Tanya Khovanova's Math Blog

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.

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.

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?

I’ve been translating a lot of linguistics puzzles lately. Now it is my turn to create a new linguistics puzzle. Here are some English phrases with their Russian translations:

• John killed Mary — Джон убил Мэри
• Mary killed Sam — Мэри убила Сэма
• Sam killed John — Сэм убил Джона

Your task is to translate into Russian the following sentences:

• John killed Sam
• Mary killed John
• Sam killed Mary

Bonus question. Have you noticed any signs that I am getting tired of linguistics?

I stumbled on a Russian linguistics competition called The Russian Little Bear. Most of the puzzles are Russian-specific; but some of them can be translated. I concentrated on puzzles for grades six through nine and used Unicode for uncoding strange characters.

Problem 1. Here are some Latin words with their English translations:

• amo — I love
• amat — He loves
• invitor — I am invited
• invitaris — You are invited
• rogas — You ask
• rogatur — He is asked

Pick the line of words from A to E that best translates these phrases into Latin: You are loved, I ask, He invites.

• (A) amas, rogo, invitat;
• (B) amaris, rogo, invitat;
• (C) amaris, rogor, invitas;
• (D) amaris, rogat, invitatur;
• (E) amaris, rogo, invito.

Problem 2. The first astronauts from India (I), Hungary (H), France (F) and Germany (G) were Bertalan Farkas (1), Sigmund Jähn (2), Rakesh Sharma (3) and Jean-Loup Chrétien (4). Match the astronauts to the countries:

• (A) I2, H1, F4, G3;
• (B) I3, H1, F4, G2;
• (C) I3, H1, F2, G4;
• (D) I1, H4, F3, G2;
• (E) I3, H2, F4, G1.

Problem 3. You do not need to know Russian to solve this puzzle. It is enough to know the modern Russian alphabet: А, Б, В, Г, Д, Е, Ё, Ж, З, И, Й, К, Л, М, Н, О, П, Р, С, Т, У, Ф, Х, Ц, Ч, Ш, Щ, Ъ, Ы, Ь, Э, Ю. Before XVIII century, numbers in Russian were denoted by letters, for example: ТЛЕ — 335, РМД — 144, ФЛВ — 532.

How was 225 written in old Russian?

(A) ВВФ; (B) ВВЕ; (C) СКЕ; (D) СКФ; (E) ВНФ.

Problem 4. Here are several Turkish words and phrases with their English translations:

• ada — an isle
• adalar — isles
• iki tas — two cups
• adam — a man
• otuz adam — thirty men
• taslar — cups

Pick the line of words from A to E that best translates these phrases into Turkish: thirty isles, men?

• (A) otuz adalar, adamlar;
• (B) otuz ada, adam;
• (C) otuz adalar, adam;
• (D) otuz ada, adamlar;
• (E) ikilar ada, adamlar.

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. All of them are from the phonetics section and I’ve kept the same problem number as in the book. I’ve used the Unicode encoding for special characters.

Problem 20. In the table below there are numerals from some Polynesian languages. Note that I couldn’t find the proper English translation for one of the languages, so I used transliteration from Russian. The language sounds like “Nukuhiva” in Russian.

Your task is to find the words that should be in the empty cells. Note that wh, ‘, and ŋ denote special consonants.

Problem 21. Below you will find words in several relative languages. You can group these words into pairs or triples of words with the same origin and the same or a similar meaning.

āk, dagr, bōk, leib, fōtr, waʐʐar, buoh, dæʒ, plōgr, hām, wæter, hleifr, pfluog, eih, heimr, fuoʐ, plōʒ.

Task 1. Divide the words into groups so that the first group has words from the same language, the second group has words from another language and so on.

Task 2. (optional) List your suggestions about the meanings of the words and about the identity of the languages.

Problem 22. These words from the Aliutor language are followed by their translations. The stresses are marked by an apostrophe in front of the stressed vowel.

• t’atul — fox
• nətɣ’əlqin — hot
• nur’aqin — far away
• ɣ’əlɣən — skin
• n’eqəqin — fast
• nəs’əqqin — cold
• tapl’aŋətkən — He sews shoes
• k’əmɣətək — to roll up
• ʔ’itək — to be
• paq’ətkuk — gallop
• n’ilɣəqinat — white (they both)
• p’unta — liver
• qet’umɣən — relative
• p’iwtak — to pour
• nəm’itqin — skillful
• t’umɣətum — friend
• t’ətka — walrus
• k’əttil — forehead
• qalp’uqal — rainbow
• kəp’irik — hold (a baby in the hands)
• təv’itatətkən — I work
• p’intəvəlŋək — attack (each other)

Your task is to put the stresses in the following words: sawat ‘lasso’, pantawwi ‘fur boots’, nəktəqin ‘solid’, ɣətɣan ‘late autumn’, nəminəm ‘bouillon’, nirvəqin ‘sharp’, pujɣən ‘spear’, tilmətil ‘eagle’, wiruwir ‘red fish’, wintatək ‘to help’, nəmalqin ‘good’, jaqjaq ‘seagull’, jatək ‘to come’, tavitətkən ‘I will work’, pintətkən ‘he attacks (someone)’, tajəsqəŋki ‘in the evening’.

Note that the vowel ə is similar to many unstressed syllables in English words, such as the second syllable in the words “taken” and “pencil”. This vowel is shorter than other vowels in the Aliutor language.

Last time Sue refinanced her mortgage was six years ago. She received a 15-year fixed loan with 5.5% interest. Her monthly payment is $880, and Sue currently owes $38,000.

Sue is considering refinancing. She has been offered a 5-year fixed loan with 4.25% interest. You can check an online mortgage calculator and see that on a loan of $38,000, her monthly payments will be $700. The closing costs are $1,400. Should Sue refinance?

Seems like a no-brainer. The closing costs will be recovered in less than a year, and then the new mortgage payments will be pleasantly smaller than the old ones. In addition, the new mortgage will last five years instead of the nine years left on the old mortgage.

What is wrong with this solution? What fact about Sue’s old mortgage did I wickedly neglect to mention? You need to figure that out before you decide whether Sue should really refinance.

So many people liked the puzzles I posted in Subtraction Problems, Russian Style, that I decided to present a similar collection of multiplication and division puzzles. These two sets of puzzles have one thing in common: kids who go for speed over thinking make mistakes.

Humans have 10 fingers on their hands. How many fingers are there on 10 hands?

This one is from my friend Yulia Elkhimova:

Three horses were galloping at 27 miles per hour. What was the speed of one horse?

Here is a similar invention of mine:

Ten kids from Belmont High School went on a tour of Italy. During the tour they visited 20 museums. How many museums did each kid go to?

Another classic:

How many people are there in two pairs of twins, twice?

Can you add more puzzles to this collection?

A stick has two ends. If you cut off one end, how many ends will the stick have left?

This pre-kindergarten math problem was given to me by Maxim Kazarian who lives in Moscow, Russia. That got me thinking about math education in the US. Actually, just about anything can get me thinking about education in our country. One of our math education patterns is to provide simplified templates and to train kids to plug numbers into them without thinking.

Math education should be about thinking. We need to give kids a lot of math problems that do not fit into standard templates, in order to encourage creative thinking. Here is another puzzle from Maxim:

A square has four corners. If we cut one corner off, how many corners will the remaining figure have?

I invite my readers to invent additional problems that sound as if a subtraction by one is needed, when, in fact, it is not. Here is my contribution:

Anna had two sons. One son grew up and moved away. How many sons does Anna have now?