Tanya Khovanova's Math Blog

I know I’m a computer addict, because:

• When I just wake up, before opening my eyes, my first thought is not that I need to run to the bathroom, but rather, “I need to check my email.”
• My computer desk is the farthest surface suitable for eating from my kitchen, still I have most of my meals there.
• To take a break from my laptop I play evil level websudoku. To take a break from sudoku, I check my email.
• I gave a name to my laptop — Richard. Now when I travel I never feel lonely — I have my own Dick with me all the time.

If you’re reading this, you might be as bad as I am. Please finish this sentence and add it to comments below: “I know I’m a computer addict, because …”

 This puzzle was given to me by John H. Conway, and he heard it from someone else:

Find a ten-digit number with all distinct digits such that the string formed by the first k digits is divisible by k for any k ≤ 10.

Surprisingly, there is a unique solution to this puzzle. Can you find this very special ten-digit number?

For the contrast, consider ten-digit numbers with all distinct digits such that the string formed by the last k digits is divisible by k for any k ≤ 10. These numbers are not so special: there are 202 of them. My puzzle is: find the smallest not-so-special number.

John Conway sent me a puzzle about wizards, which he invented in the sixties. Here it is:

Last night I sat behind two wizards on a bus, and overheard the following:

— A: “I have a positive integral number of children, whose ages are positive integers, the sum of which is the number of this bus, while the product is my own age.”
— B: “How interesting! Perhaps if you told me your age and the number of your children, I could work out their individual ages?”
— A: “No.”
— B: “Aha! AT LAST I know how old you are!”

Now what was the number of the bus?

The first time I heard about the “stopping problem,” many years ago, it was in this version: The king announces that it is time for his only daughter to marry. Shortly thereafter 100 suitors line up in a random order behind the castle walls. Each suitor is invited to the throne room in front of the eyes of the princess and the king. At this point, the princess has to either reject the suitor and send him away, or accept the suitor and marry him. If she doesn’t accept anyone from the first 99, she must marry the last one. The princess is very greedy and wants to marry the richest suitor. The moment she sees the suitor she can estimate his wealth by his clothes and his gifts. What strategy should she use to maximize the probability of marrying the richest person?

The correct strategy is to reject the first 37 suitors and then marry the one who is better than anyone else before him. Generally, if there are N suitors the number of people to skip is about N/e — slightly more than one third of the whole group.

However, the greedy princess never received a good mathematical education. It is clear that her goal should have been to maximize the expected wealth of the future husband. In this case the strategy would be different; in particular, it changes significantly at the end. Let’s imagine that when the line of leftover suitors thins, she realizes that she’s already rejected the best one. In this case, it would be in her interest to consider the second best and, closer to the end, even the third best.

In real life, marrying the second best creates a burden of regret and bitterness. Let us assume that we all want to marry the best we can. But of course in our cases, the best does not necessarily mean the wealthiest. Also, we do not know how many suitors are lining up for us. Back in Russia I heard that a woman, on average, receives ten proposals. So, we should skip the first three, then marry the best person after that.

I do not know how many proposals an average American woman gets, but nowadays she doesn’t have to wait for an official proposal before deciding whether a guy is right for her. The question becomes: which marriage candidate should a woman try to marry?

If we assume that marriage candidates are distributed evenly in time and girls are seriously hunting for husbands between ages 20 and 35, then the above math advice can be applied in the following way: From age 20 to about 25 or 26, just look around and see what life offers you. After that, marry the one who is better than all the previous ones.

This is a very mathematical piece of advice. The idea makes sense: first you sample your options then you target the best. The problem is that these assumptions do not cover our real-life situations. Let’s look at some realistic adjustments and how they affect the age at which you stop sampling and become more active.

• You might be OK without marrying at all. In this case you can afford to be much pickier and skip the first 60% of the candidates.
• Potential husbands are not spread evenly in time. In this case try to estimate the distribution and act accordingly. For example, if you expect more men around you while you are in college, your cut-off age goes down.
• You change with time. If you think that you will lose your freshness and charm with age, your cut-off age goes down. If you think that you’ll become more experienced and effective at seducing men later in life, your cut-off age goes up.
• Your values change with time. You might be interested in looks now and value a big heart later. Because of this, it is better to delay your choice until you know yourself well and your opinion of life has stabilized.
• The value of a man changes with time. A high school drop out can become a very successful businessman and the brightest student in college can become an alcoholic. You might benefit from waiting to see how he stabilizes, thus moving your cut-off age up.
• You can divorce. This allows you to make a mistake on the first try and moves your cut-off age for this first try down.

Based on the mathematics and discussion above, here is the advice I would give to my teenage daughter — if I had one:

Take your time looking around and sampling your boyfriends. Constantly analyze the pool of your boyfriends as a whole. If there are strange patterns — for example, all of them look exactly like your father or you always pay for their dinners — start psychotherapy and work it out. As soon as your boyfriends start to look different from each other — except for things that are important to you, like education — compare your dream husband to the pool. If you dream about a Nobel prize winner who is not older than 30 and on the list of the 100 sexiest men alive, you should adjust your expectations to your chances. You will choose a guy who is better than anyone before, but it is unrealistic to expect him to be much better. As soon as you get to a cut-off age, which you have estimated using the suggestions above, stop sampling and start deciding. As soon as you find someone who is better than anyone else before, go for it — marry him.

That was advice from my brain, now I will give you advice from my heart:

Use mathematics to guide you, but make the final decision with your heart.

I read an article published in US News and World Report: Alcohol in Teens Leads to Adult Woes. This article describes the discovery that teenagers who drink heavily are much more likely to become alcoholics and have mental disorders and depression when they become adults, and that they are much less likely to finish college or be satisfied with their jobs.

This correlation is not surprising. Have you ever seen a depressed alcoholic satisfied with his/her job?

For me, the interesting question is what the word “leads” in the title “Alcohol in Teens Leads to Adult Woes” means. One might interpret “leads” as indicating that alcohol in teens causes the adult woes. If we persuade our teenagers to abstain from alcohol, will they have fewer problems in their adult lives? Will it help if you install pictures of a cirrhotic liver as a screen saver for your child’s computer?

In the middle of the article, there is a sentence that correctly states:

“What these data don’t tell us is whether those kids were already predisposed to have problems or whether drinking helped cause the trouble.”

Who is the genius who came up with a title that contradicts the article? Did they even read the article? Flashy titles sell better, but such contradictions show disrespect to the reader.

The truth is that correlations are usually insufficient to prove causality; a different type of research is needed. It appears that some of it was actually done. An interesting article, “A longitudinal study of alcohol use and antisocial behaviour in young people,” describes the study that investigated the causality between alcohol and woes. In this study, they started with three hypotheses about the long-term causality:

• Alcohol use causes antisocial behavior
• Antisocial behavior causes alcohol use
• Both above statements are true: alcohol use causes antisocial behavior and the reverse

I didn’t check this study, but the fact that they are trying to compare different hypotheses is encouraging. The result of this study is that the data supports the second hypothesis — in the long run, antisocial behavior causes alcohol use. That means the correct title for the article in the US News and World Report should have been: “Teen Woes Lead to Adult Alcohol.”

So what can you do to stop your teen’s antisocial behavior? There are many studies on that subject too. I do not know if they are correct, but you might consider a fish diet for your teen or sign up your child to train dogs.

My English teacher and editor Sue Katz wrote a funny blog entry about masturbation: “Sex and the Single Hand: Stroke Your Way to Health”

I followed the link of one of the studies she mentions to the BBC article “Masturbation ‘cuts cancer risk'”, where ” … They found those who had ejaculated the most between the ages of 20 and 50 were the least likely to develop the [prostate] cancer.”

When I hear such results, my first question is, “How was the study conducted?” It appears that “Australian researchers questioned over 1,000 men who had developed prostate cancer and 1,250 who had not about their [past] sexual habits.” The problem with asking people about their sexual habits 30 years ago is that there are a large number of dead people you can’t ask. What if the most active masturbators have died from fatigue?

Should you masturbate more to reduce your cancer risk as the BBC suggests?

Prostate cancer might not be related to masturbation at all, but rather to something else that correlates with masturbation.

• It could be that men who have a higher libido have less prostate cancer.
• Or that men who have more free time have less prostate cancer.
• Or that men who are not depressed have less prostate cancer.
• Or that men who have higher speed Internet connections have less prostate cancer.

In case you are wondering how one’s Internet connection is related to all this, let me remind you of a joke about a conversation between two geeks.

— “When you look at a girl, what do you notice first?”
— “Her hair, then her eyes, then her nose, then her lips — I have dial-up.”

One thing I know for sure: women who masturbate have even less prostate cancer than men who masturbate. Hooray for masturbation!

The smallest positive index m such that the Fibonacci number F_m is divisible by the number p is called the rank of apparition of p. If p is prime, one can prove that any Fibonacci number that is divisible by p has an index divisible by m.

Even Fibonaccis have indices divisible by 3. That means that if for some p the rank of apparition of p is divisible by three, all the Fibonaccis that are divisible by p are even. Therefore, no odd Fibonacci divides p. I already discussed this subject in my previous post “9 Divided no Odd Fibonacci.”

Now let’s look more closely at the set of primes that divide no odd Fibonacci. The Fibonacci numbers obey the following identity: F_n+k = (1/2)(F_nL_k + F_kL_n), where L_n are Lucas numbers. From here F_3n = (1/2)F_n(L_2n + L_n²). Like Fibonacci numbers, exactly every third Lucas number is even. Hence, the parity of L_2n is the same as the parity of L_n. Hence, L_2n + L_n² is divisible by 2. Let us denote G_n = (1/2)(L_2n + L_n²).

As we have already discussed before, if no odd Fibonacci is divisible by p, p‘s rank of apparition is of the form 3n which means p divides F_3n and doesn’t divide F_n. Hence, p divides G_n. On the other hand, we can show that G_n = 5F_n² + 3(-1)ⁿ. Hence, the only common divisor that F_n and G_n can have is 3. Let us take any prime divisor s of G_n other than 3. We see that F_3n is divisible by s while F_n is not. The rank of apparition of s must be a divisor of 3n and not a divisor of n. Hence this rank is divisible by 3. Thus we can see that with the exception of 3, the set of prime divisors of elements of G_n is the set of primes that do not divide odd Fibonaccis.

Here’s a bit more info about the sequence G_n. It is sequence A047946 in the Online Encyclopedia of Integer Sequences. It is a recurrence: G_n=2*G_n-1+2*G_n-2-G_n-3. Thus, we have found a recursive sequence, elements of which have a set of prime divisors which with the exception of 3 is the set of primes that do not divide odd Fibonacci numbers.

Have you ever carried your mail from your mailbox directly to your recycle bin? If so, you feel my frustration with all this wasted paper. Each time I carry these catalogues up the stairs I think that my house should have a recycle bin next to my mailbox. Even better, maybe I can put my recycle bin at the post office, so my mailman doesn’t need to carry all this weight around. The real solution would be to call the catalog company and ask them not to send any more catalogs. For a long time the idea of being put on hold for a long time and the uncertainty of success, not to mention my usual laziness, prevented me from doing this.

You might imagine how high I jumped with joy when I heard about Catalog Choice at https://www.catalogchoice.org, the company that will request on my behalf a cessation of these mailings. Their website is nicely done and appropriately greenish. You just register and enter the catalogs you do not want. It doesn’t take much time and all the negotiation is done for you.

Up to this point, I’ve rejected 58 catalogs through this website. Only eight of them confirmed that they will honor my request. I even received a special confirmation letter from L.L.Bean. Sending a letter might be polite, but it contradicts my goal of reducing the waste of paper and of my time.

Unfortunately, there were three catalogs, including Newport News, that refused to honor my request. I was so angry that I decided to call Newport News and demand my removal from their mailing list. They agreed to my request. Can you guess what happened next? I received another catalog. I called them again. And I was removed from the list again. I received yet another catalog. I called them again. They told me that I am in their database marked as a person to whom they shouldn’t send catalogs. But the catalog they sent to me had a temporary customer number, so it is not in their system and therefore doesn’t count. According to Newport News’ logic, the check-mark in the database indicating that I am not supposed to receive catalogs supersedes the fact that I actually continue receiving them. They believe their database is more reliable than facts. What can I do? I will not buy from Newport News again.

Later, I made another interesting observation: I started receiving catalogs with my name misspelled. Do they really think that if they write Tonya instead of Tanya, I will suddenly become interested in their catalogs? We all know that the reason of the misspelling is so that they can pretend to honor my request, but still send me their catalog.

This situation is not right. I shouldn’t be getting catalogs against my will, especially after I’ve made my preferences clear. I think that we should fine companies that persist in doing this. Perhaps we can have a “Do Not Mail Me Catalogs” Registry that is similar to the government’s “Do Not Call” Registry.

Meanwhile, if you are like me and care about the environment, you too can sign up at the Catalog Choice.

I was very proud of myself when I started receiving tons of comments for my math blog. Many of the comments were quite flattering. For example:

You are getting better and better. Congrats, dude.

I know that in English you can sometimes replace “ladies” with “guys”, but I wasn’t sure if being called a dude constitutes a compliment.

I got suspicious, however, when I received this:

Get real! It is interesting, but you never give proofs.

This comment was placed on my Fibonacci entry, where I think I gave more than enough proofs. So I read all the small print that accompanied the comment. It appears that “sexy-girls-in-chains” were extremely excited about the divisibility of Fibonacci numbers.

I am used to my everyday winnings of Millions of Dollars in the UK lottery, but this was something new and different. They caught me off-guard. Here I was, so proud of the public’s positive reactions, only to realize that it was some automated program. Some computer sending spam caused me to have strong emotions. I was upset that I was caught. Sigh.

What can I do? Just laugh. We can laugh together at “girls-deflowered” who were interested in Quantifying Favors, at “is-your-penis-small” who commented on Losing the Lottery, at “squirting-vibrating-realistic-anal-dildo” who was impressed with my Fantasy Future and at “teenage-girls” who associated themselves with Numbers Needing Sponsors.

One of the latest comments was:

We can professionally say that this information is objective and true and of highest quality.

It was signed by “pissing-ladies.” Am I proud or what?

Here are recent additions to my math jokes collection:

* * *

During a lecture to his students, a military instructor says, “There is a 40% chance that we will hit our target.”
One student asks, “What happens if we aim away from the target?”
The instructor replies, “Logically, we would have a 60% chance of hitting the target.”

* * *

“Do you know that 67% of people are not capable of doing simple arithmetic?”
“I belong to the other 23%.”

* * *

What is so special about 6.9?
It is 69 ruined by a period.

* * * (submitted by Irene Ogievetskaya)

Teacher: Solve the equation: x + x + x = 9.
Student: x = 3, 3, and 3.

* * *

Teacher: What is 2k + k?
Student: 3000!