Tanya Khovanova's Math Blog

My guest blogger is a friend and a wordsmith Sue Kelman:

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Rizzo (Ratso to his friends) was my cage mate. We had a nifty pad at Children’s Hospital — all we could eat with no scrounging, clean beds, quiet surroundings, and plenty of activity to keep us occupied.

Rizzo’s favorite was the maze. Each week he bragged to us about how fast he made it through to the cheese. Larry over in Row-D was always the slowest. All the guys used to razz him about it. No matter how hard he tried, Larry took the wrong turn every time. I think his mother spent some time out at a psych hospital, so maybe they messed with her brain and that affected Larry. Who knows? I suspect that know-it-all visiting researcher from MIT knows what happened to Larry but he’s probably keeping it under his hat until he publishes his results in JAMA. Putz!

Okay, so one day, Rizzo just came back from one of those tests where they make us hit a little button when the red and green lights go on. Personally this is my favorite gig because of course there’s no running around, but Rizzo likes to throw a monkey wrench into the research data. So every now and then, even when he knows how we should respond, he does just the opposite. I told you, he’s one smart rodent.

Rizzo’s pretty famous, too. Oh he’s not as famous as that talking grey parrot that used to be over at Harvard, but he’s been around. For a while he was a top gun — the big performer for a group of genetics guys. He’s had his DNA tested more times than Mike Tyson.

Then they lent him out to Hematology where, I swear, the guy’s already had 15 blood transfusions. No wonder he’s healthy as a horse.

Me, I’m just your average lab rat. I know the drill: wake up, eat a few pellets, perform, eat some more pellets, doze off, and wake up to do it all over again. Not a bad life if you can stay away from those vivisection weirdos. They’re like Dr. Mengele all over again.

I’d tell you more but Rizzo’s gonna tell us about the time he got out of his cage and made it almost all the way to the Starbucks wagon before they caught him. Great story and he’s a real raconteur. None of that Stuart Little crap. We fall over laughing every time we hear it. Gotta go.

Due to the popularity of my previous posting of linguistics puzzles, I’ve translated some more puzzles from the online book Problems from Linguistics Olympiads 1965-1975. I’ve kept the same problem number as in the book; and I’ve used the Unicode encoding for special characters.

Problem 180. Three Tajik sentences in Russian transliteration with their translations are below:

• дӯсти хуби ҳамсояи шумо — a good friend of your neighbor
• ҳамсояи дӯсти хуби шумо — a neighbor of your good friend
• ҳамсояи хуби дӯсти шумо — a good neighbor of your friend

Your task is to assign a meaning to each out of four used Tajik words.

Problem 185. For every sequence of words given below, explain whether it can be used in a grammatically correct English sentence. If it is possible show an example. In the usage there shouldn’t be any extra signs between the given words.

1. could to
2. he have
3. that that
4. the John
5. he should
6. on walked
7. the did

Problem 241. In a group of relatives each person is denoted by a lower-case letter and relations by upper-case letters. The relations can be summarized in a table below:

The table should be read as following: if the intersection of the row x and the column y has symbol Z, then x is Z with respect to y. It is known that e is a man.

You task is to find out the meaning of every capital letter in the table (each letter can be represented as one English word).

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.