Tanya Khovanova's Math Blog

Nowadays, supporting a database is cheap. Computer storage is cheap, too. This gives companies an opportunity to collect more and more information about us. If you think, as I do, that this is an invasion of your privacy, here are some ways to resist.

When you buy something over the Internet or through a catalog, they ask for both your email address and telephone number. They may need a way to contact you in case something happens with your order, but they do not need both. When you are ordering online and their default contact is by email, they do not need your phone number. If the website requires your phone number, you can put in a fake number. Of course, you are a nice person and do not want to provide some innocent soul’s phone number instead of yours. Here is the perfect solution. Put the number 555-5555 as your home number with any area code.

The phone numbers of the format 555-xxxx are reserved for the movie industry. That is, if Hugh Grant calls Julia Roberts in a movie, there would be hundreds of bored or not very smart people who would try to call the same number Hugh dialed hoping to talk to Julia Roberts. For these situations, the movie industry reserves all the numbers of the form 555-xxxx. This way they guarantee that all of these fans will not bother a real person. So you can use these numbers without any guilt.

If you are ordering by phone, they might see your number on the caller ID. In this case, you can always say that you do not have an email address. You can also use a one-time email address offered through Sneakmail or AddressGuard at Yahoo.

Store shopping cards also scare me very much. When you use your store shopping card, they know exactly what and when and in what amounts you are buying. If you do not want anyone to know that you are buying 100 Tylenol pills a month, do not use your store card, and consider paying cash.

My friend Sam Steingold suggested I try card swapping. You have a CVS card and your friend has a CVS card — you can swap them. CVS’s database will register that you quit buying Tylenol in Boston, but started buying cigarettes in Atlanta. If you continue swapping, CVS’s database will be totally confused. The good part of this idea is that if someone tries to hold your purchases against you, you have a way to prove that you are not responsible.

The disadvantage of card swapping, is that for the transition time you lose targeted coupons. Your friend in Atlanta will get all the Tylenol coupons he doesn’t need. But you still will be able to buy sale items with discounts.

Here’s what I did – I put another last name on my CVS card. They didn’t notice. If they were to notice, I would have told them that I am in process of changing my last name to my newly acquired husband’s last name and would ask for newlyweds’ coupons.

Sometimes when you buy things, they ask you for your phone number at the cash register. It is even worse than shopping cards. They have your information on file without giving you your discounts. Just remember: you can always refuse. Or if you’re not comfortable refusing, let us all agree to give the same number: (area code)-555-5555. Let their analysts wonder why the same person is buying morning-sickness pills in one store and condoms in another.

My Number Gossip page is on hold. I am out of work and can only afford to spend time on things that can bring me my next job. As a result, numbers suffer. If you would like to support the website, consider donating to Number Gossip. With your donations, I will be able to spend more time on the website. There are many things I would like to do. Here are the three most important and fun areas that I would chose to work on.

1. Do you know that 40(forty) is the only number whose constituent letters appear in alphabetical order? I have about 1,000 unique number properties like this on my number gossip website. I also have about 600 properties in my database that need to be checked before adding them to the website. Being a perfectionist I double-check and recheck all the properties. As a result, my webpage is quite rigorous and reliable. As a downside I need a lot of time to add new unique properties to the website and 600 of such properties are waiting impatiently to join their comrades in the public view.
2. Have you heard about aspiring or untouchable numbers? What about practical or perfect numbers? There 50 famous properties like that that my website currently allows you to check for your favorite number. For example, 33 is deficient, evil, lucky and odd at the same time. There are many other properties I plan to add in the future. If you would like to know whether or not your favorite number is brilliant, fortunate or primeval, consider donating to Number Gossip.
3. Currently the limit for numbers you can input is 10,000. Some of the basic properties are very difficult to check. The expansion is not trivial and will require significant tuning of my algorithms. Even so, I would like to do that.

I’ve devoted many years to this project, and now I need some financial help. If you know a person or a company who wants to sponsor numbers, please ask them to contact me at (tanyakh at yahoo).

Alice and Bob are good friends. Bob caught a cold and called Alice for help. He wanted Alice to go to a pharmacy and bring him some cold medicine. Alice did that and I would like to assign a number to this act of giving. How can we quantify this favor?

First, we need to choose a scale. Usually favors cost us in time, money and emotions. Alice spent half an hour driving around, plus $5 on the medicine (we’ll skip the cost of gas for simplicity). It also cost her emotionally, especially because the traffic was really bad.

Measuring everything on three scales is complicated. I would like to convert everything to one scale, because in the future I intend to compare this act of giving to other favors Alice does. For example, Alice knows for sure that this favor for Bob was a less costly favor than her phone call yesterday to her ex-mother-in-law, even though the phone call took only five minutes and didn’t cost any money.

We probably can convert everything to dollars, but I am trying to resist this money-driven society that measures everything in dollars. So, I prefer to use points. Each dollar translates to one point, but time and emotion are more subjective.

Alice makes two calculations in her head: what she really spent and what she is owed.

Here’s what she spent: Alice counts 5 points for the medicine. She also views her time as money. She charges $100 an hour for consulting and values all her time at this rate. Hence, she adds 50 points for time spent. Traffic was bad, but not so bad. She thinks that her traffic stress cost her 15 points. Since she also had to cancel her date with her boyfriend, she estimates her annoyance with this at 100 points. On the other hand, she got this warm feeling from helping Bob and she was happy to see him. So she thinks that she got back 30 points. Adding all this up, we get a total of 140 points. This is how much Alice thinks she spent for this particular favor.

Does it mean that Alice thinks that Bob owes her 140 points? Usually not. The calculation of how much Alice thinks Bob actually owes her is completely different. She thinks that he owes her 5 points for the cost of medicine. Also, she knows that Bob earns much less than she does and values time differently, so she think that he owes her 30 points for her time. Since Bob is not responsible for traffic, she doesn’t add traffic points. Also, she never told Bob that she had to sacrifice her date for him, so she doesn’t think it’s fair to want Bob to be thankful for the sacrifice he doesn’t know about. At the same time she hopes that one day Bob will sacrifice something for her. She can’t ignore this sacrifice completely, so she adds 10 points for that. Altogether she thinks that Bob owes her 45 points.

Do you think Bob feels as if he owes Alice 45 points? Like Alice, he also has two numbers in his mind. One number is the amount of points he received as a result of this favor and the other number is how many points he officially owes Alice.

He actually was planning to ask his neighbor to buy the medicine, but for some reason he called Alice first and she offered help. Alice was delayed at her work and arrived at Bob’s place much later than he expected. She also brought the worst flavor of the syrup. Bob doesn’t value time as much as Alice, so he thinks that Alice spent 10 points driving and 5 points on the medicine. Bob felt ill throughout Alice’s visit and did not enjoy seeing her. Combining that with her late arrival with the wrong syrup, he thinks that he was annoyed for about 15 points. So he thinks that he got zero points from this transaction.

At the same time he wants to be fair. Bob knows that Alice did her best to help him; besides he never specified the flavor he likes. As a result, he doesn’t count his annoyance in how much he owes Alice. So he thinks that he owes Alice 15 points. What Bob really did to thank Alice, I will discuss in a later blog entry.

In conclusion, let me remind you of my system. I measure all favors in points. And for each favor I assign four numbers:

• the giver’s official favor value (in our example 45 points)
• the giver’s hidden favor value (in our example 140 points)
• the receiver’s official favor value (in our example 15 points)
• the receiver’s hidden favor value (in our example 0 points)

My favorite puzzle at 2008 MIT Mystery Hunt was the puzzle named Functions. Here is this puzzle:

36 -> 18      A,B
2 -> 1        A,C,G,H,K,L,O
512 -> 256    A,C,H
4 -> 2        A,G,H,Q
320 -> 160    A,R
411 -> 4      B,E,Q
13 -> 3       B,G,K
88 -> 11      C,D
45 -> 9       C,D,F,J,L
48 -> 6       C,G,M,P,Q
4 -> 1        C,K,L,N,O
36 -> 9       D,E,F
66 -> 8       D,E,G,I
10 -> 3       D,G,L
1 -> 3        D,L
150 -> 15     D,M
3 -> 2        E,H,J,K
25 -> 3       E,K,L,N,Q
9477 -> 14    E,M
129 -> 4      E,N,P
55 -> 10      F,J
411 -> 6      F,K,L,M,N
2002 -> 4     F,O,Q
79 -> 8       G,I,L,P
25 -> 20      H,M
176 -> 80     H,R
3665 -> 8     I,N,Q
7 -> 3        K,Q
11 -> 5       L,M
501 -> 2      L,O,P,Q
8190 -> 5     M,O
180 -> 3      O,P
50 -> 10      R

? -> (?)      F,R
(?) -> ?      J,L
(?) -> ?      A,F
(?) -> ?      N,O,Q
? -> (?)      A,D,J
(?) -> ?      D,H
(?) -> ?      G,K,Q
? -> (?)      B,D,M
(?) -> ?      E,H
? -> (?)      D,F,G,L
? -> (?)      C,G,P

I had a dream that sometime in the future I am babysitting my two-year old granddaughter-to-be Inna.

Me: Here is one apple; here is another apple. How many apples will you get when we put them together?
Inna: Two.
Me: So, 1+1 is… ?
Inna: Two.
Me: Good girl.
Inna: How much is 2+2?
Me: Shhh. We can’t talk about that.
Inna: Why? Will a big bad wolf come and eat us?
Me: Sort of. It is copyrighted and I do not have enough money for the private use license.
Inna: Did you spend all your money on a 1+1 license?
Me: No, honey. Google owns the rights and they released it for public use.
Inna: What about 3+3?
Me: We might be able to talk about it in a couple of years. The government is discussing the purchase of the rights, though it would be half of their annual education budget.
Inna: What about 4+4?
Me: 4+4 is approximately 8.
Inna: Don’t you know if it is 8? Do you think it could be 7?
Me: No, I know exactly how much it is. But the copyright has a loophole. You can’t say the exact sentence, but it doesn’t forbid variations. Have you heard that Stephen Colbert is being sued for saying how much 4+4 is on his show? Colbert argues that his grimace constitutes a complete reverse in meaning.
Inna: What about 5+5?
Me: Your father’s brother’s nephew’s cousin’s former roommate is a lawyer and he says that the 5+5 license doesn’t permit the answer to be in the same sentence as the statement. So, to be on the safe side, you should always go like this: “5+5 is a number. I like ice cream. My favorite flavor is chocolate. The number I’ve mentioned several sentences ago is 10.”
Inna: Can this lawyer find a loophole in the 2+2 license?
Me: The copyright agreement itself is copyrighted and too expensive for him.

Later my son, Alexey, comes to pick up his daughter. I continue the conversation with him.

Me: Your daughter is gifted in math. Is there any chance that her public school can teach her how to add 10+10?
Alexey: I know that our public school bought a limited license. They can discuss additions only in a designated room on Mondays from 11:00am to noon and only after 8th grade.
Me: Why the heck wait until 8th grade?
Alexey: They are required to study copyright laws first and pass the state exams.
Me: Have you considered private schools? Inna is so gifted — she might even get a scholarship.
Alexey: Our private school was able to copyright only three questions for their scholarship evaluations. And everyone knows that the answers are A, D and D.
Me: I have an idea. I am subscribed to Russian TV. They have a channel that broadcasts an educational math show in English.
Alexey: How could that be? The U.S. blocks all foreign non-copyrighted broadcasts in English.
Me: Their English is so bad, everyone thinks it is French.
Alexey: Ah, I was wondering where my boss’s son got his new horrifying accent.

At Inna’s next visit, Inna came up with some ideas.

Inna (in a low voice): I think I know how much 2+2 is.
Me: You can’t tell me that. But maybe you have a new favorite number. You can tell me your favorite number.
Inna: My favorite number is 4.
Me: Do you know how much 3+3 is?
Inna: I changed my mind. My new favorite number is 6 now.
Me: Good girl.
Inna: How come we are talking about addition and you never told me what the number after 10 is?
Me: Shhh. We can’t talk about that… —

This is my first non-mathematical entry. But I invented this diet myself two days ago and I wanted to share it with you. It is a variation on my son Alexey’s “Am I hungry?” diet.

The only restriction for my diet is that you are not allowed to eat while your brain is busy with something else, like watching TV or playing sudoku.

If you are driving to your office while reading a newspaper and talking on the phone, you are allowed to have your morning donut, but only if you stop the car and put away your newspaper and phone. This diet is based on having an undistracted dialogue with your food.

Here how this diet works. Each time you open your mouth to take a bite, you should look at your food and ask yourself, “Why the heck do I need this bite?” This is it. Just look and ask. Nothing more.

It is better if you say it aloud. But if you are on a first date you are allowed to pronounce it in your head.

Here is what happened to my cake yesterday. On the first “Why the heck do I need this piece of cake?” I just ate one bite. On the second “Why the heck do I need this piece of cake?” my inner voice told me, “Shut up. I just want it.” On the third “Why the heck do I need this piece of cake?” my inner voice said: “Well, I am stressed out and I really crave some sugar. Besides, today is my last day at work, so I am allowed to celebrate.” On the fourth “Why the heck do I need this piece of cake?” I just put the piece of cake back in the fridge. I didn’t want it anymore. Altogether, I ate a third of my usual portion of cake. It works.

Try it. This diet is free. It is easy to remember. You do not need to change your lifestyle, go to the store to buy fresh vegetables or adopt new recipes. It might increase your morning commute time by one minute. But you can recover this minute by cutting down on exercise, since you won’t need it quite as much.

There is a big difference between evaluating exercise DVDs and reviewing movies. You are supposed to use exercise DVDs many times. So the value of the DVD changes over time. An exercise DVD that is too difficult at the first try could become a lot of fun later. Alternatively, one that explains everything in detail can be great at the beginning, but it will become boring after several viewings.

Smart DVD producers probably know some common rules. The number of people who use a DVD for the first time is much bigger than the number of people who use it for the hundredth time. Users often post reviews and ratings of products they have bought. Therefore, the proportion of reviews by the first-time watchers is much higher than by the hundredth-time watchers. This means that to get better ratings the DVD producers should target the first-time watchers. Is this why we have so many boring exercise DVDs?

In my opinion, exercise DVDs should have two parts. One part explains everything by breaking the routine down into elements and the other part allows people who have learned the routine to do it without interruption.

Keep your eyes open for my upcoming web page with reviews of dance exercise DVDs that I own. These reviews will address both first-time users and every-day-for-a-year users.

Everyone knows that math education in public schools in this country is pathetic. If you looked at this problem from an economics point of view, the first question would be, “Qui prodest?.”

Who profits from bad math education? I know one place — the lottery. People who understand how the lottery works rarely buy tickets. They might buy an occasional ticket as entertainment, but never as an investment. No wonder they say that the lottery is a tax on people bad at math.

Huge money from lotteries goes to states and towns, and a big portion of that goes to education. That means towns, schools and math teachers have direct financial incentive not to provide good math education. This conflict of interest creates a situation in which, in the long run, it is profitable for schools to hire very poor math teachers or cut their math programs.

The situation is unethical. I think that lottery organizers should at least pretend that they are resolving this conflict and spend part of the lottery money to educate people not to play the lottery.

I did it. I handed in my resignation letter to my boss. I’m resigning effective Jan 3, 2008. If you want to know why I’m waiting until next year, I can give you several reasons.

• First, Christmas-time is usually the most enjoyable work time because no one is there. It is quiet.
• Second, I’m superstitious: I believe the way I greet the New Year determines how the New Year is going to go, so I want to be employed at the strike of the midnight clock.
• Last but not least, it appears that to get my company’s annual profit-sharing bonus, I have to be employed on December 31^st.

I am happy and sad at the same time. In four and a half years I’ve made a lot of friends and accomplished a lot professionally. Now it is my time to move forward. Where is forward? It is in the direction of a cemetery, but I would rather be doing something more meaningful to me than battle management while I am slowly crawling there.

Do you know that 1210 is the smallest autobiographical number? You probably do not know what an autobiographical number is. You are right if you think that such a number should be a pompous self-centered number whose only purpose in life is to describe itself.

Here is the formal definition. An autobiographical number is a number N such that the first digit of N counts how many zeroes are in N, the second digit counts how many ones are in N and so on. In our example, 1210 has 1 zero, 2 ones, 1 two and 0 threes.

Let us find all autobiographical numbers using the “zoom-in” method.

1. By definition, the autobiographies can’t have more than 10 digits. It is nice to know that these egotistical numbers can’t be too grand.
2. The sum of the digits in an autobiography equals the number of the digits. Consequently, the sum of the digits will not be more than 10.
3. The first digit is the number of zeroes. As you know, self-respecting integers do not start with a zero. Hence, the number of zeroes is not a zero.
4. Subtracting statement “c” from statement “b” above, we get a resulting statement that the sum of all the digits, except for the first one, is equal to the number of non-zero digits plus 1.
5. That means, other than the first digit, the set of all other non-zero digits consists of several ones and 1 two.
6. Furthermore, the number of ones is either 0, 1 or 2.

Now we continue zooming in in three different directions depending on the number of ones. In this blog entry, I will consider only the case in which there are no ones; I leave the other two cases to the reader.

• If the number of ones is zero, then the only non-zero non-first digit of such a number is 2.
• This 2 should be included in the autobiography; since the third digit of the number is not zero, it must be 2.
• The number has 2 twos.
• It must be 2020.

Here is the full set of autobiographical numbers: 1210, 2020, 21200, 3211000, 42101000, 521001000, 6210001000.

This is the sequence A104786 in the Online Encyclopedia of Integer Sequences (OEIS), where I first encountered the autobiographical numbers.

Autobiographical numbers are very cute numbers. But there is a problem with their name. If there is a notion of an autobiography of a number, then it would be logical to expect that there is a notion of a biography of a number. What would be the logical candidate for a biography of a number? Let us say that given a number N, its biography is another number M such that the first digit of M is the number of zeroes in N, the second digit of M is the number of ones in N and so on.

Of course, for a number to have a biography, we need to assume that none of its digit is present more than nine times. Still there are several problems with the definition of a biography.

The first problem is that if N doesn’t have zeroes, its biography starts with a zero. As numbers don’t start with 0, that biography is not a number! Furthermore, if N starts with 0, it can have a biography but N is not a number. Luckily for this article, a digit string starting with zeroes can’t be an autobiographical string, because the number of zeroes is not a zero. It is a relief that those illegitimate strings that are trying to pretend to be numbers can’t actually be autobiographical.

The second problem with biographies is that a number can have many biographies. Indeed, if a number doesn’t have nines, you can remove or add zeroes at the end of a biography to get another biography of the same number. Since mathematicians like to define things uniquely, we might consider it a problem if a number has several biographies. In real life it is possible to have many biographies of a person. So the second problem is not a big problem. I will call the shortest possible biography of a number the curriculum vitae and the longest possible biography the complete life story.

The third problem is that numbers with the same digits in different permutations have the same biographies. So in a sense a biography follows the life not of a number, but rather the set of its digits.

Suppose for now we allow a biography to start with 0. Also, let us choose the curriculum vitae — the shortest biography in case there could be several. Let us build a sequence of CVs. As an example, we start with 0. Zero’s CV is 1, one’s CV is 01, continuing that we get the following sequence: 0, 1, 01, 11, 02, 101, 12, 011, 12, 011, 12, …. You can see that the CVs’ sequence fell into a cycle in this case. I tried sequences of CVs starting with many numbers. I found that they fall into two cycles. One cycle is described above and another one is: 22, 002, 201, 111, 03, 1001, 22. Can you find another cycle or, alternatively, can you prove that all the numbers that allow the sequence of CVs converge to only these two cycles?

Let us build the sequence of complete biographies, that is, life stories, starting with 0: 0, 1000000000, 9100000000, 8100000001, 7200000010, 7110000100, 6300000100, 7101001000, 6300000100, …. We see that this sequence falls into a cycle of length two. The members of this cycle are legitimate numbers. These numbers are too shy to advertise themselves. But Alice praises Bob, because Bob praises Alice. It’s a very advantageous flattery pattern! I will call such a pair a mutually-praising pair. We’ve already seen mutually-praising strings: 12 and 001. Two other examples of number pairs thriving on each others’ compliments are, first, 130 and 1101, and second, 2210 and 11200.