Tanya Khovanova's Math Blog

My Number Gossip website is becoming very popular. Recently the number of visits increased significantly. I decided to use my Google Analytics software to check the sources of this traffic.

The first spike in visits came after the link to my website appeared on the site PointlessSites.com. Never underestimate bad publicity.

But the real increase was due to the following description of my website at CollegeHumor.com:

Don’t even get me started on number 7! Number gossip. All the dirt on your favorite numbers, like 5.

The author of that quote is right. I did dig out some dirt on 5. Prime number 5 is secretly preparing to apply for a Nobel Prize in genetics. It is hiding its two twin brothers, 3 and 7, from the public eye, so that no one will notice that although 3 and 7 are twin brothers of 5, they are not twin brothers of each other. This situation is unique. While 5 has always been a member of two pairs of twin prime brothers, none of the tabloids has picked up on this.

And don’t even get me started on number 7! I haven’t put it on my website, but everyone knows that 7 is a cannibal, because seven eight nine.

After the link on CollegeHumor appeared, the number of visits jumped 20-fold. The dirt is as good as gold. Maybe I should rethink my approach and put more scandal into my website. Better yet, I should add sex to it. Actually, I already have sex on my website — as part of the word “sextuple”.

No, I won’t cheapen my numbers just for publicity. I will not add numerical sex positions 69, 71, and 88 to my website. If you’re interested in those, check out the cartoon above, which is the work of the famous geek Randall Munroe in his xkcd comic. But he’s such a true geek that he doesn’t really know what 71 is and probably never heard of 88. (Feel free to call me, Randall.)

My website is about math. I admit that I have sprinkled a few entries on my site that might be considered non-mathematical, but those are my guilty pleasures. Numerical sex positions, however, are not one of them.

Here is a baby puzzle. On Monday the baby said A, on Tuesday AU, on Wednesday AUUA, on Thursday AUUAUAAU. What will she say on Saturday?

You can see that this very gifted baby increases her talking capacity twice each day. The first half of what she says repeats her speech of the day before; and the second half is like the first half, but switches every A and U. If the baby continues indefinitely, her text converges to an infinite sequence that mathematicians call the Thue-Morse sequence (A010060). Of course, mathematicians use zeroes and ones instead of A and U, so the sequence looks like 0110100110010110100….

This sequence has many interesting properties. For example, if you replace every zero by 01 and every one by 10 in the Thue-Morse sequence, you will get the Thue-Morse sequence back. You can see that this is so if you code A in the baby’s speech by 01 and U by 10. Thus the Thue-Morse sequence is a fixed point under this substitution. Moreover, the only two fixed points under this substitution are the Thue-Morse sequence and its negation (A010059).

The Thue-Morse sequence possesses many other cool properties. For example, the sequence doesn’t contain substrings 000 and 111. Actually any sequence built from the doubles 01 and 10 can’t contain the triples 000 and 111 because we switch the digit after every odd-indexed term of such a sequence. A more general and less trivial statement is also true for the Thue-Morse sequence: it doesn’t contain any cubes. That is, it doesn’t contain XXX, where X is any binary string.

I stumbled upon this sequence when I was playing with evil and odious numbers invented by John H. Conway. A number is evil if the number of ones in its binary expansion is even, and odious if it’s odd. We can define a function, called the odiousness of a number, in the following way: odiousness(n) is one, if n is odious and 0 otherwise. We can apply the odiousness function to a sequence of non-negative integers term-wise. Now I can describe the Thue-Morse sequence as the odiousness of the sequence of non-negative numbers. Indeed, the odiousness of the number 2ⁿ + k is opposite of the odiousness of k, if k is less than 2ⁿ. That means if we already know the odiousness of the numbers below 2ⁿ, the next 2ⁿ terms of the odiousness sequence is the bitwise negation of the first 2ⁿ terms. So odiousness is built the same way as the Thue-Morse sequence, and you can easily check that the initial terms are the same too.

Let me consider as an example the sequence which is the odiousness of triangular numbers A153638: 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0…. What can we say about this sequence? We can say that the number of zeroes is infinite, as all the terms with indices of the form 2^n-1(2ⁿ+1) are zeroes. Also, the number of ones is infinite because all the terms with indices of the form 2^2n-1(2^2n-1-1) are ones.

Obviously, we can define the evilness of a number or of a sequence with non-negative terms. Namely, the evilness of a number is 1 if the number is evil, and 0 if it is not. The evilness is the bitwise negation of the odiousness. The evilness of the sequence of non-negative integers is the negation of the Thue-Morse sequence. The odiousness sequence of any sequence of zeroes and ones is the sequence itself, and the evilness sequence is its negation.

I would like to define an inverse odiousness operation on binary sequences. Many different sequences can have the same odiousness sequence. In such a case mathematicians usually define the inverse operation as a minimal non-negative sequence whose odiousness is the given sequence. Obviously, the minimal inverse of a binary sequence is the sequence itself, and thus not very interesting. I suggest that we define the inverse as a minimal increasing sequence. In this case the odiousness inverse of the Thue-Morse sequence is the sequence of non-negative numbers.

For example, let me describe the inverse odiousness of the sequence of all ones. Naturally, all the numbers in the sequence must be odious, and by minimality property this is the sequence of odious numbers A000069: 1, 2, 4, 7, 8, 11, 13, 14, 16, 19…. Analogously, the odiousness inverse of the sequence of all zeroes is the sequence of evil numbers A001969: 0, 3, 5, 6, 9, 10, 12, 15, 17, 18, 20….

Let us find the odiousness inverse of the alternating sequence A000035: 0, 1, 0, 1, 0, 1…. This is the lexicographically smallest sequence of numbers changing putridity. By the way, “putridity” is the term suggested by John Conway to encompass odiousness and evilness the same way as parity encompasses oddness and evenness.

The odiousness inverse of the alternating sequence is the sequence A003159: 0, 1, 3, 4, 5, 7, 9, 11, 12, 13…. By my definition we can describe this sequence as indices of terms of the Thue-Morse sequence that are different from the previous term. This sequence can be described in many other ways. For example, the official definition in the OEIS is that this sequence consists of numbers whose binary expansion ends with an even number of zeroes. It is fun to prove that this is the case. It is also fun to show that this sequence can be built by adding numbers to it that are not doubles of previous terms.

Let us look at the first differences of the previous sequence. This is the sequence A026465: 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2… — the length of n-th run of identical symbols in the Thue-Morse sequence. As we know that the Thue-Morse sequence doesn’t contain three ones or three zeroes in a row, we can state that the terms of this sequence will continue to be ones or twos.

You can define putridity sequences for any non-negative sequence. Which of them are interesting? I do not know, but I know which of them are not very interesting. For example, the putridity of pronic (oblong) numbers sequence is the same as the putridity of the triangular numbers sequence. This is because pronic numbers are twice triangular numbers and putridity is independent of factors of two. Another uninformative putridity sequence is the odiousness of the powers of two. This sequence consists only of ones.

I bet that my readers can find putridity sequences that are interesting.

I bought Richard a year and a half ago, and I bought him for his looks. Richard is slim, black and shiny. Richard is my first laptop, an HP Pavilion dv2000.

I stopped actively dating a while ago — I am quite happy with my life. So, I named my laptop Richard and he is my everyday male companion. Besides, Richard never complains that I didn’t do the dishes; in fact, he never even visits my kitchen.

Recently, Richard died. He just stopped booting. I went to Best Buy to try to fix him, but they told me that I could get a new one for the cost of his likely medical bills.

Why had I gone for the looks? Hadn’t I learned this lesson already with my men? This time I got a hold of myself and decided that, for my next laptop, I’m going to ignore looks altogether. My two priorities were reliability and compatibility with my operating system Ubuntu. Everyone suggested Lenovo Thinkpad. I went shopping. Even though Pavilion was still the most attractive, I turned my back on it and bought the Thinkpad.

I decided to name my new laptop Bond, James Bond. Now whenever I’m working, I am actually bonding. And, of course, his looks can be changed. I dressed up James. I gave him striking wallpaper featuring Pierce Brosnan, the most charming James Bond actor. Now James’ looks are improved, plus he is very reliable: he aims his weapon at me any time I want — all I need to do is close the windows.

While I was getting acquainted with this new man in my life, I stumbled upon an article about extended warranties from credit cards. Though my HP warranty expired, my extended warranty from the Visa card I used to buy Richard, was still valid. So I went to MicroCenter for a repair estimate. To my surprise it appeared that Richard could be recalled.

I probably should have returned James, but I wasn’t absolutely sure that Richard would be resurrected. Besides, I was scheduled to give a talk in Dallas for which I needed a laptop. So I kept James.

In a couple of weeks I got my Richard back. He is alive and well and without any hint of amnesia: no file was lost.

As I sit with my two men on my lap, I think back on all my relationships. I realize that often when I meet a new man, a second one unexpectedly appears in my life. In trying to find a reason for that, I came up with two different explanations:

1. When I am dating someone I am happy and glowy and, hence, I am more attractive.
2. When I first start dating someone, I feel insecure and nervous that I might be dumped. Thus, I need a backup to relieve my anxiety about potential pain. So, I make an extra effort and bring a second man into my life.

I have had many problems in my life, especially with men, but I went through two years of psychotherapy and changed my old patterns. I really do not need two men at the same time any more. I am not sure I need one man.

So why on earth did I end up with two laptops? Is this a sign that I need to go back to therapy? Let’s hope not. I just need to reject either Richard or James. But, whom?

Richard is an old friend. James is new and more powerful. Unfortunately, both of them are not without character flaws. Richard has energy problems: something is wrong with the battery. And James has communication problems: his wireless card is not working. I have two laptops and neither of them satisfies me fully. Maybe I should go back to therapy after all.

Recently I conducted an experiment. I wrote an essay “What’s Hidden?” in which I claimed that the essay had a hidden secret message in it. I coded the message using a very simple method — to read it you need to combine together all the capital letters in the essay.

The goal of the experiment was to audit intelligence agencies of different countries. I wanted to check if this essay would draw any special attention.

Intelligence agencies should crawl around the web and check places that might have secret messages. They might also want to sieve Internet data through some standard coding techniques and check if there are coded messages out there. But the Internet is so vast that most agencies might not have the resources to parse through all the web pages. They probably only analyze suspected pages.

Anyway, I wanted to see if my traffic for this essay would be different from the usual. I have a tool for that — Google Analytics, which provides aggregated geographical data of my traffic. Looking at the results I can see that the visits to this particular essay were mostly from the United States, with a few from Europe. The total number of visits was small, especially compared to my essay on masturbation.

If an intelligence agency has any intelligence it should hide its visits from Google Analytics and crawl around the web without being registered. For example, they can use cached Google pages.

So my intention in this experiment was to check for any agency that had so much time and money on their hands that they were monitoring the entire web and, at the same time, was dumb enough to leave a trail. I am happy to conclude that there is no such agency, with only one potential exception: my home country — the United States.

Disclaimer. I chose the jokes for analysis, not for entertainment. Some of them might be unpleasant to read. Proceed at your own risk.

Russians love jokes and especially politically incorrect ones. They have anecdotes about many nations, but when I was growing up back in Russia, Americans were treated nicely in these jokes. Americans were used as a background for Russians to laugh at themselves. Here is a very old joke about Russian political realities; this was the version that was told during Jimmy Carter’s presidency:

A Russian and an American are discussing freedom of speech. The American says, “We have freedom of speech. I can stand in front of the White House and shout that Jimmy Carter is an idiot and nothing will happen to me.c
The Russian replies, “I can also stand in front of the Kremlin and shout that Jimmy Carter is an idiot and nothing will happen to me either.”

Or this joke about harsh living conditions:

A Russian and an American describe their houses to each other. The American says, “I have a modest house, three bedrooms, a dining room, a living room, a family room, a kitchen, and a guest room.”
The Russian replies, “I have the same thing, just without the inner walls.”

The following old joke is becoming less and less amusing as the airline service in America increasingly resembles what Russians have always had:

An American is flying on a Russian flight. A beautiful stewardess approaches him and asks, “Would you like to have dinner?”
The American replies, “What are my choices?”
“Yes or No.”

What I am interested to learn from these jokes is whether Russia was preparing for a war with the US during the cold war period. In most cultures, killing a person is a bad thing to do. That means you need an argument to give to solders when you’re asking them to kill other people. Usually governments are not proposing that killing is a good thing to do in general, but rather rationalizing why this particular enemy needs to be killed. For example:

• We need to defend ourselves.
• It is needed for the greater good. For example, they are enemies of the state.
• They are less human than we are. For example, they are pigs or infidels.
• God told us to do this.

Looking at Russian jokes from that period, it’s clear that they weren’t trying to turn Russians against Americans. I therefore conclude that the USSR was not preparing an open attack on the US. Except in the case of a defensive war, one country planning to attack another would need to employ a process of preliminary distancing from their enemy.

The USSR government was capable of manipulating the populace to think that a war is a defensive war when in reality it was initiated by Russia. But no matter how treacherous they were, it would have been difficult to stage this trick across an ocean. My conclusion is that the USSR wasn’t really preparing any open attacks on the US. The whole cold war was a waste of energy, money and resources on both sides.

Now let’s move to modern times. There are still some American jokes in which Russians are laughing at themselves. For example, this one reminds us of the fate of Mikhail Khodorkovsky:

Question: Name the question to which Americans would answer “Sure!” and Russians “God, no!”
Answer: Would you like to change places with the richest person in the country?

The following joke appeared several years ago, and at first I classified it as Russians being over-proud of themselves:

“Can you describe a math department at an American university?”
“Yes, this is a place where Russian professors teach Chinese students.”

This situation changed recently. It moved from Russians being proud of their math traditions to stereotyping Americans as complete idiots. Here is a very recent joke:

Question: What do you call an intelligent person in the US?
Answer: A tourist.

Here is another joke that assumes complete cluelessness of Americans in geography and history:

Question: How do you recognize an American?
Answer: He is proud that the US won the Vietnam war with Hitler in Iraq.

Not only is the stereotype of Americans that they are stupid, but that they are stupider than everyone else:

Question: Dad, why Americans say “eighteen-ninety three” instead of “one thousand eight hundred ninety three”?
Answer: They still have eight-bit processors in their heads.

The situation with jokes about Americans is getting much worse in Russia. There are so many jokes about Americans that many joke websites formed a special category for American jokes. The speed at which the folklore changed from neutral to offensive was very fast. There is no doubt that it is supported from above. It is very useful for the Russian government to redirect the minds of their citizens from internal problems. Americans serve as a scapegoat.

Another thing Russians laugh a lot about is the idea of political correctness:

The new politically correct rules for chess in the US allowed blacks to make the first move.

Of course, a particular anecdote might not reflect the common opinion or, for that matter, the majority opinion. What is interesting is that there is a new trend. Here is a joke about consumerism:

Do you know that according to the latest American research the average speed of a woman walking around a shopping mall is $200 an hour?

The sad part is that the laugh about consumerism extends to a vision of the US government as being completely corrupt:

US — the government of the money, by the money, for the money.

Of course many jokes originated here and were translated from English. It is interesting which ones Russians chose to translate into Russian:

CNN said that after the war, there is a plan to divide Iraq into three parts … regular, premium and unleaded.

I do think that this newly found hatred comes from Putin and the present Russian administration. Russians are very much against the invasion of Iraq and many anecdotes reflect that:

Question: Mr. President, can you prove that Iraq has weapons of mass destruction?
Answer: Yes. We kept the receipts.

In general, the US is perceived as an aggressor.

George Bush found a way to invade every country — he declared war on Gypsies.

Is Russia preparing for war with the US? I do not know. Such a war seems like a ridiculous idea, but if you look at present-day anecdotes, it seems like Russia is doing more preparation than it did during the cold war period. First, Russia has been finding things to blame Americans for:

— What is the most powerful American chemical weapon?
— McDonalds.

Second, Russians are distancing themselves from Americans. Here is a very recent joke, where for the first time I saw that Americans are called “those people”:

Wild turkeys saved pilgrims from hunger. To commemorate this event, every year Americans kill and eat millions of turkeys as a thank you. God save us from doing anything good for those people.

For a long time I have checked the Russian joke websites before I go to bed. But in the last couple of years, I have become uncomfortable with the increasing antagonism towards America, so I decided to write this piece. Now that I finished it, I realized that, it’s sad, but we deserve many of the jokes.

I ran an experiment. I copied multiple choices from the 2007 AMC 8 into a file and asked my son Sergei to try to guess the answers, looking only at the choices. I allowed him to keep several choices. The score I assigned depended on the number of leftover choices. If the leftover choices didn’t contain the right answer, the score for the problem was zero. Otherwise, it was scaled according to the number of choices he left. For example, if he had four choices left and the right answer was among them he got 1/4 of a point. Here are the choices:

1. 9, 10, 11, 12, 13.
2. 2/5, 1/2, 5/4, 5/3, 5/2.
3. 2, 5, 7, 10, 12.
4. 12, 15, 18, 30, 36.
5. 24, 25, 26, 27, 28.
6. 7, 17, 34, 41, 80.
7. 25, 26, 29, 33, 36.
8. 3, 4.5, 6, 9, 18.
9. 1, 2, 3, 4, cannot be determined.
10. 13, 20, 24, 28, 30.
11. Choose picture: I, II, III, IV, cannot be determined.
12. 1:1, 6:5, 3:2, 2:1, 3:1.
13. 503, 1006, 1504, 1507, 1510.
14. 5, 8, 13, 14, 18.
15. a+c < b, ab < c, a+b < c, ac < b, b/c = a.
16. Choose picture: 1, 2, 3, 4, 5.
17. 25, 35, 40, 45, 50.
18. 2, 5, 6, 8, 10.
19. 2, 64, 79, 96, 131.
20. 48, 50, 53, 54, 60.
21. 2/7, 3/8, 1/2, 4/7, 5/8.
22. 2, 4.5, 5, 6.2, 7.
23. 4, 6, 8, 10, 12.
24. 1/4, 1/3, 1/2, 2/3, 3/4.
25. 17/36, 35/72, 1/2, 37/72, 19/36.

It is clear that if you keep all choices, your score will be 5, which is the expected score for AMC if you are randomly guessing the answers. Sergei’s total score was 7.77, which is noticeably better than the expected 5.

There were two questions where Sergei felt that he knew the answer exactly. First was question number two with choices: 2/5, 1/2, 5/4, 5/3, 5/2. All but one of the choices has a 5 in it, so 1/2 must be wrong. Numbers 2/5 and 5/2 are inverses of each other, so if organizers expect you to make a mistake by inverting the right answer, then one of these choices must be the right answer. But 5/4 and 5/3 are better suited as a miscalculation of 5/2, than of 2/5. His choice was 5/2, and it was correct. The second question for which he was sure of the answer was question 19, with his answer 79. I still do not have a clue why.

Sergei’s result wasn’t too much better than just guessing. That means that AMC 8 organizers do a reasonably good job of hiding the real answer. You can try it at home and see if you can improve on Sergei’s result. I will publish the right answers as a comment to this essay in a week or so.