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Apr
21
comment Betting on the digits of a real number
I've updated the question in response to your comments.
Apr
20
comment Betting on the digits of a real number
@joriki: Yes, I see that I formulated the question slightly incorrectly. I think I meant "Is there a strategy such that there is no $x$ that 'defeats' it in the long run". My instinct initially was that there might be choices for $x$ that 'defeat' every strategy (but you have shown that to not be the case), which is why I made the error, I think.
Apr
20
comment Betting on the digits of a real number
@carlop: $0.5 - 2|x-0.5|$ is not always greater than zero for $x$ between 0 or 1, take $x=1$, for example.
Apr
20
comment Betting on the digits of a real number
@joriki: But there is not a unique $x$ given an initial string of digits, so what does "correct guess" even mean? I see the problem you are bringing up, but I guess when I'm talking about strategy, I mean some rule or algorithm to play this game no matter what $x$ is.
Apr
20
comment Betting on the digits of a real number
I see. Does it clarify things if we define strategies to be functions mapping strings of digits (all the $nN$ revealed digits that I know at turn $n$) to guesses $p$?
Apr
20
comment Betting on the digits of a real number
Thanks for the comment. $x$ is kept secret, so the choice of $p$ must be made based only on what digits have been revealed in play. I have edited the question to clarify this.
Feb
7
comment An approximate relationship between the totient function and sum of divisors
Thanks. I find these kinds of facts remarkable. Do you know what are the prerequisites for Hardy and Wright?
Aug
12
comment Recognizing and Using Chaitin's Constant
Great answer, thanks.
Aug
11
comment Recognizing and Using Chaitin's Constant
Thank you all for your replies!