Tanya Khovanova's Math Blog

What is the answer to this question?

Round 1 of Who Wants to Be a Mathematician had the following math problem:

Bob and Jane have three children. Given that one child is their daughter Mary, what is the probability that Bob and Jane have at least two daughters?

In all such problems we usually make some simplifying assumptions. In this case we assume that gender is binary, the probability of a child being a boy is 1/2, and that identical twins do not exist.

In addition to that, every probability problem needs to specify the distribution of events over which the probability is calculated. This problem doesn’t specify. This is a mistake and a source of confusion. In most problems like this, the assumption is that something is chosen at random. In this type of problem there are two possibilities: a family is chosen at random or a child is chosen at random. And as usual, different choices produce different answers.

The puzzle above is not well-defined, even though this is from a contest run by the American Mathematical Society!

Here are two well-defined versions corresponding to two choices in randomization:

Bob and Jane is a couple picked randomly from couples with three children and at least one daughter. What is the probability that Bob and Jane have at least two daughters?

Mary is a girl picked randomly from a pool of children from families with three children. What is the probability that Mary’s family has at least two daughters?

Now, if you don’t mind, I’m going to throw in my own two cents, that is to say, my own two puzzles.

Harvard researchers study the influence of identical twins on other siblings. For this study they invited random couples with three children, where two of the children are identical twins.

1. Bob and Jane is a couple picked randomly from couples in the study with at least one daughter. What is the probability that Bob and Jane have at least two daughters?
2. Mary is a girl picked randomly from a pool of children participating in the study. What is the probability that Mary’s family has at least two daughters?

The puzzle Family Ties was written for the 2013 MIT Mystery Hunt, but it never made it to the hunt. Here’s your chance to solve a puzzle no one has seen before. I wrote the puzzle jointly with Adam Hesterberg. The puzzle is below:

Mathematics professor S. Lee studies genealogy and is interested in the origins of life.

1. Alexei Mikhailovich Ivanov
2. Alexei Petrovich Ivanov
3. Amminadab
4. Anna of Moscow
5. Arador
6. Arahad II
7. Arassuil
8. Arathorn I
9. Arathorn II
10. Aravorn
11. Argonui
12. Asger Thomsen
13. Caecilia Metella Dalmatica
14. Egmont
15. Eldarion
16. Ellesar
17. Endeavour
18. Faustus Cornelius Sulla
19. Henry Frederick
20. Hezron
21. Isaac
22. Ivan the Great
23. Ivan the Terrible
24. Jacob
25. James I and VI
26. James V
27. Jens Knudsen
28. John Francis
29. Joseph Patrick
30. Joseph Patrick
31. Jørgen Jensen
32. Judah
33. Knud Nielsen
34. Margaret Stuart
35. Maria Donata
36. Mary Stuart
37. Matthew Rauch
38. Mikhail Ivanovich Ivanov
39. Niels Møller
40. Ole Pedersen
41. Peder Petersen
42. Peter Jørgensen
43. Petr Alexeyevich Ivanov
44. Pharez
45. Ram
46. Robert Francis
47. Rose Elizabeth
48. Søren Thomsen
49. Thomas Olsen
50. Ursula Gertrud
51. Vasily I of Moscow
52. Vasily II of Moscow
53. Vasili III of Russia
54. Yuri of Uglich
55. Zerah

I already wrote about my favorite problem from the 2015 All-Russian Math Olympiad that involved tanks. My second favorite problem is about coins. I do love almost every coin problem.

A coin collector has 100 coins that look identical. He knows that 30 of the coins are genuine and 70 fake. He also knows that all the genuine coins weigh the same and all the fake coins have different weights, and every fake coin is heavier than a genuine coin. He doesn’t know the exact weights though. He has a balance scale without weights that he can use to compare the weights of two groups with the same number of coins. What is the smallest number of weighings the collector needs to guarantee finding at least one genuine coin?

I already wrote how I build a magic SET hypercube with my students. Every time I do it, I can always come up with a new question for my students. This time I decided to flip over two random cards, as in the picture. My students already know that any two cards can be completed to a set. The goal of this activity is to find the third card in the set without trying to figure out what the flipped-over cards are. Where is the third card in the hypercube?

Sometimes my students figure this out without having an explicit rule. Somehow they intuit it before they know it. But after several tries, they discover the rule. What is the rule?

Another set of questions that I ask my students is related to magic SET squares that are formed by 3 by 3 regions in the hypercube. By definition, each magic SET square has every row, column, and diagonal as a set. But there are four more sets inside a magic SET square. We can call them super and sub-diagonal (anti-diagonal) wrap-arounds. Can you prove that every magic SET square has to have these extra four sets? In addition, can you prove that a magic SET square is always uniquely defined by any three cards that do not form a set, and which are put into places that are not supposed to from a set?

Professor Bock came to his office at the Math Department of Deys University and discovered that someone had broken in. Luckily he had a lunch scheduled with his friend Detective Radstein. Bock complained to the detective about the break-in and the detective agreed to investigate.

Nothing was stolen from the office. It looked like somebody had just slept on Professor Bock’s couch. The couch was bought recently and it was positioned so that it was impossible to see it through the tiny window of the office door, which had been locked. Interestingly, Professor Bock’s office was the only one with a couch. The detective concluded that the person who broke in knew about the couch and thus was from the Math Department. In addition, the couch was very narrow and couldn’t possibly sleep more than one person.

Investigating crimes at the Math Department of Deys University was very easy. This was due to a fact discovered by Detective Radstein on a previous case: every member of the department was one of three types:

• A dodger who always tells the truth and answers the question directly, with one exception. If such a person is guilty and asked to confirm that, then s/he remains truthful, but dodges the question.
• A liar who’s every statement is a lie. A liar might dodge or not dodge the question.
• A pathological truth-teller who always answers the question directly and truthfully.

Thus solving a crime at the department is very easy. Detective Radstein just needs to ask every person two questions, “How much is 2+2?” and “Are you guilty?” Only a guilty dodger would answer “4” and dodge. Only a guilty truth-teller would answer “4” and “yes”. Only a guilty liar would answer something other than “4” and “no”.

Detective Radstein decided to enjoy himself by making this investigation more challenging. He asked everyone only one question “Are you guilty of breaking-in?” These are the first three replies:

—Albert: I did it.
—Bill: Albert didn’t do it.
—Connie: Albert did it.

It was amusing how many people knew where Albert slept, but these answers were enough for Detective Radstein to figure out the culprit. After this issue was resolved he asked Professor Bock, “While I am here, what other problems do you have at the department?”

“Well,” said the professor, “In fact, I would be grateful if you could help us resolve two more issues. This semester our expenditure for tea increased three times. It is clear someone is stealing tea. It’s also clear that this could only be someone who started working at the department this semester. There are two people who fit this description: David and Eve. So Detective Radstein asked each of them, “Are you stealing the tea?” He got the following answers:

—David: Eve steals the tea.
—Eve: Only one person steals the tea.

Having solved the Math Department’s tea problem Detective Radstein then asked Professor Bock, “What else?” “Well,” said Professor Bock, “one more thing. We have a lot of blackboards around the department. Every day a famous three-letter Russian curse word appears on the blackboards. The handwriting is always the same, so it is one person. Luckily the word starts with XY, so most people assume that this is some math formula. Anyway, I want to look into the eyes of the person who does this. I left the department late yesterday; only Fedor, Grisha and Harry were here. The blackboards were clean. This morning when I opened up the Math Department, the vulgar swearword had been written on the blackboards. I don’t suspect someone of sneaking into the department at night to scribble on our blackboards. I am certain it is one of the three people I mentioned.”

Once again Detective Radstein asked the suspects whether they did it.

—Fedor: Grisha did it.
—Grisha: Fedor did it.
—Harry: I do not speak Russian.

And again Detective Radstein solved the case. He was surprised that everyone in the department knew what everyone else was doing. Only his friend Professor Bock seemed clueless.

If you were the detective, would you be able to help Professor Bock?

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.

Kvant was a very popular science magazine in Soviet Russia. It was targeted to high-school children and I was a subscriber. Recently I discovered that a new magazine appeared in Russia. It is called Kvantik, which means Little Kvant. It is a science magazine for middle-school children. The previous years’ archives are available online in Russian. I looked at 2012, the first publication year, and loved it. Here is the list of the math puzzles that caught my attention.

The first three problems are well known, but I still like them.

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.)

Problem 2. If it is raining at midnight, with what probability will there be sunshine in 144 hours?

Problem 3. How can you fill a cylindrical pan exactly half-full of water?

I like logic puzzles, and the next two seem especially cute. I like the Parrot character who repeats the previous answer: very appropriate.

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?

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?

The next three problems are a mixture of puzzles.

Problem 6. Olga Smirnov has exactly one brother, Mikhail, and one sister, Sveta. How many children are there in the Smirnov family?

Problem 7. Every next digit of number N is strictly greater than the previous one. What is the sum of the digits of 9N?

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.