Tanya Khovanova's Math Blog

I am scared of that old German gentleman. Forgot his name. Oh, yeah. Alzheimer.

I started to lose my brain processing speed a long time ago. By my estimates I am about 100 times slower than I used to be. From time to time someone gives me a puzzle I remember I solved in 30 seconds years ago, but now it takes me 30 minutes. And it is getting worse. When I moved to my new apartment recently, it took me a year to remember my own phone number.

On the positive side, I feel quite famous, according to one of the signs of success. I am often greeted by people I don’t know.

My moment of action came when I was in my basement doing my laundry and couldn’t remember how to turn on my washing machine. This is after 10 years of heavily using this damn machine. I started to look for the button to push, but there were no buttons. There were only knobs, and I couldn’t find the word “on” anywhere. After struggling with my memory and with those knobs, I pulled out one of the knobs, and the machine started.

I panicked. I am afraid of Alzheimer’s Disease. I do not want to become demented. I do not want to forget how to count or to stop recognizing my children. I do not want to become a drain on my children’s time, emotions and money.

I had complained about my memory to my doctor before, but the only thing he ever found was anemia. This time I was more insistent. I had an MRI that ruled out tumors. I had more extensive blood tests that confirmed anemia and showed a Vitamin D deficiency. But then he sent me to a neurologist who suggested a sleep study. Finally I got a diagnosis of severe sleep apnea. I am so happy now. I might not have Alzheimer’s Disease. In the worst case scenario, I might die in my sleep. In comparison, this doesn’t sound so bad.

I want to come back to a middle-school Olympiad problem I posted a while ago.

Streamline School Olympiad 2000 (8th grade). You have six bags of coins that look the same. Each bag has an infinite number of coins and all the coins in the same bag weigh the same amount. Each different bag contains coins of a different weight, ranging from 1 to 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 you need to weigh the coins in order to confirm that the labels are correct?

The answer is unpretentious: one weighing is enough. We can take one 5-gram coin, two 4-gram coins, three 3-gram coins, four 2-gram coins and five 1-gram coins for the total of 35 grams. This number is not divisible by 6, so we can add one more 1-gram coin and weigh all of them against six 6-gram coins. I leave it to the reader to show that this solution works and to extrapolate the solution for any number of bags.

My new challenge is to find a weighing for the above problem using the smallest number of coins. What is the number of coins in such a weighing for a given number of bags?

I manually calculated this number for a small number of bags, but I would like to get a confirmation from my readers. Starting from 6 bags, I don’t know the answer. Can you help me?

The 2015 Intel Science Talent Search results are out. This year they divided the prizes into three categories: basic research, global good, and innovation. All three top prizes in basic research were awarded to our PRIMES students:

• First place: Noah Golowich, Resolving a Conjecture on Degree of Regularity, with some Novel Structural Results
• Second place: Brice Huang, Monomization of Power Ideals and Generalized Parking Functions
• Third place: Shashwat Kishore, Multiplicity Space Signatures and Applications in Tensor Products of sl₂ Representations

PRIMES’ success in this year’s Siemens competition is even more impressive. Unlike Intel, Siemens didn’t divide the projects into three groups. We took the first and second overall individual prizes.

• First place: Peter Tian, Extremal Functions of Forbidden Multidimensional Matrices
• Second place: Zoseph Zurier, Generalizations of the Joints Problem

PRIMES is the place for high school math research. Congratulations to all our students—and to me (and my colleagues) for a job well done!

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.

* * *

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

* * *

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

* * *

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

* * *

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)