Tanya Khovanova's Math Blog

I lead recitations for a Linear Algebra class at MIT. Sometimes my students are disappointed with their grades. The grades are based on the final score, which is calculated by the following formula: 15% for homework, 15% for each of the three midterms, and 40% for the final. After all the scores are calculated, we decide on the cutoffs for A, B, and other grades. Last semester, the first cutoff was unusually low. The top 50% got an A.

Some students who were above average on every exam assumed they would get an A, but nonetheless received a B. The average scores for the three midterm exams and for the final exam were made public, so everyone knew where they stood relative to the average.

The average scores for homework are not publicly available, but they didn’t have much relevance because everyone was close to 100%. However, a hypothetical person who is slightly above average on everything, including the homework, should not expect an A, even if half the class gets an A. There are two different effects that cause this. Can you figure them out?

One day I received a call on my home line. I do not like calls from strangers, but the guy knew my name. So I started talking to him. I assumed that it was some official business. He told me that their company monitors Internet activities, and that my computer is emitting viruses into the Internet traffic degrading Internet performance. All I need to do is to go to my computer and he will instruct me how to get rid of my viruses.

While he was saying all this, I covered my phone’s microphone and made a call to the police from my cell phone. I was hoping the police could trace the call and do something while I kept the line to the guy open. The police told me to hang up. They said there is nothing they can do.

Meanwhile, the guy on the phone kept directing me to my Start button while I kept telling him that I can’t find it. After talking to the police, I got so angry that I told the guy that I wasn’t actually looking for the Start button, but talking to the police. So the guy asks, “What does the police say?”

These people are laughing at us. They know that the police do nothing. And then continued instructing me about my Start button.

Have you ever solved a CalcuDoku puzzle, or a MathDoku puzzle? Maybe you have, but you do not know it. Many incarnations of this puzzle are published under different names. The MIT’s Tech publishes it as TechDoku. What distinguishes this puzzle type from most others is that it is trademarked. The registered name is KenKen. So anyone can invent and publish a KenKen puzzle as long as they do not call it KenKen.

In this variously named puzzle you need to reconstruct a Latin square, where cells of a square are grouped into regions called cages. Each cage has a number and an operation (addition, subtraction, multiplication, division) in the upper-left corner of the cage. The operation applied to the numbers in the cage must result in a given value. For non-commutative operations (subtraction and division), the operation is applied starting from the largest number in the cage.

These are my two NOT KenKen puzzles. I will call them TomToms, the name for this puzzle used by Tom Snyder in his The Art of Puzzle LINK blog. In the TomTom variation, cages without a number in a corner are allowed and the operation might be missing, but it has to be one of the standard four. The first puzzle I call Three Threes and the second is a minimalistic version where only one number without the operation is given.

But my goal today is not to discuss KenKen or its ekasemans. Ekaseman is the reverse of the word namesake. My son Alexey invented the term ekaseman to denote a different name for the same thing. My real goal is to discuss a new type of puzzle that can be called Crypto KenKen. In this puzzle the digits in the corner are encrypted using a substitution cipher: each digit corresponds to its letter. I first saw this puzzle at Tom Snyder’s blog, where it is called TomTom (Cipher). I think the crypto version of this puzzle deserves its own name. So I suggest PamPam: it is an encryption of KenKen as well as TomTom. And it would be nice to have a female name for a change.

It is time to report on my weight loss progress. Unfortunately, the report is very boring; I am still stuck at the same weight: 225. What can I do? Let’s laugh about it. Here are some jokes on the subject.

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After the holidays I stepped on my scale. After an hour I tried again and had a revelation: tears weigh nothing!

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I am on a miracle diet: I eat everything and hope for a miracle.

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Ideas to lose weight: A glass of water three days before your meal.

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I wanted to lose five pounds by this summer, now I have only ten pounds to lose to reach my goal.

I wrote a paper with my son, Alexey Radul, titled (Not so) Much Ado About Nothing. As the title indicates, nothing is discussed in this paper. It’s a silly, humorous paper full of puns about “nothing.” We submitted the paper to the arXiv two months ago, and it has been on hold since then.

This reminds me of an earlier paper of mine that the arXiv rejected because it didn’t have journal references. (Not so) Much Ado About Nothing is done in proper style. It follows all the formal rules of math papers, and contains references, acknowledgements, an introduction with motivation, and results. However, the results amount to nothing. The fact that this paper is not accepted is a good sign. It means the arXiv doesn’t just look at the papers formally; they look at the content as well.

On the other hand, the paper is submitted as a paper in recreational mathematics, and it is humorous, so it could have been accepted, since nothing is more recreational than nothing.

Neither rejection nor acceptance would have surprised me. The only thing I do not understand is why it is on hold. Why hold on to nothing?

I know how to walk. Everyone knows how to walk. Or so I thought. Now I am not sure any more.

I’ve been taking ballroom dance lessons on and off for many years. But at some point I stopped progressing. I got stuck at the Silver level. I know many steps and am a good follower, but I often lose balance and my steps are short.

Then I met Armin Kappacher, an unusual dance coach for the MIT Ballroom dance team. I would like to share some of his wisdom with you, but Armin doesn’t have much presence on the web. He only wrote one article for Dance Archives: A Theoretical and Practical Approach to “Seeing The Ground of a Movement.”

Although I’ve been attending his classes for several years, I haven’t been able to understand a word. He might say, “Your right arm is disconnected from your chest center.” But what does that mean? Others seemed to understand him, because they greatly improved under his guidance. But I was so out of touch with my body that I couldn’t translate his words into something my body could understand. Being a mathematician, I lived my whole life in my brain. I never tried to listen to my body. I was never aware of whether my forehead was relaxed or tensed, or if my pelvic floor was collapsed. I grasped that it was my own fault that I failed to understand Armin. I stuck with his classes.

After three years of group classes, I asked Armin for a private lesson with an emphasis on the basics. He instructed me to take three steps and quickly discovered my issues, which included:

• I was too collapsed.
• My spine was too curved.
• My butt stuck out too much.
• My weight was not forward enough.
• My head was too forward.
• I didn’t sway.

So now I am taking walking classes from Armin. I am slowly starting to feel what Armin means when he says that my pelvic floor is collapsed. I feel better now. Whenever I pay attention to how I am walking, my posture improves. As a result, I feel more confident, my mood approves, and I feel like more oxygen is getting to my brain. My friends have noticed a change. For example, my son Sergei got married recently, and I was sitting under the Chuppah during the ceremony (see photo). Afterwards, several friends told me that I looked like a queen.

I have to give some credit to my earrings. They were too long and were getting caught on my dress. So I was constantly trying to wiggle my head up—using Armin’s techniques.

I proudly brought this photo to Armin to show him my queen-like posture. He told me that I look okay above the chest center. But below the chest center my spine is still collapsed. Next time I will take sitting lessons with Armin.

Skyscraper puzzles are one of my favorite puzzles types. Recently I discovered a new cute variation of this puzzle on the website the Art of Puzzles. But first let me remind you what the skyscraper rules are. There is an n by n square grid that needs to be filled as a Latin square: each number from 1 to n appears exactly once in each row and column. The numbers in the grid symbolize the heights of skyscrapers. The numbers outside the grid represent how many skyscrapers are seen in the corresponding columns/row from the direction of the number.

The new puzzle is called Skyscrapers (Sum), and the numbers outside the grid represent the sum of the heights of the skyscrapers you see from this direction. For example, if the row is 216354, then from the left you see 8(=2+6); and from the right you see 15(=4+5+6).

Here’s an easy Skyscrapers (Sum) puzzle I designed for practice.

The Art of Puzzles has four Skyscrapers (Sum) puzzles that are more difficult than the one above:

• Skyscrapers (Sum) by Grant Fikes
• Skyscrapers (Sum) by Thomas Snyder
• Dr. Sudoku Prescribes #146 — Skyscrapers (Sum)
• Dr. Sudoku Prescribes #69 — Skyscrapers (Sum)

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A button of unknown functionality should be pressed an even number of times.

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When I tell you that I am closer to 30 than to 20, I mean to tell you that I am 42.

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If a car with a student-driver sign gets its windshield wipers turned on, then the car is about to turn.

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I always learn from the mistakes of people who followed my advice.

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A traffic policeman stops a car:
—You’re going 70 in a 35 miles-per-hour zone.
—But there are two of us!

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The most popular tweet, “Live your life so that you do not have time for social networks.”

I recently wrote a post about the ApSimon’s Mints problem:

New coins are being minted at n independent mints. There is a suspicion that some mints might use a variant material for the coins. There can only be one variant material. Therefore, fake coins weigh the same no matter where they’ve been minted. The weight of genuine coins is known, but the weight of fake coins is not. There is a machine that can precisely weigh any number of coins, but the machine can only be used twice. You can request several coins from each mint and then perform the two weighings in order to deduce with certainty which mints produce fake coins and which mints produce real coins. What is the minimum total of coins you need to request from the mints?

The post was accompanied by my paper Attacking ApSimon’s Mints.

Unfortunately, both the post and the paper contain wrong information. They both state that the number of coins as a sequence of the number of mints is 1, 2, 4, 8, 15, 38, 74. This is wrong. I took this data from the sequence A007673 in the OEIS database. The sequence had incorrect data lying dormant for 20 years. I believe that the sequence was generated from the paper of R. K. Guy and R. J. Nowakowsky, ApSimon’s Mints Problem, published in Monthly in 1994. To the credit of Guy and Nowakowsky, they never claimed to find the best solution: they just found a solution, thus providing a bound for the sequence. Someone mistook their solution for the optimal one and generated the sequence in the database.

After my post, my readers got interested in the problem and soon discovered the mistake. First Konstantin Knop found a solution for 6 mints with 30 coins and for 7 mints with 72 coins. Konstantin is my long-time collaborator. I trust him so I was sure the sequence was flawed. Then someone located a reference to a paper in Chinese A New Algorithm for ApSimon’s Mints Problem. Although none of my readers could find the paper itself nor translate the abstract from Chinese. But judging from the title and the formulae it was clear that they found better bounds than the sequence in the database. My readers got excited and tried to fix the sequence. David Reynolds improved on Konstantin’s results with a solution for 6 mints with 29 coins and for 7 mints with 52 coins. David did even better on his next try with 28 and 51 coins correspondingly. He also found a solution with 90 coins for 8 mints. Moreover, his exhaustive search proved that these were the best solutions.

Now the sequence in the database is fixed. It starts 1, 2, 4, 8, 15, 28, 51, 90.

For future generations I would like to support each number of the sequence by an example. I use the set P(Q) to represent the sequence of how many coins are taken from each mint for the first(second) weighing. For one mint, only one coin and one weighing is needed. ApSimon himself calculated the first five values, so they were not in dispute.

• a(2) = 2: P=(1,0) and Q=(0,1). Found by ApSimon.
• a(3) = 4: P=(0,1,2) and Q=(1,1,0). Found by ApSimon.
• a(4) = 8: P=(0,1,2,3) and Q=(1,0,2,2) or P=(0,1,1,4), Q=(2,0,1,1). Found by ApSimon.
• a(5) = 15: P=(0,1,1,4,5) and Q=(2,1,2,5,0). Found by ApSimon.
• a(6) = 28: P=(1,2,2,5,5,0) and Q=(0,1,2,1,8,10). Found by Robert Israel, Richard J. Mathar, and David Reynolds,
• a(7) = 51. P=(2,3,7,2,8,12,0) and Q=(0,2,1,7,7,12,12). Found by David Applegate and David Reynolds.
• a(8) = 90. P=(4,6,6,7,3,13,15,3) and Q=(4,0,1,6,12,12,1,27). Found by David Applegate and David Reynolds.

There is a solution for nine mints using 193 coins that is not confirmed to be optimal. It was found by David Reynolds: P=(1,2,4,12,5,4,20,39,43) and Q=(0,1,3,3,25,33,34,18,27). In addition, David Reynolds provided a construction that reduces the upper bound for n mints to (3⁽ⁿ⁺¹⁾−2n−3)/4. The following set of coins work: P=(1,3,7,15,…,2ⁿ−1) and Q=(1,4,13,40,…,(3ⁿ−1)/2).

Recently I wrote that my blog is under attack by spam comments. Most of the comments were caught by my spam-filter Akismet, the best-known filter for WordPress. I was receiving about 50,000 comments a day and 200 of them were sneaking through this filter. I had to moderate those and delete them. This was an extreme waste of my time. But I can understand that the bots achieved some goal. None of their comments made it to my website, but at least I myself was made aware of opportunities for hair removal in Florida.

The comments crashed my server and I had to install CAPTCHAs. I was happy that the number of comments that I had to moderate went down, but the total number of comments was still so high that my server kept crashing. Now that the comments are blocked from human view, why are they still pouring in? One software package is trying to inform the other software package about weight-loss wonder drugs. I am convinced that Akismet is not interested.

My hosting provider couldn’t handle the traffic and asked me to upgrade to a more advanced hosting package. It’s annoying that I have to pay a lot of money for these bots’ attempts to sell Akismet fake Louis Vuitton bags.

The upgrade was too expensive, so I tried a different solution. I closed comments for older posts. It didn’t help. The bad software continued trying to leave comments that can’t be left. They especially like my post about Cech cycles, called A Mysterious Bracelet. My weblog tells me that every second someone downloads this page and tries to leave a comment. But no one will ever see these comments. Even Akismet will never know what it is missing: it might have had a chance to make $5,000 a day from home.