majority of mathematicians waited for the judgment of the experts, as did Perelman. The experts were busy and much slower than Perelman expected. The conjecture was extremely difficult, and it was a high-profile situation — after all, $1 million was attached to its solution. So the experts were very cautious in their pronouncements.

Finally, instead of congratulating Grigory, they said that the proof seemed to be correct and that they had not yet found any mistakes. If like Perelman, I was certain of my proof, I would have found this a painfully under-whelming conclusion.

Perelman expected to feel proud, but instead he probably felt unappreciated and attacked. Instead of the parade he may have hoped for, he had to wait for a long time, only to face disappointment and frustration. This reminds me of an old joke:

A genie is trapped in a lantern at the bottom of the sea. He vows, “I will give one million dollars to the person who frees me.” One thousand years pass. He changes his vow, “I will give any amount of money to the one who frees me.” Another thousand years pass. He ups the ante, “I will give any amount of money and two more wishes to the person who frees me.” Another thousand years pass. He promises, “I will kill the one who frees me.”

Third explanation. Perelman was profoundly disappointed in the math community. Unlike the genie, Perelman didn’t want to kill anyone, but he did want to express his disillusionment. Perhaps that is why he rejected a million dollars.

Did You Notice?

I recently posted a short article on plagiarism. Did you notice that not a word of it was mine?

Infinite Deductible

I have an idea for a start-up medical insurance company for Massachusetts. My insurance will have an infinite deductible. That means you pay your own bills. The cost of insurance can be very low, say $100 a year, as I do not need to do anything other than to send you a letter confirming that you have medical insurance. People who otherwise will be fined up to $900 for being uninsured will run in droves to buy my insurance.

I have an even better idea. For an extra fee, I will negotiate with doctors so that you will pay the same amount as medical insurance companies pay to them, which is often three times less than you would pay on your own.

Who am I kidding? I am not a business person, I can’t build a company. But I am looking to buy the insurance I just described.

Internet-Search-Friendly Names

When you name your child there are many considerations to take into account. For example, you should always check that your kids’ initials don’t embarrass them. For example, if the Goldsteins want to name their son Paz, because it means golden in Biblical Hebrew, the middle name shouldn’t be Isaak, or anything starting with I.

Contemporary culture adds another consideration: how easy would it be to find your child on the Internet? I personally find it extremely convenient to have a rare name, because my fans can find my webpage and blog just by googling me. Parents need to decide whether they want their children to be on the first page of the search engine or hidden very far away when someone googles them.

When I named my son Sergei, I knew that there was another mathematician named Sergei Bernstein. But I didn’t think about the Internet. As a result, I confused the world: is my son more than a hundred years old or did Sergei Natanovich Bernstein compete at Putnam?

Jewish Problems

I already gave an example of the kinds of problems that were given to Jewish people at the oral entrance exam to the math department of Moscow State University. In fact, I have a whole page with a collection of such problems, called Jewish problems or Coffins. That page was one of the first pages I created when I started my website more than ten years ago.

When my son Alexey was in high school, I asked him to help me type these problems into a file and to recover their solutions from my more than laconic notes, and solve the problems that I didn’t have notes for. He did the job, but the file was lying dormant on my computer. Recently I resurrected the file and we prepared some of the solutions for a publication.

The problems that were given during these exams were very different in flavor: some were intentionally ambiguous questions, some were just plain hard, some had impossible premises. In our joint paper “Jewish Problems” we presented problems with a special flavor. These are problems that have a short and “simple” solution, that is nonetheless very difficult to find. This way the math department of MSU was better protected from appeals and complaints.

Try the following problem from our paper:

Find all real functions of real variable F(x) such that for any x and y the following inequality holds: F(x) − F(y) ≤ (x − y)2.

I will give a talk on the subject for UMA at MIT on October 18, at 5pm.

This is Not a Consultation

In my essays The Oral Exam and A Math Exam’s Hidden Agenda, I gave some examples of math problems that were used during the entrance exams to Moscow State University. The problems were designed to prevent Jewish and other “undesirable” students from studying at the University. My readers might have supposed that an occasional bright student could, by solving all the problems, get in. Here is the story of my dear friend Mikhail (Misha) Lyubich; it shows that being extremely bright was not enough.

Misha passed the first three exams and was facing his last exam: oral physics. He answered all the questions. None of his answers were accepted: all of them were declared wrong. Misha insisted that he was right and requested that the examiners explain themselves. Every time their reply was the same:

This is not a consultation, it’s an exam.

Misha failed the exam. The solution to the last problem was a simple picture: a document that seemed to be impossible to deny, so Misha decided that he had grounds for an appeal. The person in charge denied the appeal. When Misha requested an explanation, can you guess the answer?

This is not a consultation, it’s an appeal.

Misha ended up studying at Kharkov State University. Now he is a professor at Stony Brook and the director of the Institute for Mathematical Sciences at Stony Brook.

The Rings of Power

My son Alexey taught me to always plug unused power strips into themselves, so that we can call them “The Rings of Power.” These are my Borromean Rings of Power:

Borromean Rings of Power

Star Trek TNG Science Quiz

Question 1. Holodeck. After a long and difficult assignment on an uninhabited planet, Commander Riker went to Holodeck III to unwind. While there he ate three cheeseburgers generated by the holodeck program. Is Commander Riker hungry after he ends the program?

Question 2. Relativity. We know that speed in space is relative, there is no absolute speed. What does Captain Picard mean when he orders a “full stop”?

Question 3. The Replicator. Captain Picard approached a replicator and requested: “Tea, Earl Grey. Hot.” The replicator immediately created a glass with hot Earl Grey tea. How much energy would the Enterprise have saved in seven years if they used a dish-washing machine, rather than creating glasses from atoms each time and dissolving them afterwards?

Question 4. Contractions. Commander Data hasn’t mastered contractions in English speech. In what year do you think the first program was written to convert formal English into English with contractions?

Question 5. Data. Commander Data is fully functional and absolutely superior to a vibrator. Given that there are more than a thousand people on board the Enterprise, estimate how many times a year on average Data will receive sexual requests.

The next two questions are related to particular episodes.

Question 6. “Up The Long Ladder”. Mariposans reproduce by cloning. Why do all the identical sets of clones appear to be the same age? Does it mean that upon the reproduction the clone is the age of the host? If so, they all should be 300 years old.

Mariposans steal sample DNA from Commander Riker and Dr. Pulaski. If Riker and Pulaski didn’t destroy their maturing clones what age would those clones be? Would they know how much two plus two is when they awaken? If clones awaken as adults, what is their life span?

Question 7. “Force of Nature”. Serova sacrifices herself to save her world from the effects of warp drive, but in doing so, she herself creates the rift that will destroy her world. Explain the logic.

Too Good at Spider Solitaire

Have you ever been punished for being too good at spider solitaire? I mean, have you ever been stuck because you collected too many suits? Many versions of the game don’t allow you to deal from the deck if you have empty columns, nor do they allow you to get back a completed suit. If the number of cards left on the table in the middle of the game is less than ten — the number of columns — you are stuck. I always wondered what the probability is of being stuck. This probability is difficult to calculate because it depends on your strategy. So I invented a boring version of spider solitaire for the sake of creating a math problem. Here it goes:

You start with two full decks of 104 cards. Initially you take 54 cards. At each turn you take all full suits out of your hand. If you have less than ten cards left in your hand, you are stuck. If not, take ten more cards from the leftover deck and continue. What is the probability that you can be stuck during this game?

Let us simplify the game even more by playing the easy level of the boring spider solitaire in which you have only spades. So you have a total of eight full suits of spades. I leave it to my readers to calculate the total probability of being stuck. Here I would like to estimate the easiest case: the probability of being stuck before the last deal.

There are ten cards left in the deck. For you to be stuck, they all should have a different value. The total number of ways to choose ten cards is 104 choose 10. To calculate the number of ways in which these ten cards have different values we need to choose these ten values in 13 choose 10 ways, then multiply by the number of ways each card of a given value can be taken from the deck: 810. The probability is about 0.0117655.

I will leave it to my readers to calculate the probability of being stuck before the last deal at the medium level: when you play two suits, hearts and spades.

No, I will not tell you how many times I played spider solitaire.