Tanya Khovanova's Math Blog

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.

Of course you can. Can you do it in Russian? You do not need to know Russian to do it; you just need to solve my puzzle. Below are some numerals written in Russian. You have enough information to write any number from 1 to 99 inclusive in Russian.

• 1 — один
• 10 — десять
• 11 — одиннадцать
• 12 — двенадцать
• 13 — тринадцать
• 14 — четырнадцать
• 15 — пятнадцать
• 18 — восемнадцать
• 22 — двадцать два
• 31 — тридцать один
• 33 — тридцать три
• 40 — сорок
• 44 — сорок четыре
• 46 — сорок шесть
• 55 — пятьдесят пять
• 88 — восемьдесят восемь
• 97 — девяносто семь
• 99 — девяносто девять

If you are too lazy to write all the Russian numerals I requested, try the most difficult ones: 16, 17, 19, 67 and 76.

If you know Russian, then I have a back-up puzzle for you. Do the same thing for French:

• 1 — un
• 10 — dix
• 11 — onze
• 12 — douze
• 13 — treize
• 14 — quatorze
• 16 — seize
• 17 — dix-sept
• 21 — vingt-et-un
• 22 — vingt-deux
• 31 — trente-et-un
• 33 — trente-trois
• 40 — quarante
• 44 — quarante-quatre
• 46 — quarante-six
• 48 — quarante-huit
• 55 — cinquante-cinq
• 61 — soixante-et-un
• 71 — soixante et onze
• 72 — soixante-douze
• 75 — soixante-quinze
• 79 — soixante-dix-neuf
• 80 — quatre-vingts
• 81 — quatre-vingt-un
• 91 — quatre-vingt-onze
• 98 — quatre-vingt-dix-huit

And again, if you are lazy, you can concentrate on translating 15, 18, 19, 41, 51, 56, 65, 78 and 99 into French.

I invite my readers to create similar puzzles in all languages.

I recently read an article titled “Think having children will make you happy?” that discusses studies correlating happiness and having children. Some studies show that parents and non-parents have the same level of happiness. But other studies show that non-parents are happier. So, do children make us less happy?

There are two major reasons that kids might make people less happy in a long run. First, children require a lot of resources; they put a strain on our budget, time and careers. As my friend Sue Katz puts it: parental unhappiness could stem from poverty, illness, fighting the educational institutions, feeling stuck in a violent relationship because of the kids — a million things, depending on class and options.

Second, children might not live up to our expectations. Parents often dream that their children will have wonderful careers, be supportive of their parents later in life and most importantly be good people. But in reality, children choose their own careers, not necessarily a path approved by their parents. Plus they might live at a distance or the relationship might be strained. They might even develop completely different values from their parents.

The article claims that on average kids will bring more problems than joy to our lives. Do not rush to cancel unprotected sex with your spouse tonight yet.

My friend Peggy Boning suggested that the study should have separately checked parents who wanted children and parents who didn’t. It could be that parents who didn’t want children are less happy than parents who wanted them. Which means that if you do not want children, make sure you have protected sex. If you do want children, you might be happier with children than without.

Anyone who has studied statistics knows that correlation doesn’t mean causality. An individual who wants to have children might be happier as a result, and at the same time the statistics data may well be true. I’d like to find arguments that can make peace between these two suppositions.

• Younger people are more often childless than older people. If studies do not differentiate by age and younger people are generally happier than older people, than we might see parents less happy, because they are older on average.
• I am sure that suicidal people are more likely to actually kill themselves if no one depends on them. Thus, the most unhappy segment of childless people will have died out, while unhappy people with children will drag on.
• Some very happy people might be self-centered and do not want children.

Feel free to add your own ideas.

In Psilvania no one knows English, except for one retired professor Mary Bobs. That is why every year the organizers of the linguistics Olympiad in Psilvania beg Mary to design a puzzle in English. Kids in Psilvania know other languages — which gives individuals an advantage if the puzzle is in those languages. An English puzzle would create a level playing field.

Here is the puzzle that Mary proposed. I’m omitting the Psilvanian text, because the characters do not match anything in Unicode tables.

Professor Bobs provided the following sentences in English, accompanied by their translations into Psilvanian. She called these sentences Raw Materials:

• Kate is devouring a pencil.
• A laptop is being devoured by Paul.
• A fig is eating Kate.
• Kate is dating a fig.
• Jane is defenestrating Paul.
• Pete is being defenestrated by Paul.

The first task that she required was to translate the following sentences into Psilvanian:

• Paul is being dated by a laptop.
• Jane is being devoured by Paul.

Professor Mary Bobs had quit smoking that very week and she couldn’t concentrate. It seems that she may have given more information than is necessary. Is it possible to remove any of the Raw Materials (one or more translated sentences) and keep the puzzle solvable? If so, what is the largest number of Raw Materials you can eliminate? Explain.

Her second task was to translate some sentences from Psilvanian into English, and the answers she hoped the students would calculate were:

• A fig is being eaten by Paul.
• A pencil is being devoured by a laptop.
• A laptop is being defenestrated by Pete.

For each of the three English sentences above, decide whether the participants of the Olympiad will be capable of getting this particular answer. If for any of these three sentences you suspect that they will not be able to arrive at the correct answer, explain why.

Just out of curiosity I googled two phrases: “male mathematician” and “female mathematician”. The results for these phrases on May 3, 2009 were:

• male mathematician — 824
• female mathematician — 5,680

Why do you think that female mathematicians are more popular than male mathematicians? I think it is because when people hear the word mathematician, by default they picture a man, so the phrase “male mathematician” is perceived as pleonastic.

I decided to look at some of the 824 sites talking about “male mathematicians”. Many web-pages containing the words “male mathematician” are actually pages about female mathematicians, where there is a need to mention a mathematician of the opposite sex. Many other sites are dating pages where a mathematician looks for a partner, and it is wise to start with a description of the sex of the seeker.

Speaking of dating, did I ever mention that I am a female mathematician who was married three times, and all of my spouses were male mathematicians?

One of the best inventions of recent years is toilet seat covers. So whenever I visit a bathroom without seat covers, I curse and make my own out of toilet paper. This is not very effective, as toilet paper doesn’t stay nicely in place and it takes a lot of time, when time might be of the essence. Besides it wastes a lot of tissue.

When I come into a toilet and see a lot of long pieces of clean toilet paper lying around, I know that someone else tried to create a seat cover before me.

So here is my idea. Why not make individual packages with folded toilet seat covers, like kleenex packets? When I couldn’t find toilet seat covers in my local pharmacy, I started wondering how to patent my idea and dreaming about a lot of money. But before I let myself get too excited, I checked the Internet. My new toilet-seat-covers-to-go were already available. So I bought some. Now, every time I flush them, I watch my potential patent going down the toilet.

I would like to discuss the solution to one of the linguistics puzzles I posted a while ago. Here is problem number 211 from the online book Problems from Linguistics Olympiads 1965-1975:

You are given words in Swahili: mtu, mbuzi, jito, mgeni, jitu and kibuzi. Their translations in a different order are: giant, little goat, guest, goat, person and large river. Make the correspondence.

First, lets say that a giant is a large man. The Swahili translation of “giant” may have elements of Swahili words for a “man” and a “large river”. Next we notice that each of these Swahili words naturally divides into two parts. We can put them in a table such that the first part is the same for every row and the second part is the same for every column.

When I gave this problem to my students, they loved the idea that the word “giant” is comprised of the two words “large” and “man”, so they assumed that in Swahili a “guest” would also have a two-part translation, such as a “man who visits.” In the list of words we have three different types of “man”: man, giant and guest. Once they noticed that “m” appears three times, they concluded that “m” must mean a man. Therefore, the object must be the first part of a Swahili word, while the second part contains its description.

Next, they noted that the first part “ji” appears twice. They decided that “ji” must be a goat and thus “ki” must be a river. All of this gives us sufficient information to derive the translations: “mgeni” a guest, “kibuzi” a large river, “mbuzi” a giant, “mtu” a man, “jitu” a goat and “jito” a little goat.

My students were very proud of themselves, but I was dissatisfied with this solution. Here are the problems I’ve identified:

• If “buzi” means large, then what does “tu” mean?
• If “tu” means normal size, then what is the size of the guest?
• If everything is about sizes, then the descriptive part “geni” is an odd one out.
• In a real language what part should be smaller: the one describing the size of the object or the one describing the object itself?

I would suggest a different approach. Let’s say that the puzzle is about sizes, and we have three objects (man, guest, goat) of normal size, two large objects (giant and large river), and one small object (little goat). That means “m” must mean normal, and the size description is in front. If the first part is the size, then “ki” is small, “ji” large. From here “mgeni” is a guest, “kibuzi” a little goat, “mbuzi” a goat, “mtu” a man, “jitu” a giant, and “jito” is a large river.

I love this puzzle because it teaches us to continue pondering, even after everything seems to fit. If you stumble upon the first solution you need to go back and think some more. Only after you discover the second solution does it become clear that the second one is right.

My son, Alexey Radul, married Rebecca Frankel recently. They had a nerdy wedding. You can judge just how nerdy by this poem written and presented by Gremio (Gregory Marton), the best man:

In this summation, may there be no subtraction;
May you multiply blissfully, and find no division;
May the roots of the power of your love run deep;
May the logs of your joys be exponentially steep;
May you derive greatest pleasure from integrating your lives,
And well past your primes, retain composite rhymes.
This group gathered here, on that lush green field:
May we help you build proofs on the vows you just sealed.

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.

A long ago my son Sergei went to the Streamline School Olympiad. Some of the problems were really nice and I asked the organizer, Oleg Kryzhanovsky, where he took the problems from. It seems that he himself supplied all the problems, many of which are his original creations. He told me that he can invent a math Olympiad problem on demand for any level of difficulty on any math topic. No wonder that he is the author of almost all math problems at the Ukraine Olympiad.

The following is a sample of his problems from the Streamline Olympiad. For my own convenience I have chosen problems without figures and equations. Note: I edited some of them.

1998 (8th – 9th grade). Find three numbers such that each of them is a square of the difference of the two others.

1999 (9th – 10th grade). The positive integers 30, 72, and N have a property that the product of any two of them is divisible by the third. What is the smallest possible value of N?

1999 (9th – 10th grade). You have 6 coins weighing 1, 2, 3, 4, 5 and 6 grams that look the same. The number (1, 2, 3, 4, 5, 6) on the top of each coin should correspond to its weight. How can you determine whether all the numbers are correct, using the balance scale only twice?

1999 (11th – 12th grade). In how many ways can the numbers 1, 2, 3, 4, 5, 6 be ordered such that no two consecutive terms have a sum that is divisible by 2 or 3.

2000 (6th – 7th grade). Let A be the least integer such that the sum of all its digits is equal to 2000. Find the left-most digit of A.

2000 (8th grade). You have six bags with coins that look the same. Each bag has an infinite number of coins and all coins in the same bag weigh the same amount. Coins in different bags weigh 1, 2, 3, 4, 5 and 6 grams exactly. There is a label (1, 2, 3, 4, 5, 6) attached to each bag that is supposed to correspond to the weight of the coins in that bag. You have only a balance scale. What is the least number of times do you need to weigh coins in order to confirm that the labels are correct?