Tanya Khovanova's Math Blog

Each time I see John Conway he teaches me something new. At the Gathering for Gardner he decided to quiz me on how well I know a regular six-sided die. I said with some pride that the opposite sides sum up to 7. He said, “This is the first level of knowledge.” So much for my pride. I immediately realized that the next level would be to know how all the numbers are located relative to each other. I vaguely remembered that in the corner where 1, 2, and 3 meet, the numbers 1, 2, and 3 are arranged in counter-clockwise order.

Here’s how John taught me to remember every corner. There are two types of corners. In the first type numbers form an arithmetic progression. John calls such numbers counters. He chose that name so that it would be easy to remember that counters are arranged in counter-clockwise order. The other numbers he calls chaos: their increasing sequence goes clockwise.

Once I grasped that, I relaxed thinking that now I know dice. “What about the third level?” he asked. “What third level?” “Now that you know which number goes on which side, you need to know how the dots are arranged.” Luckily, there are only three sides on which the dots are not placed with rotational symmetry: 2, 3, and 6. And they all meet in a corner, which John calls the home corner. The rule is that the diagonals formed by the dots on the sides with 2, 3, and 6, meet in the home corner. You might argue that 6 doesn’t have a diagonal. But if you look at 6, you can always connect the dots to form the letters N or Z, depending on the orientation of the die. When you lay the letter N on its side, it becomes the letter Z. Thus they define the same diagonal. This diagonal has to meet the diagonals from 2 and 3 in the corner.

When I came home from the conference I picked up a die and checked that the rules work. There are 8 corners. It is enough to remember one corner of numbers to recover the other numbers by using the opposite sum rule. But it is nice to have a simple rule that allows us to bypass the calculation. Four of the corners have numbers in arithmetic progression: 1:2:3, 1:3:5, 2:4:6, and 4:5:6. They are counters and they are arranged counter-clockwise. The other four corners are: 1:2:4, 1:4:5, 2:3:6, and 3:5:6, and they are arranged clockwise.

I wanted to provide a picture of a die for this post and went online to see if I could grab one. Many of the graphic images of dice, as opposed to photographs, were arranged incorrectly. Clearly these visual artists did not study dice with John Conway.

Then I decided to check my own collection of dice. Most of them are correct. The ones that are incorrect look less professional. Here is the picture. The ones on the right are correct.

I started my Yellow Road a year ago on February 9, 2013, when my weight was 245.2 pounds. My system worked for eight months. I lost 25 pounds. Then I went to two parties in a row and gained four pounds. According to my plan, I was supposed to eat only apples after lunch. It was too difficult to stick to that, and I got off-target. My target weight continued decreasing daily, as per my plan, while I got stuck. The growing difference between my real weight and my target weight was very discouraging, so I lost my momentum.

I decided to reset the target weight and restart the plan. I changed my plan slightly to incorporate the lessons I had learned about myself.

On February 9, 2014, I started my New Yellow Road. I weighed 223.2 pounds. So I reset my target weight to be 223.2 on February 9. Each day my target weight goes down by 0.1 pounds. I weigh myself each morning. If I am within one pound of my target weight, I am in the Yellow zone and I will eat only fruits and vegetables after 5:00 pm. If I am more than one pound over my target weight, I am in the Red Zone and will eat only apples after 5:00 pm. If I am more than one pound below my target weight, I can eat anything.

I want to tell you the story of my “successful” salary negotiation. The year was 2003 and I had a temporary visiting position at Princeton University. I wanted to move to Boston and my friend showed my resume to Alphatech. The interview went well and they offered me a position as Lead Analyst with a salary of $110.000. This was particularly good news considering that the tech job market was very weak in 2003.

At that time, I was working very hard on building my self-esteem. I read lots of books and was in therapy for two years. I decided to practice what I had learned to try to negotiate a bigger paycheck. While speaking to the HR guy on the phone, I was standing up, as I was taught, and projecting my voice firmly with my chest opened up. I could hardly believe it when I heard myself ask for a $10,000 increase.

Despite my wonderful posture, the human resource person refused. However, he remembered that they had forgotten to give me a moving bonus. He asked me about my living conditions. I told him that I lived in a four-bedroom house. I didn’t elaborate: it was a tiny four-bedroom house made out of a garage. I made a counter offer: forget the moving bonus, but give me my salary increase as I asked. He agreed.

For the first year, there was no real difference, because the salary increase was equal to the moving bonus. But I was planning to stay with the company for a long time, so by the second year, my clever negotiation would start to pay off. My negotiations were a success. But were they?

Things change. The boss who hired me and appreciated me stopped being my boss. The company was bought by BAE Systems who were not interested in research. To my surprise, I started getting non-glowing performance reviews. Luckily, by that time I had made a lot of friends at work, and one of them not only knew what was going on, but was willing to tell me. My salary was higher than that of other employees in the same position. Salary increases were tied to performance. They wanted to minimize my increases to bring my salary into the range of others at my level. So to justify it, they needed a negative performance review. After one negative review it is difficult to change the trend. A negative review stays on the record and affects the future reputation.

In a long run I am not sure that my salary negotiations were a success.

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.

* * *

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

* * *

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

* * *

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.

* * *

— 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.

* * *

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.

* * *

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: