by Julian Havil. I opened this book to a random page and it was Chapter 13. That was not such a big deal by itself, but in addition Chapter 13 was titled "Friday the 13th".
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I was waiting for a train in Newark. On the platform, there was an LCD screen that flashed advertising. The ad I was staring at was titled, "Reasons to take NJ Transit to Prudential Center, #15." I was impressed that the NJ Transit sales people were working so hard to invent that many reasons.
I waited for my train for half an hour. It turned out that the NJ Transit advertising people were not working hard after all. The screen was flipping between four reasons, numbered 3, 6, 12 and 15. This is a case of false advertising. You look at reason number 15 and think that there must be a lot of reasons. They fool with your head. Cheaters.
I hope you noticed that I did the same thing with this posting — purely in order to illustrate my point.
Here is my simplified version of Conway's wizards puzzle.
Last night when I was coming home from my writing class with Sue Katz, I sat behind two wizards on the bus, and overheard the following:
— Wizard 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 the amount of dollars I have in my pocket."
At this point I interrupted the wizard. "Excuse me, professor, I overheard your conversation and can't resist asking you a question. Usually when a father says 'my children' everyone assumes that he has at least two children. Can I assume that?"
— Wizard A: "No. I stated my assumptions up front. A positive integral number of children means one or more."
I started thinking. If I were to explain this to a non-mathematician who assumes that 'my children' means more than one child, I would need to change the wizard's statement into the following:
"I have at least one child. The ages of my one-or-more children are all positive integers. The sum of the ages of my children or the age of my only child is the number of this bus. The product of the ages of my children or my only child's age is the amount of dollars I have in my pocket."
Hmm. I like that mathematicians use 'my children' to indicate any number of children. Makes puzzles faster to type.
Anyway, the wizards continued their discussion:
— Wizard B: "How interesting! Perhaps if you told me the number of your children, I could work out their individual ages"
— Wizard A: "No."
— Wizard B: "Aha! AT LAST I know how many children you have!"
If I were John Conway, I would have asked you next, "What is the number of the bus?" As I am not John Conway, I'll ask you, "Why do we presume that Wizard A hasn't cheated on his wife?"
The answer is that all wizards are notorious for making precise statements. If he cheats a lot, he would have started the conversation with, "The number of children I know about is a positive integer." Or maybe, more discreetly, "My wife and I have a positive integral number of children."
If you have already figured out the number of the bus, the bonus question is, "Why did I change the 'age of the first wizard' in Conway's original puzzle into the 'amount of dollars' in my puzzle?"
When I left the bus, I started wondering why on earth anyone would ever want to sum up the ages of their children. And I remembered that I once did it myself. I was trying to persuade my sister to apply for U.S. citizenship. My argument was that by moving here the life expectancy of her children would increase by 30 years. Indeed, she has two sons and the male life expectancy in Russia and the U.S. has an astonishing 15-year difference. I have to admit that my argument is not very clean, as we do not know the causes for this difference and, besides, the data is for life expectancy at birth and it changes while our kids age. My sister dismissed my argument, saying that the low male life expectancy in Russia is due to alcoholism and that her family is not in the high-risk group.
So, there could be a reason to sum up the ages of your children, but why would anyone ever want to multiply the ages of their children? In any case, if the first wizard continues to keep the amount of dollars equaling the product of the ages of his children in his pocket, his pocket will do better than mutual funds for the next several years.
I know I'm a computer addict, because:
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.
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:
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 &mdash 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.
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 Fm 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: Fn+k = (1/2)(FnLk + FkLn), where Ln are Lucas numbers. From here F3n = (1/2)Fn(L2n + Ln2). Like Fibonacci numbers, exactly every third Lucas number is even. Hence, the parity of L2n is the same as the parity of Ln. Hence, L2n + Ln2 is divisible by 2. Let us denote Gn = (1/2)(L2n + Ln2).
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 F3n and doesn't divide Fn. Hence, p divides Gn. On the other hand, we can show that Gn = 5Fn2 + 3(-1)n. Hence, the only common divisor that Fn and Gn can have is 3. Let us take any prime divisor s of Gn other than 3. We see that F3n is divisible by s while Fn 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 Gn is the set of primes that do not divide odd Fibonaccis.
Here's a bit more info about the sequence Gn. It is sequence A047946 in the Online Encyclopedia of Integer Sequences. It is a recurrence: Gn=2*Gn-1+2*Gn-2-Gn-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 http://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!
I stumbled upon an article in the Boston Globe (February 29, 2008) titled "Study: Spanking children affects their sex lives as adults." Here are some quotes:
New research by a University of New Hampshire domestic abuse expert says spanking children affects their sex lives as adults. …[C]hildren who are spanked are more likely as adults to coerce partners to have sex, to have unprotected sex and to have masochistic sex.
The classical method to prove scientifically that spanking affects something is to find many parents of newborn identical twins and persuade them to treat their children the same way, with the exception that they spank one and do not spank the other. The researchers should then compare these twins in their adulthood. Such a project is impossible, as parents are likely to start feeling guilty towards the child they spank, for you can't separate spanking from the whole package of how parents treat their children.
I decided to study the study. I found a more detailed description of the study in the Concord Monitor. As I suspected, the study was a survey. The survey found a correlation between spanking and "undesirable" sexual behavior. As every statistician knows, correlation doesn't prove causality.
Here's another quote from the study: "The best-kept secret in child psychology is that children who were never spanked are among the best behaved." Did it occur to anyone that the best behaved children do not need spanking?
Could it be that parents who spank their children are tired, impatient and less loving? Could it be that not being loved as a child affects your sexual behaviour as an adult much more than spanking?
Could it be that parents who spank their children are more aggressive in general? Could it be that they pass their aggressive genes to kids and their kids' aggressive behavior is related not only to upbringing, but to genetics?
Do not get me wrong. I do think that spanking is bad. I am saying that the study doesn't prove that spanking is affecting anyone's future sex life.
I am surprised that so many magazines republished the article without thinking. Now all the country is fooled into believing that the easy way to improve their kids' future sex life is to stop spanking.
Go ahead! Stop spanking. Love your children too.
I tried to enroll on a website recently, but they didn't allow me to continue without choosing five security questions out of about ten samples they supplied. I started in good faith to do what they asked.
Question: What is your father's middle name?
Answer: They do not have middle names in Russia; they have something called "otchestvo" and I know seven different ways to spell my father's.
Question: What is the name of the street on which you were born?
Answer: I am glad it was not Lenin Street, but it was equally bad. Besides, it was renamed and I am not sure which name to choose.
Question: What is the name of your high school?
Answer: Finally, an easy question. In Russia we didn't have names, but rather numbers for schools. I happily entered 444, and oops — the applet wouldn't accept numbers.
I couldn't find five questions that I could answer uniquely and reliably. I felt that the designers of these questions were clueless and disrespectful to other cultures. Then I thought about whether I really wanted some creepy database to know the name of my best friend. No, I didn't.
Now I have established a file where I put the answers to security questions and I can have all the fun I want with my new biography. I can name my first dog Tom Cruise and have my wedding date be 20 years before I was born. I can name my husband Freedom Of Speech and my city of birth IHateSecurityQuestions. Maybe next time I will switch: Freedom Of Speech will be my dog and Tom Cruise my husband.
If you are lazy like me, you can choose your questions so you have the same answer for everything. This way you do not need to type much into your file. For example, you can name your city, your cat and your best friend George Washington. Or, if you are really lazy, God.
I just read the following in Women's Health Magazine (March, 2008; page 54): Visa conducted a study of 100,000 fast food restaurant transactions. They found that people who pay with credit cards spend 30% more on food than people who pay cash.
The article concludes with the suggestion to pay cash, so you spend less and lose weight.
My question is: Who is more incompetent, Visa or Women's Health Magazine?
Perhaps people who do not have credit cards are poorer and more price-conscious; hence, they spend less on food. This might explain the correlation. Here's another possible explanation: people who are ordering for large groups might prefer to pay with a credit card. Or, maybe stores do not like using credit cards for small transactions, so they encourage people to pay cash for modest orders.
The main rule of statistics is that correlation doesn't mean causality.
There are several possible answers to my question about incompetence:
It could well be that paying cash makes you stingier, or at least more price-conscious, but I can't trust Women's Health Magazine any more. One thing I know for sure is that math can help you lose weight. Math allows you to differentiate a good study from a dumb study.
Recently I was buying gas at a gas station. I was paying with my credit card and the machine asked for my zip code. If you read my previous post, you know what happened: I got furious. First, I tried 00000 as a zip code. The machine was smart enough to tell me that it was invalid. No matter what combination I tried, it wouldn't accept another zip code.
Finally, the machine got frustrated with me and printed on its screen that I needed to talk to the cashier. Then, I argued with the cashier. He could have suggested that I pay cash, but he didn't. I left without gas.
Luckily, for me I had dinner that day with my son, Alexey Radul, and his friend, Grem. Grem explained to me that gas stations are not collecting customers' zip codes. They use zip codes as a security measure for checking that the credit card is not stolen. I guess that means I behaved exactly like a person who stole a card. Protecting your own privacy sometimes makes you look like a thief.
After dinner, I went back to the same gas station to conduct some experiments. This time I was armed with several valid zip codes. It didn't work. Grem was right — only the zip code corresponding to the card could have worked. I looked like a thief again. I paid for my gas with cash and left.
The next day I called my credit card company and asked them about this. The manager I talked to told me the same thing as the manager Grem talked to. Apparently credit cards do not give your zip code information to gas stations. Your zip code is used instead of a pin number to verify that your credit card is not stolen. So in this case you do not need to worry that you are providing extra information, since credit card companies know your zip code anyway.
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.
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:
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.
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.
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.
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). The 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.
I would like to become a professor of mathematics. How can I get to academia? I was told that applicants are measured by the number of papers they write. They expect about 3 papers per year starting after the Ph.D. I got my Ph.D. 20 years ago. I published 6 papers after my Ph.D. papers. That means I urgently need to come up with 54 papers.
There are several problems with working in industry and trying to publish at the same time:
Because of these obstacles, the papers I have started at my job are on hold. It's unlikely that I'd be allowed to finish them during work hours.
So I started writing non-job-related papers on my weekends. I started doing this seriously a year ago. It goes very slowly and I hope to publish three papers soon, but my speed needs to be much higher than that to catch up with the 54 papers I didn't have time to write while being a single mom and providing for my family.
So, I came up with this idea: to quit my job and write papers. I do not have enough money to support this idea for very long. Certainly, not enough time for 54 papers. We probably can survive on my savings for half a year. My goal is to write as many papers as I can in half a year and see what my real speed is. This way I can at least prove to myself that I am a mathematician for real.
The only problem is that my savings were meant for a down payment on my first house. I've been asking myself for awhile: What is more important− a dream job or a dream house? I just realized today that I will never be happy if I am not happy at my job and I am quite happy with the apartment I am renting now. I guess this is it − I just have to take the plunge.
Are you afraid of Friday the 13th? Here is my only Friday the 13th story.
It was Friday the 13th and I was listening to the psychologist Joy Browne's show. Joy asked her listeners to call in with stories of interesting things that happened to them on Friday the 13th. I wondered why I didn't remember any stories about Friday the 13th.
At the end of her show, I went to pick up my mail, where I found a book kindly sent to me by Princeton University Press named Nonplussed!: Mathematical Proof of Implausible Ideas
One of the things Julian Havil discussed in the chapter is how often the 13th of the month falls on different days of the week. You might remember from your elementary school education that the Gregorian calendar repeats an entire identical day-of-the-week cycle every 400 years. Hence, it is just a matter of calculation to check on which day of the week the 13th falls the most often.
Can you guess the answer? I am sure you can apply some meta-thinking and derive that there is one special day of the week on which the 13th most frequently falls. You might even guess by now that that day is Friday. Otherwise, why would I write this blog entry? Or what would Mr. Havil have to say in the whole chapter of the aforementioned book?
As we can see, this calculation increases the worry for people who suffer from paraskevidekatriaphobia — the fear of Friday the 13th. The 13th falls on Friday more often than on any other day.
Should we be worried?
Mathematically, the difference between the number of Fridays the 13th and, say, Thursdays the 13th is so small that it can only be observed when we look at the 400 year lifetime of the Gregorian calendar. Many countries have yet to experience the full cycle of the Gregorian calendar. For example, Russia adopted the calendar only in the 20th century. Is this why I am not so very afraid?
On second thought, for people of my generation, who are unlikely to live until the year 2100, the situation is slightly different. In the years between 1901 and 2099 our calendar has a days-of-the-week cycle of 28 years. You can calculate and check that in the period of 28 years, the 13th falls on any day of the week with the same probability. Hence, in events happening around my life time, there is not much to worry about, because Friday is no more special than any other day.
On third thought, a particular individual might see more Fridays on the 13th in his lifetime depending on the exact date of his birth. In my own life up to today, Monday is the most frequently occurring 13th. Maybe that's why I do not like Mondays.
For 25 years my children were my priority. I made several decisions in my life that benefited my family, but "harmed" my mathematical career. I do not regret any of my choices. After all, being a single mom made me a more confident, stronger person. Maybe this will help my career in a long run.
Now that my youngest son is 16 years old, my life can't revolve around him anymore. Now I must think about the meaning of my life, beyond bringing up children. The only thing I want to do is mathematics. I am actually doing some math on weekends, but I really want to do it full-time. My tasks at my job are getting further and further from mathematics and research. In short, I feel that my job doesn't fit me at this stage of my life.
I really should find another job. I am somewhat scared of change though. I think that the first thing to do is to try to turn around the situation at my current job. There is a lot of interesting mathematics in battle management. The problem is to match a math problem to a charge number. That is, I would need to convince my management that the algorithms we design need a sound mathematical basis.
Here is my decision: I will try to find some tasks at my work that include mathematics and see how I can change my situation there by the end of the year. If I don't succeed, I will have to think of something else. Let the Web be my witness. I will report the results to you soon.
I stumbled upon the following sentence in the MathWorld article on the Fibonacci numbers: "No odd Fibonacci number is divisible by 17." I started wondering if there are other numbers like that. Of course there are — no odd Fibonacci number is divisible by 2. But then, an odd number need not be a Fibonacci number in order not to be divisible by 2.
So, let's forget about 2 and think about odd numbers. How do we know that the infinite Fibonacci sequence never produces an odd number that is divisible by 17? Is 17 the only such odd number? Is 17 the smallest such odd number? If there are many such odd numbers, how do we calculate the corresponding sequence?
We'll start with a general question: How can we approach puzzles about the divisibility of Fibonacci numbers? Suppose K is an integer. Consider the sequence aK(n) = Fn(mod K), of Fibonacci numbers modulo K. The cool thing about this sequence is that it is periodic. If it is not immediately obvious to you, think of what happens when a pair of consecutive numbers in the sequence aK(n) gets repeated. As a bonus for thinking you will get an upper bound estimate for this period.
Let us denote the period of aK(n) by PK. By the way, this period is called a Pisano period. From the periodicity and the fact that aK(0) = 0, we see right away that there are infinitely many Fibonacci numbers divisible by K. Are there odd numbers among them? If we trust MathWorld, then all of the infinitely many Fibonacci numbers divisible by 17 will be even.
How do we examine the divisibility by K for odd Fibonacci numbers? Let us look at the Fibonacci sequence modulo 2. As we just proved, this sequence is periodic. Indeed, every third Fibonacci number is even. And the evenness of a Fibonacci number is equivalent to this number having an index divisible by 3.
Now that we know the indices of even Fibonacci numbers we can come back to the sequence aK(n). In order to prove that no odd Fibonacci number is divisible by K, it is enough to check that all the zeroes in the sequence aK(n) have indices divisible by 3. We already have one zero in this sequence at index 0, which is by divisible by 3. Because the sequence aK(n) is periodic, it will start repeating itself at aK(PK) . Hence, we need to check that PK is divisible by 3 and all the zeroes up to aK(PK) have indices divisible by 3. When K = 17 it is not hard to do the calculations manually. If you'd like, try this exercise. To encourage (or perhaps to discourage) you, here's an estimate of the scope of the work for this exercise: the Pisano period for K = 17 is 36.
After I checked that no odd Fibonacci number is ever divisible by 17,
I wanted to find the standard solution for this statement and followed
the trail in MathWorld. MathWorld sent me on a library trip where I
found the proof of the statement in the book Mathematical
Gems III by Ross Honsberger
The method we developed for 17 can be used to check any other number. I trusted this task to my computer. To speed up my program, I used the fact that the Pisano period for K is never more than 6K. Here is the sequence calculated by my trustworthy computer, which I programmed with, I hope, equal trustworthiness:
The sequence shows that 9 is the smallest odd number that no odd Fibonacci is ever divisible by, and 17 is the smallest odd prime with this same property. Here is a trick question for you: Why is this property of 17 more famous than the same property of 9?
Let us look at the sequence again. Is this sequence infinite? Obviously, it should include all multiples of 9 − hence, it is infinite. What about prime numbers in this sequence? Is there an infinite number of primes such that no odd Fibonacci number is divisible by them? While I do not know the answer, it's worth investigating this question a little bit further.
From now on, let K be an odd prime. Let us look at the zeroes of the sequence aK(n) more closely. Suppose a zero first appears at the m-th place of aK(n). Then aK(m+1) = aK(m+2) = a. In this case the sequence starting from the m-th place is proportional modulo K to the sequence aK(n) starting from the 0-th index. Namely, aK(n+m) = a*aK(n) (mod K). As a is mutually prime with K, then aK(n+m) = 0 iff aK(n) = 0. From here, for any index g that is a multiple of m, aK(g) = 0. Furthermore, there are no other zeroes in the sequence aK(n). Hence, the appearances of 0 in the sequence aK(n) are periodic with period m.
By the way, m is called a fundamental period; and we just proved that the Pisano period is a multiple of the fundamental period for prime K. Hence, the fact that no odd Fibonacci number is divisible by K is equivalent to the fact that the fundamental period is not divisible by 3. It is like saying that if the smallest positive Fibonacci number divisible by an odd prime K is even, then no odd Fibonacci number is divisible by K.
If the remainder of the fundamental period modulo 3 were random, we would expect that about every third prime number would not divide any odd Fibonacci numbers. In reality there are 561 such primes among the first 1,500 primes (including 2). This is somewhat more than one third. This gives me hope that there is a non-random reason for such primes to exist. Consequently, it may be possible to prove that the sequence of prime numbers that do not divide odd Fibonacci numbers is infinite.
Can you prove that?
Why didn't I think of this before?
I want to share some ideas about mathematics and about my life as a mathematician. In this blog, you'll read about such things as the properties of numbers and sequences and how mathematicians approach practical things.
I started my life as a genius girl mathematician, winning silver and gold medals at the International Math Olympiad (IMO) as a teenager. My PhD is from Moscow State University. When I got married, I wanted to have a family and mathematics at the same time, but being a woman, this affected my mathematics career. Now my kids are growing up and mathematics is becoming more and more important in my life. This is why I decided to start this blog.
Last revised February 2008