Tuesday, November 14, 2006

Paul R. Halmos (1916-2006)

Prof. Paul R. Halmos died on October 2, 2006. George Pólya Award, for articles of expository excellence published in the College Mathematics Journal, was given to him in 1983 for his article The Thrills of Abstraction.

From The Thrills of Abstraction:

My wife and I were invited to a party recently, a party attended by four other couples, making a total of ten people. Some of those ten knew some of the others, and some did not, and some were polite, and some were not. As a result a certain amount of handshaking took place in an unpredictable way, subject only to two obvious conditions: no one shook his or her own hand and no husband shook his wife's hand. When it was all over, I became curious and I went around the party asking each person: " How many hands did you shake? ... And you? ... And you?" What answers could I have received? Conceivably some people could have said "None", and others could have given me any number between 1 and 8 inclusive. That's right, isn't it? Since self-handshakes and spouse-handshakes were ruled out, 8 is the maximum number of hands that any one of the party of 10 could have shaken.

I asked nine people (everybody, including my own wife), and each answer could have been any one of the nine numbers 0 to 8 inclusive. I was interested to note, and I hereby report, that the nine different people gave me nine different answers; someone said 0, someone said 1, and so on, and, finally, someone said 8. When it was all over, my curiosity was satisfied: I knew all the answers. Next morning, I told the story to my colleagues at the office, exactly as I told it now, and I challenged them, on the basis of the information just given, to tell me how many hands my wife shook.

If you try to find the answer for that question, you will find a kind of similarity, in approach, with the following problem.

C thinks of two consecutive numbers between 1 and 10. C tells one number to A, and the another to B. Now for the conversation between A and B:

A: I do not know your number
B: I do not know your number
A: I do not know your number
B: I know your number
A: I too know your number

What are those numbers?

References

1. An obituary at the website of the Mathematical Association of America
2. A brief biography: http://www-history.mcs.st-and.ac.uk/history/Biographies/Halmos.html
3. Wikipedia http://en.wikipedia.org/wiki/Paul_Halmos
4. Paul R. Halmos, The Thrills of Abstraction, Two-Year Coll. Math. J. 13 (1982), 243-251.
5. Number puzzle

6 comments:

Anonymous said...

Sad news indeed.

One of the books that's been on my reading list that I've never gotten around to reading was Paul Halmos's "Naive Set Theory".

divide_by_zero said...

i guess (A = 7,B = 8) and symmetrically A = 4, B = 3 might work?

Shencottah said...

Answer for I don't know puzzle:

The two numbers are 5 and 6 but we will not know if C tells 5 to A or B.

divide_by_zero said...

Won't that be the case only if there is one more "I dont know your number" response from each guy?

Shencottah said...

The first I don't know removes 2 and 9 (1 and 10 are not there as the numbers are between 1 and 10).

The second I don't know removes 3 and 8.

The third I don't know removes 4 and 7.

Only two numbers are left after this. Hence the statement I know your number follows.

Anonymous said...

I prefer the edited wording for this puzzle that's found by following the link you provided:

A teacher thinks of two consecutive numbers between 1 and 10 (Edit: 1 and 10 included).