Tanya Khovanova's Math Blog

There is a famous puzzle about three light bulbs, that is sometimes given at interviews.

Suppose that you are standing in a hallway next to three light switches that are all off. There is another room down the hall, where there are three incandescent light bulbs—each light bulb is operated by one of the switches in the hallway. You can’t see the light bulbs from the hallway. How would you figure out which switch operates which light bulb, if you can only go to the room with the light bulbs one and only one time?

This puzzle worked much better in the past when we only had incandescent light bulbs and so didn’t need to specify the type of bulbs. Unfortunately, the standard solution only works with incandescent bulbs and the word “incandescent” nowadays needs to be stated. But the use of “incandescent” is a big hint. Indeed, incandescent light bulbs generate heat when they are on, so the standard solution is to turn on the first light switch, to keep the second switch off, and to turn the third switch on for five minutes before turning it off. In the room, the light bulb corresponding to the first switch will be lit, and out of the two unlit bulbs, the one corresponding to the third switch will be warm.

It’s a cute solution, but there could be so many other approaches:

• You can ask a friend to help you. You can come to the room with the light bulbs and shout to her which switch to turn on.
• You can place mirrors from the room to the hall, so you can see bulbs through the multiple reflections. You might not need to enter the room at all.
• Before playing with the switches, you can place a video camera in the room to transmit the scene in real time. You will not be allowed to enter the room again to get the camera back, but what the heck: the job will be done.
• You can put timers on the switches and set them to turn on at different times.
• You can attach strings to switches and turn them on or off from a distance.

I invite my readers to invent other methods to solve this problem. Be creative. After all, if I were to interview you for a job, I would be more impressed by a new solution than the one that is all over the Internet.

You arrive at an archipelago of many islands. On each island there are two villages. In one village truth-tellers live, and they always tell the truth. In the other village liars live, and they always lie. The islanders all know each other.

On the first island you stumbled upon three islanders and you ask each of them your question:

How many truth-tellers are there among you?

Here are their answers:

A: One.
B: A is wrong.
C: A and B are from the same village.

Can you determine who is a truth-teller and who is a liar?

This island is called a classic island, where all behave as if they were in a standard logic puzzle. It is a perfectly nice puzzle but B and C didn’t answer the question: B ratted on A, and C went on a tangent. When I was younger, I never cared about the motivations of A, B, or C. Their answers are enough to solve the puzzle. But now that I am older, I keep wondering why they would choose these particular answers over other answers. So I invented other islands to impose rules on how the villagers are allowed to answer questions.

Now you travel to the next island that is called a straightforward island, where everyone answers your question exactly. You are in the same situation, and ask the same question, with the following result:

A: One.
B: One.
C: Ten.

Can you determine who is a truth-teller and who is a liar?

Once again we wonder about their motivation. This time C told an obvious lie, an answer that is impossible. Why on earth did he say 10? Isn’t the goal of lying to deceive and confuse people? There is nothing confusing in the answer “ten.”

Now you come to the third island, which is a straightforward inconspicuous island. To answer your question, a liar wouldn’t tell you an obvious lie. For this particular situation, the liar has to choose one of the four answers that are theoretically possible: zero, one, two, or three. You are again in the same situation of asking three people how many truth-tellers are among them, and these are the answers:

A: Two.
B: Zero.
C: One.

Can you determine who is a truth-teller and who is a liar?

When you think about it, a truth-teller cannot answer zero to this question. So although zero is a theoretically possible answer, we can deduce that the person who said it is a liar. If liars are trying to confuse a stranger, and they’re smart, they shouldn’t answer “zero.”

The next island is a straightforward inconspicuous smart island. The liars on this island are smart enough not to answer zero. You are in the same situation again and ask the same question with the following outcome:

A: Two.
B: Two.
C: One.

Can you determine who is a truth-teller and who is a liar? You shouldn’t be able to. There are three possibilities. There are two truth-tellers (A and B), one truth-teller (C), or zero truth-tellers.

Let us assign probabilities to liars’ answers. Assume that liars pick their answers randomly from the subset of wrong answers out of the set: one, two, three. If two of these answers are incorrect, they pick a wrong answer with probability one half. If all three of the answers are incorrect, they pick one of them with probability one-third. Suppose the people you meet are picked at random. Suppose that the probability that a random person is a truth-teller is 1/2. Given the answers above, what is more probable: that there are two truth-tellers, one truth-teller, or zero truth-tellers?

A movie passes the Bechdel Test if these three statements about it are true:

• There are at least two named women in it
• Who talk to each other
• About something besides a man.

Surely there should be a movie where two women talk about the Bechdel test. But I digress.

The Bechdel test website rates famous movies. Currently they have rated 4,683 movies and 56% pass the test. More than half of the movies pass the test. There is hope. Right? Actually they have a separate list of the top 250 famous movies. Only 70 movies, or 28%, from this list pass the test.

My son Alexey suggested the obvious reverse Bechdel test, which is more striking than the Bechdel test. A movie doesn’t pass the test if it

• Has at least two named men characters
• Whenever they talk to each other
• They only talk about women.

I can’t think of any movie like that. Can you?

Update (Sep 2023): I should have included the possibility that there are no two named male characters.

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A Roman walks into a bar, holds up two fingers, and says, “Five beers, please.”

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To understand what a recursion is, you must first understand recursion.

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A guy is complaining to his mathematician friend:
— I have a problem. I have difficulty waking up in the morning.
— Logically, counting sheep backwards should help.

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— Can I ask you a question?
— You can, but you have already just done that.
— Darn, what about two questions?
— You can, but that was your second question.

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The Internet ethics committee worked hard to generate a list of words that should never be used on the Internet. The problem is, now they can’t post it.

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Quantum entanglement of a pair of socks: As soon as one is designated as the left, the other instantly becomes the right.Share:

The last MIT Mystery Hunt was well-organized. It went smoothly—unlike the hunt that my team designed the year before. Sigh. As I do every year, here is the list of 2014 puzzles related to math.

• Build Your Own Sudoku. As it says: Sudoku.
• Oyster Card. Nikoli puzzles related to the London tube.
• A Rose by Any Other. Paint by numbers. I enjoyed this puzzle during the hunt.
• Please Remain Seated. A roller-coaster puzzle, where the extraction needs some math.
• Marking Territory. A Shikaku variation.
• SAT III. Several optimization problems.
• Growth and Fixed Costs. Some financial mathematics.
• Magic Mushrooms. A very cool puzzle with variations of Skyscrapers, Nurikabe, Hashi, Heyawake, Corral, Filomino, Hitori, and Slitherink. I discovered it after the hunt, and enjoyed solving individual puzzles. The extraction is difficult, though.
• Glazing Over. A broken Sudoku puzzle.

There were also several puzzles requiring decoding or having a CS flavor.

• The Spy Who Read Me. Several cipher texts.
• A Puzzle with the Answer CLEMENS. Some encrypted words.
• A Puzzle with the Answer CRONIN. A stream of data.
• A Puzzle with the Answer RICE. Unraveling a huge list of instructions.
• Callooh Callay, World! Programs in a fictional programming language.
• Some Nutty Lines. My favorite decoding puzzle.
• WAT? Some crazy looking cryptograms.
• SKI Trees. Computer sciencey ski slopes.

I want to mention one non-mathematical puzzle.

• Operator Test. It is based on puzzles from the previous years and one of them was Wordplay, co-written by me.

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You visit an island of three towns: Trueton, Lieberg and Alterborough. Folks living in Trueton always tell the truth. Those who live in Lieberg, always lie. People from Alterborough alternate strictly between truth and lie. You meet an islander who says:

Two plus two is five. Also, three plus three is six.

Can you determine which town he is from?

It should be easy. He made two statements: the first one is false, the second is true. So he must be from Alterborough.

But what about “also”? How should we interpret this transition? There are many ways to interpret this “also.” On one hand it could mean: In addition to the previous statement I am making another statement. On the other hand it could mean: The previous pause shouldn’t be considered as the end of the statement; the whole thing should be interpreted as one statement. Besides this person was speaking not writing. Are we sure that the first period was not meant to be a comma or a semi-colon? If we assume that the quote is one statement, then the speaker might be either a liar or an alternator.

Here is a puzzle for you from the same island:

One night a call came into 911: “Fire, help!” The operator couldn’t ID the phone number, so he asked, “Where are you calling from?” “Lieberg.” Assuming no one had overnight guests from another town, is there an emergency? If so, where should help be sent? And was it a fire?

Now find the ambiguities.Share:

The MIT Mystery Hunt starts on Friday. My old team—Manic Sages— fell apart after last years’ hunt. My new team—Death and Mayhem—started sending us daily practice puzzle to prepare for the hunt.

Today’s puzzle was written by Paul Hlebowitsh. As usual, the answer is a word or a phrase.

Puzzle 3: Humans

“After a long day taking care of the animals, it’s good to unwind by letting the taps flow.”

Tap 1: Sierra Nevada Pale Ale
Tap 2: Lagunitas Pale Ale
Tap 3: 21st Amendment “Brew Free! or Die IPA”
Tap 4: Wachusett Blueberry
Tap 5: Harpoon UFO

Alice, Bob, Carol, Danny, Erica, Fred, and Gregario, the seven children of Noah, went to a local bar before the flood to get drinks. Accounts of night vary, and no one remembers exactly what happened, but some facts have become clear:

1. Everyone had two drinks, in some order.
2. Danny liked his first drink so much he had it again. Everyone else had drinks from different breweries.
3. Erica was the only person to drink an IPA.
4. Four people had the Wachusett Blueberry as their first drink.
5. One person had the Sierra Nevada Pale Ale as their first drink.
6. Alice only drank beers with headquarters in California, in order to spite Bob and Danny.
7. Two Harpoon UFOs were ordered, as well as two Sierra Nevada Pale Ales. 8
8. Alice’s second drink was the same as Fred’s first drink.
9. Bob and Danny hate California and refuse to drink any beer from a company that’s headquartered there.
10. Alice and Erica had the same first drink.
11. Fred had a Harpoon UFO.

Who ordered which drinks and in what order?Share:

I stopped losing weight. My Yellow Road plan stopped working. If you recall I draw a line on the weight/time plane which I call my target weight. If I’m more than one pound above my target weight, then I’m in the red zone and must restrict my evening food to apples. The hour at which the evening starts depends on how many pounds I’m above my target weight.

And now, back to my bump.

First I stopped following the plan exactly. I realized that I didn’t need to restrict myself to eating only apples in the evening when I am in the red zone. I can use salad or anything light.

One day I found myself in the red zone weighing two pounds over my target weight. I was invited to dinner that evening. I decided to accept the invitation and skip the plan for one day. At the party the food was so good I couldn’t resist it. The next day I was four pounds over my target weight.

My Yellow Road plan requires me in this situation to eat only apples from 2:00pm onward, but I knew that applying this restriction after 6:00pm worked for me. I decided not to torture myself and started to restrict my food only from 6:00pm. The weight didn’t go down. Even though I went to bed very hungry for two weeks, it didn’t work.

After these two weeks, I started to feel hungry all the time and even began dreaming about food. As a result, my food intake increased. Now I am seven pounds over my target weight. I’ve reached a plateau. For the last two months I’ve been stuck at 220 pounds.

There is some good news: I now have a partner on this journey. After I started my program, I received an email from Natalia Grinberg from Germany. She offered to join forces. We send each other weekly updates on our progress and cheer each other along. Natalia’s path wasn’t smooth from the start, so she tried to supplement her diet with Almased, which is very popular in Europe. Because Natalia likes it, I looked into it. While I am afraid of pills and chemical ingredients, Almased seems to be okay. It contains soy, yogurt, honey, and vitamins. I bought one can. It is expensive and tastes awful. I’ll experiment with cinnamon or pepper and see if that helps. Will Almased help me get over the Bump?Share:

I collect geeky jokes. I think I’ve heard most of them. So I was surprised to stumble upon a website with many new ones: 50 People On ‘The Most Intellectual Joke I Know’. These are some of them:

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Q: What does the “B” in Benoit B. Mandelbrot stand for?
A: Benoit B. Mandelbrot.

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Entropy isn’t what it used to be.

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There are two types of people in the world: those who can extrapolate from incomplete data sets

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This sentence contains exactly threee erors.

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There’s a band called 1023MB. They haven’t had any gigs yet.

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A logician’s wife is having a baby. The doctor immediately hands the newborn to the dad. His wife asks impatiently, “So, is it a boy or a girl”? The logician replies, “Yes.”

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The barman says, “We don’t serve time travelers in here.”
A time traveler walks into a bar.

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The first rule of Tautology club, is the first rule of Tautology club.

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A woman walks in on her husband, a string theorist, in bed with another woman. He shouts, “I can explain everything!”

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What do you get when you cross a joke with a rhetorical question?Share:

I used to love arXiv. I’ve long thought that it was one of the best things that happened to mathematics. arXiv makes mathematical research available for free and without delay. Moreover, it is highly respected among mathematicians. For example, Grigori Perelman never submitted his proof of the Poincaré conjecture to any journal: he just posted it on arXiv.

When I came back to mathematics, all my math friends explained to me that I should submit my paper to arXiv on the same day that I submit it to a journal. As my trust in arXiv grew, I started submitting to arXiv first, waiting one week for comments, and then submitting to a journal.

Now it seems that arXiv might not love its contributors as much as they used to. arXiv moderators seem to be getting harsher and harsher. Here is my story.

In June of 2013, I submitted my paper “A Line of Sages” to arXiv. This paper is about a new hat puzzle that appeared at the Tournaments of the Towns in March 2013. The puzzle was available online at the Tournaments of the Towns webpage in Russian. After some thought I decided that it is better to cite the Tournament itself inside the body of the paper, rather than to have a proper reference. Online references in general are not stable, and this particular one was in Russian. Very soon this competition will be translated into English and the puzzle will appear in all standard math competition archives.

arXiv rejected my paper. A moderator complained that I didn’t have a bibliography. So I created a bibliography with the link to the puzzle. My paper was rejected again saying that the link wasn’t stable. Duh. That’s the reason why I didn’t put it there from the start. I Goggled the puzzle and I still didn’t find any other links.

I argued with my moderator that the standards for papers in recreational mathematics are different from the standards for purely research papers. Short recreational notes do not require two pages of history and background, nor a long list of references. A recreational paper doesn’t need to have theorems and lemmas.

Meanwhile, the moderator complained that the paper was “not sufficiently motivated to be interesting to the readership.”

I got tired of exchanging emails with this moderator and submitted my paper to The Mathematical Intelligencer, where it was immediately accepted. So I dropped my submission to arXiv.

Now my paper is not available for free on arXiv. But anyone can freely buy it from Springer for $39.95.Share: