You are in LImbo. The administrator challenges you to a game. He will think of a natural number, and you can have one guess at it each day. (Once he has thought of his number, he will never change it.) If you guess it, he'll let you out. Will you get out? (Remember - you have eternity...) Explain.
What if he instead told you that he was going to pick an integer - now will you get out? What about if he chose a rational number? A real number?
No. There can be no 'days' in eternity;as such, it is not measurable in counting terms. Therefore, you would only get one guess, and it would be highly unlikely that the guess would be correct.
ReplyDeleteI'm sure that the question really would like us to assume day-after-day-after-day, ad infinitum. In that case, the guess can be achieved eventually if the number remained unchanged.(Jim May)
Good answer, Jim! :-)
DeleteDepending on your religious stance, eternity may be days, or seconds (e.g., to God a day may feel like a thousand years, and a thousand years like a day) - so since you have eternity, and you feel that there are days, and we're talking integer numbers, I'd say yes, you'd eventually get out. But if you talk about real numbers, I doubt it, because there is an infinite count of real numbers between 0 and 1, so unless he says he gives you a real number between 0 and 1, it is unlikely that you will get out. :-)
ReplyDeleteAlright, so I've taken my share of math classes, and this seems to be a question about countability. There is an easy algorithm for finding your opponent's number if he has selected from the Natural Numbers: Start with 1, increase your guess by 1 each day, and no matter what his number is you'll eventually guess it (on day n, if he has selected the number n). Integers and Rationals are both Countably Infinite, in that there exists a one-to-one mapping between them and the natural numbers; define such a mapping, and you've got an algorithm for your guesses on those sets as well. The Reals, however, are not countably infinite; they are in fact uncountable, since there exists no one-to-one mapping between them and the naturals. Since your days (and thus guesses) are a sequence of natural numbers, there is no mechanism for covering the entire set of Reals with (even infinite) guesses proffered in such a manner.
ReplyDeleteLong story short: Yes, you can guarantee you'll eventually guess his number if you know it is from the Naturals, the Integers, or even the Rationals; but no, you cannot make such a claim if he selects from the Reals.
I love how everyone gets involved and works these questions in ways I couldn't even imagine!
ReplyDeleteYou can guess any number, even real one, as long as its name can be written on a piece of paper:
ReplyDeleteEvery day you'll write a new string on paper, and hand it over or read it out-load...
On a finite time you will cover all strings with one character, on other finite time you will cover all strings with two characters etc'...
For example "seven and one third" is a 19 letters string, "Pie times nine" is a 14 letters string.
That said, you don’t really need actual paper...
As an added bonus you may answer many other riddles during this process... `)