Tanya Khovanova's Math Blog

My mom used to tell me “You are born to teach.” My mom was a very shy person with no desire to be a teacher. Somehow she ended up teaching chemistry all her life. She also told me that teaching was our family business. All my ancestors were teachers or priests. Given that I lived in the Soviet Union, priesthood was not an option. (Given that I’m a woman, priesthood was not an option anywhere else.)

After my postdocs, I ended up quitting academia and working for industry. As a result, I didn’t teach much. Until 2008 my only teaching experience was a course in Discrete Mathematics that I taught at Bar-Ilan University as part of my postdoc. Somehow Bar-Ilan University wanted five people to teach this same course in parallel and wanted me to teach it in Russian to attract fresh Russian immigrants. By the end of my course, I had way more people coming to my class than at the beginning, many of whom had transferred from the other parallel courses.

I had another clue that I was doing a good job. During one of my classes, the door was left open. A guy was walking down the hall, but when he heard me, he stopped. He stood there listening intently until the end of my class.

At the end of 2007 I resigned my industry job to go back to mathematics. I needed some financial support to do so, so my friends tried to arrange a math coach position for me at AMSA (Advanced Math and Sciences Academy Charter School). I had my interview in April, 2008, for a position for the following academic year. My interviewers were skeptical: they had had a series of PhDs who had proven incapable of teaching. I told them that deep down in my heart I knew that I would be a great teacher. And I made them an offer: I will start working in April and work until the end of the school year in June. Hiring me for two months was not a great risk, but gave them a basis on which to decide whether to give me a contract for the following year. I’ve been working at AMSA ever since.

My class is optional and there are no grades, but most of my students stick with me for all five years. (I teach grades 8 to 12.) My students show that they like me in many ways. I’m especially thrilled when they tell me that I am the best teacher they ever had.

Informally they call my class Tanya Time.

One evening Detective Radstein visited Professor Bock. He was hanging out in the kitchen and overheard a conversation between Professor Bock and his wife. It appeared that Mrs. Bock had discovered that all the cutlets she had prepared for the next day were missing. She asked her husband:

—Did you eat the cutlets?
—I ate soup, the professor replied.
—Oh well, the children were probably very hungry.

Detective Radstein smiled to himself. Many mathematicians have trouble making false statements. Some of them adjust to social situations by learning how to lie while formally telling the truth. Radstein calls them dodgers. Fortunately, dodgers only dodge a question when they are threatened. It was obvious that Professor Bock was such a dodger. He implied that he didn’t eat the cutlets, because he had soup for lunch. The detective was ready to bet $10,000 that, in truth, the soup was just an appetizer to Professor Bock’s meal of cutlets.

Detective Radstein talked to his friend Professor Bock about this obsession mathematicians have to make true statements. Professor Bock agreed that many mathematicians are like that. In fact, all the faculty members in his department are either dodgers or pathological truth-tellers. Ironically, all other staff members at Professor Bock’s math department are liars.

A pathological truth-teller is another term that Detective Radstein uses to describe people who tell the truth no matter what. They never dodge. They answer a question exactly, often disregarding the context and purpose of the question. For example, when someone enters an elevator and asks a pathological truth-teller, “Is this elevator going up or down?” the answer s/he gets is “Yes.”

One day Professor Bock asked Detective Radstein for help, following a series of laptop thefts at his department. It was clear that the thefts were committed by someone working at the department and that the criminal acted alone.

Detective Radstein decided that the easiest starting point would be to ask everyone the same question: Did you steal the laptops? If a pathological truth-teller is the perpetrator, s/he would admit to the crime. A dodger would evade the question, but only if they are guilty. A liar is flexible: s/he might either answer the question with a lie or dodge with a lie. These are the first three answers the detective got from members of the department:

—Alice: No, I didn’t steal the laptops.
—Bob: Alice stole the laptops.
—Clara: Alice didn’t steal the laptops.

Who stole the laptops?

Yesterday I went online to check out the problems at the 2015 All-Russian Math Olympiad. I was stunned to discover a problem about tanks and fighter jets. I have never seen such a militarized problem before. The problem setup tells me something about the mood of people in Russia. It tells me that the mood has changed—alarmingly for the worse.

On the other hand, mathematically it was my favorite problem. So here it is.

The field is a 41 by 41 square made of square cells. A tank is camouflaged and hidden in one of the cells. A pilot flying a fighter jet above knows that the tank is there and wants to destroy it. If there is a tank in a cell, it will be hit by the shot directed at this cell. The pilot also knows that two hits are needed to destroy the tank. The tank will move to a neighboring cell immediately upon being hit, without breaking its camouflage. Other than that, the tank won’t move. What is the smallest number of shots required to guarantee that the tank is destroyed?

My post with Kvantik’s 2012 problems for middle school was a success. So I scanned the 2013 issues and found 7 more problems that I liked. Here are two cute problems I’ve seen before:

Problem 1. In the equation 30 − 33 = 3 move one digit to make it correct.

Problem 2. A patient got two pills for his headache and two pills for his cough. He was supposed to use one of each type of pill today and do the same tomorrow. The pills all looked the same. By mistake, the patient mixed up the pills. How should he use them so that he follows the prescription exactly?

Now logic and information-theoretical problems:

Problem 3. One strange boy always tells the truth on Wednesdays and Fridays and always lies on Tuesday. On other days of the week he can lie or tell the truth. He was asked his name seven days in a row. The first six answers in order are Eugene, Boris, Vasiliy, Vasiliy, Pete, Boris. What was his answer on the seventh day?

Problem 4. Ten people are suspected of a crime. It is known that two of them are guilty and eight are innocent. Suspects are shown to a psychic in groups of three. If there is a guilty person in the group the psychic points him out. If there are two guilty people in the group, the psychic points to one of them. If all of them are innocent, the psychic points at one of the three.

1. How would you find at least one guilty person in four séances?
2. How would you find both criminals in six séances?

Problem 5. There are four balloons: red, blue, green and black. Some of the balloons might be magical. There is also a detector box that can say how many balloons out of the ones put inside are magical. How can you find all the magical balloons using the detector box not more than three times?

I conclude with two miscellaneous problems.

Problem 6. Three runners started their loops at the same time at the same place on the same track. After some time they ended at their starting point together. The fastest runner passed the slowest runner 23 times. Assuming each runner has a constant speed, how many times was one runner passed by another runner in total?

Problem 7. Given a point inside a circular disk, cut the disk into two parts so that you can put them back together into a new disk such that the given point is the center.

I recently posted a set of problems from a Russian magazine for middle school children. Now it’s time for solutions.

Problem 1. There are 6 glasses on the table in a row. The first three are empty, and the last three are filled with water. How can you make it so that the empty and full glasses alternate, if you are allowed to touch only one of the glasses? (You can’t push one glass with another.)
Solution. Pour the water from the fifth glass to the second glass.

Problem 2. If it is raining at midnight, with what probability will there be sunshine in 144 hours?
Solution. In 144 hours it be midnight. Assuming we are not close to the earth’s poles, there will be no sunshine.

Problem 3. How can you fill a cylindrical pan exactly half-full of water?
Solution. Fill it with water, then tilt so that the water slowly runs out until you are about to see the rim of the floor of the pan.

Problem 4. The Jackal always lies; the Lion always tells the truth. The Parrot repeats the previous answer—unless he is the first to answer, in which case he babbles randomly. The Giraffe replies truthfully, but to the previous question directed to him: his first answer he chooses randomly.
The Wise Hedgehog in the fog stumbled upon the Jackal, the Lion, the Parrot, and the Giraffe, although the fog prevented him from seeing them clearly. He decided to figure out the order in which they were standing. After he asked everyone in order, “Are you the Jackal?” he was only able to figure out where the Giraffe was. After that he asked everyone, “Are you the Giraffe?” in the same order, and figured out where the Jackal was. But he still didn’t have the full picture. He started the next round of questions, asking everyone, “Are you the Parrot?” After the first one answered “Yes”, the Hedgehog understood the order. What is the order?
Solution. On the question “Are you the Jackal?” the Jackal and the Lion have to answer no. The Giraffe has to say “yes.” The Parrot could not have said “yes,” because in this case the Hedgehog, knowing where Giraffe is, could have figured out where the Parrot is. That means, in the first round there was only one “yes.” We also can conclude that the Parrot is not after the Giraffe. On the question “Are you the Giraffe?” the Lion would have said “no,” and the Jackal would have said “yes.” The argument here is similar to the previous discussion: the Parrot couldn’t have said “yes.” We also know that the Parrot is not after the Jackal. So the Parrot is either the first or is after the Lion. After the Hedgehog’s last question, we know that the Lion can’t be the first. That means the Parrot is the first. Which means the previous animal said “yes,” implying that the Jackal is the last. So there are two possibilities: The Parrot, the Lion, the Giraffe, the Jackal, or the Parrot, the Giraffe, the Lion, the Jackal. But in the second arrangement, the Hedgehog would have differentiated the Lion from the Parrot before the last question, as the Parrot couldn’t have been after the Giraffe. The answer is: the Parrot, the Lion, the Giraffe, the Jackal.

Problem 5. There are 12 cards with the statements “There is exactly one false statement to the left of me,” “There are exactly two false statements to the left of me,” …, “There are 12 false statements to the left of me.” Pete put the cards in a row from left to right in some order. What is the largest number of statements that might be true?
Solution. Suppose there are more than six statements that are true. That means one of the cards with a number more than six is true, meaning that there are at least 7 false statements. This is a contradiction. Suppose there are 6 true cards. They have to be cards with numbers one through six. In addition, no pair of adjacent cards can both be true. Arranging the cards in the order 7, 1, 8, 2, 9, 3, 10, 4, 11, 5, 12, 6 works. Notice that we can permute the cards with numbers more than 6 and it will still work.

Problem 6. Olga Smirnov has exactly one brother, Mikhail, and one sister, Sveta. How many children are there in the Smirnov family?
Solution. I shouldn’t have posted this problem. In Russian it is clear that Olga is a girl and the answer is 3 children. In English this problem might be confusing.

Problem 7. Every next digit of number N is strictly greater than the previous one. What is the sum of the digits of 9N?
Solution. Remember that 9N = 10N − N. That means the i-th digit of 9N is the (i+1)-th digit of N minus the the i-th digit of 9N. The next to last digit of 9N is the last digit of N minus the next to last digit of N minus 1 (because of the carry). The last digit of 9N is 10 minus the last digit of N. Summing this up, the answer is 9.

Problem 8. Nine gnomes stood in the cells of a three-by-three square. The gnomes who were in neighboring cells greeted each other. Then they re-arranged themselves in the square, and greeted each other again. They did this one more time. Prove that there is at least one pair of gnomes who didn’t get a chance to greet each other.
Solution. Color the square in checkerboard colors. There are 8 different ways to assign colors to gnomes for the three rounds. That means there are two gnomes who stood on the same colors all three times. They couldn’t have greeted each other.

John Conway told me a story about the book Winning Ways for Your Mathematical Plays that he wrote jointly with Elwyn Berlekamp and Richard Guy. The book includes the birthdays of each of the book’s three authors. The book has short author bios, each of which mentions that author’s birthday. But in addition, each birthday is hidden inside the book.

Elwyn Berlekamp was born on September 6, 1940, and this date is pictured on a tombstone on page 318 of Volume 2. That chapter is about suiciding moves.

John Conway was born on December 26, 1937. The Doomsday algorithm to calculate the day of the week for a given date is discussed on page 903 of Volume 4. Boxing Day in 1937 is chosen as an example. Boxing Day is a British holiday that originated when wealthy people gave gift boxes to servants the day after Christmas. John Conway was born on Boxing Day. Hidden behind this calculation that it was a Sunday, is the fact that it was Conway’s day of birth.

Richard Guy was born on September 30, 1916. He often uses his initials RKG for Richard Kenneth Guy. Maybe because of that he got the nickname Archangel. If you look into the index pages of Volumes 3 and 4, you will find an entry for Archangel that refers to pages 9, 30, 1916. Not only are there no mentions of “Archangel ” on pages 9 and 30, the book only has 1004 pages.

The trio of Berlecamp, Conway, and Guy will be the topic of an upcoming MOVES (Mathematics Of Various Entertaining Subjects) conference at the Museum of Mathematics.

The last time I visited John Horton Conway, he showed me a book, Genius At Play: The Curious Mind of John Horton Conway by Siobhan Roberts. It had nothing to do with him wanting to brag about the book. Nothing, nothing at all.

He just wanted to show me a picture from the book. While he was browsing this book about himself, I took a photo of him (featured on the left). The picture he was looking for was on page 314. I am reproducing it here (Photo by Michael A. Stecker, courtesy of Stephen D. Miller).

When we finally found the picture, he asked me how many tennis balls were in it. I smelled a trick question right away and didn’t bite. I shrugged. It appears that one of the balls at the far corner at the base of the pyramid had rolled away. So there was one less ball than I would have calculated. So, how many balls are there?

Without using a calculator, can you tell how much 75² is? If you are reading my blog, with high probability you know that it is 5625. The last two digits of the square of a number ending in 5 are always 25. So we only need to figure out the first two digits. The first two digits equals 7 times 8, or 56, which is the first digit of the original number (7) multiplied by the next number (8).

I was good at mental arithmetic and saved myself a lot of money back in the Soviet Union. Every time I shopped I calculated all the charges as I stood at the cash register. I knew exactly how much to pay, which saved me from cheating cashiers. To simplify my practice, the shelves were empty, so I was never buying too many items.

When I moved to the US, the situation changed. Salespeople are not trying to cheat, not to mention that they use automatic cash registers. In addition, the US sales taxes are confusing. I remember how I bought three identical chocolate bars and the total was not divisible by 3. So I completely dropped my at-the-cash-registers mental training.

Being able to calculate fast was somewhat useful many years ago. But now there is a calculator on every phone.

John H. Conway is a master of mental calculations. He even invented an algorithm to calculate the day of the week for any day. He taught it to me, and I too can easily calculate that July 29 of 1926 was Thursday. This is not useful any more. If I google “what day of the week is July 29, 1926,” the first line in big letters says Thursday.

One day a long time ago I was watching a TV show of a guy performing mental tricks: remembering large numbers, multiplying large numbers, and so on. At the end, to a grand fanfare, he showed his crowned trick. A member from the audience was to write down a 2-digit number, and to raise it to the fifth power using a calculator. The mental guy, once he is told what the fifth power was, struggled with great concentration and announced the original number.

I was so underwhelmed. Anyone knows that the last digit of a number and its fifth power are the same. So he only needs to guess the first digit, which can easily be estimated by the size of the fifth power.

I found the description of this trick online. They recommend memorizing the fifth powers of the numbers from 1 to 9. After that, you do the following. First, listen to the number. Next, the last digit of the number you hear is the same as the last digit of the original number. To figure out the first digit of the original number, remove the last 5 digits of the number you hear. Finally, determine between which fifth powers it fits. This makes sense. Or, you could be lazy and remember only meaningful digits. For example, 39⁵=90,224,199, and 40⁵=102,400,000. So if the number is below 100,000,000, then the first digit is less than 4.

You do not have to remember the complete fifth powers of every digit. You just need to remember significant ranges. Here they are:

Besides, for this trick the guy needed to guess one out of one hundred number. He had a phenomenal memory, so he could easily have remembered one hundred fifth powers. Actually he doesn’t need to remember all the numbers, only the differentiated features. On this note, there is a cool thing you can do. You can guess the number before the whole fifth power is announced. One caveat: a 1-digit number x and 10x when taken to the fifth power, both begin with the same long string of digits. So we can advise the audience not to suggest a 1-digit number or a number divisible by 10, as that is too easy (without telling the real reason).

Now we can match a start of the fifth power to its fifth root. Three first digits are almost always enough. Here is the list of starting digits and their match: (104,16), (107,64), (115,41), (116,65), (118,26), (12,66), (130,42), (135,67), (141,17), (143,27), (145,68), (147,43), (156,69), (161,11), (164,44), (17,28), (180,71), (184,45), (188,18), (19,72), (2051,29), (2059,46), (207,73), (221,74), (229,47), (23,75), (247,19), (248,12), (253,76), (254,48), (270,77), (282,49), (286,31), (288,78), (30,79), (33,32), (345,51), (348,81), (370,82), (371,13), (38,52), (391,33), (393,83), (40,21), (4181,53), (4182,84), (44,85), (454,34), (459,54), (470,86), (498,87), (50,55), (51,22), (525,35), (527,88), (53,14), (550,56), (558,89), (601,57), (604,36), (62,91), (64,23), (656,58), (659,92), (693,37), (695,93), (71,59), (73,94), (75,15), (77,95), (792,38), (796,24), (81,96), (84,61), (85,97), (902,39), (903,98), (91,62), (95,99), (97,25), (99,63).

Or, you can come to the mental guy’s performance and beat him by shouting out the correct answer before he can even finish receiving the last digit. This would be cool and cruel at the same time.

* * *

Do you know a statistics joke?
Probably, but it’s mean!

* * *

Twelve different world statisticians studied Russian roulette. Ten of them proved that it is perfectly safe. The other two scientists were unfortunately not able to join the final discussion.

* * *

A statistician bought a new tool that finds correlations between different fields in databases. Hoping for new discoveries he ran his new tool on his large database and found highly correlated events. These are his discoveries:

• The most correlated fields were the title and the gender. If the title is Mr., then the gender is male.
• The children have the same last names as parents.
• The children are much younger than the parents.
• The main cause of divorces is weddings.

* * *

Scientists discovered that the main cause of living ’till old age is an error on the birth certificate.

* * *

Scientists concluded that children do not really use the Internet. This is proven by the fact that the percentage of people saying ‘No’ when asked ‘Are you over 18?’ is close to zero.

* * *

— Please, close the window, it is cold outside.
— Do you think it will get warmer, after I close it?

Some of my friends collect volumes of The Best Writing on Mathematics published by Princeton University Press. The first annual volume appeared in 2010. In 2014, one of my papers, Conway’s Wizards, was chosen to be included in the volume The Best Writing on Mathematics 2014.

All my life I have hated writing. My worst grades in high school were for writing. I couldn’t write in Russian, and I was sure I would never be able to write in English. I was mistaken. I am a better writer than I gave myself credit for being.

Writing in English is easier for me than writing in Russian because when I make a mistake, I have an excuse. As my written English gets better and better, maybe one day I’ll get the courage to try writing in Russian.

I would like to use this opportunity to thank Sue Katz, my friend, English teacher, and editor. Sue edits most of my blog essays. Originally I wrote about Conway’s wizards on my blog. When I had three essays on the subject I combined them into a paper. Sue edited the essays and the paper. I am honored to be included in this respected volume and I want to share this honor with Sue.