It began with a strange email.
On that day, I arrived at my office in the university and went to the computer room, as usually after I left my bag on a sofa, to check up email. Facing to a workstation and typing my login name and password quickly, I heard a little sound of a hard disk A red pilot lamp is brinking to indicate accesss to the network. At last, several windows appeard on the display, at a corner of which the icon of a American style post box, looking like a Quonset hut, is staying. It rised up its flag with "Pip!" This told me the reception of new emails.
I had been happy to receive an email from someone just when I had got an email account. However, email is enough popular now, which is forceing me to receive a lot of unpleasant things. Actually I am very delightful when a mathematician in an abroad country send a message to ask me about my research whlie most of emails on that day gave me troublesome jobs. But, one of them was a strange email with different flavor.
The sender of this email was Mr. S, who is working on graph theory, the same field as I am working, but I won't write his real name to keep his honor. However, everyone working in this field knows him as a good guy who likes such a playful thing.
That email included the following message with subject "hanoi":
I don' t know what the phrase "Hanoi Tower fell down!" means, but I imagned immediately this was a kind of chain mails, which was popular and called "the Letter of Unhappyness" when I was a boy. If you recieved such a letter, you have to send the same message to three other persons; otherwise, you will be unhappy. Conversely, I am sure that there also was "the Letter of Happyness", which was to create a chain of friends and make you happy.
Anyway, if I forwarded this email to two mathematicians I know, and if they followed this instruction, too, then the number of persons who received this message would increase exponetially, as growth of rats. The first guy who began this game sends it two persons and each of the two persons send it two other persons. Furthermore, 2 x 2 = 4 persons send it and so one.
Precisely speaking, there might be a person who received the same email twice or more. However, if we suppose that there was no such guy, the number of persons who received this message will increase as follows.
Since this is what is called a geometric progression, The sum of the first to the nth terms is:
Explain this formula briefly for a person who doesn't know it. Add 1 to the left hand of the formular temporarily and sum up the fist few terms.
It is the point that the first two terms always are the same and that this happens again after unifying those two terms. Repeating this process, we get 2n when we reach to the last 2n-1. Recall that we added the extra 1 at first. Subtracting this from that, we obtain 2n - 1.
However, it is tedious to say 2n - 1 every time. So I will just say 2n, omitting "-1". Thus, if this strange email reached me via n persons, there have been 2n mathematicians, including me, who received this message.
For example, if it passed through 10 persons, that number is 210 = 1024. This is not so big since it is only about 1000. However, it will be surprisingly if through 100 persons.
You know, 2100 is a number consisting of 31 digits.
It is beyond our imagination how big this number is. This number is much bigger than the number of all stars in the universe and cannot compare with the number of mathematicians on the earth. I wrote many thins about the wonder of 2100 in my book "Refreshing! three stories of 2100" (published from YUSEISHA). Refer to this book.
So, this message passed through very few people from the first guy to me. It depends on how to define a mathematician. However, since Japan Mathematical Society consists of less than 10,000 members, if I played this game and forwerded this message, the day when all mathematicians in Japan receive this message will come soon.
Finally, my decision was to stop this game. Hanoi Tower fell down. How important is this message? Tower of Hanoi is just one of puzzles, as I willl explain later. This is exaggerated. What does it matter with a broken toy of the tower? It might be a nice phrase with good wit, but we cannot smile with it at all in our Japanese sense.
People might be happy to receive such a messagel if they are beginners for email, as well as I was. But, email is just a tool for communication in office. So I decided that I would stop to make a seed irritating many people who are forced to sit down at workstations and personal computers everyday.
Anyway, why did the first guy send the message "Hanoi Tower fell down" to the world? How does my decision to stap the game work if there is a big will with this message? In either case, he must not be able to know whether or not I recevied his message after he sent it.
But, there remains the fact that I received the massage "Hanoi Tower fell down" even if I stoped the game. I couldn't imagine at that time how it will become a big problem.
Furthermore, Mr. S confessed that he had sent this message to his friend living in New Zealand. So, it is possible that the only persons who received the message "Hanoi Towe fell down" in Japan are Mr. S and me. In either case, the number of such persons will be very small.