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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
21
revised Betting on the digits of a real number
changes in response to commenters
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.
Apr
20
awarded  Editor
Apr
20
revised Betting on the digits of a real number
clarification of question after a comment
Apr
20
asked Betting on the digits of a real number
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?
Feb
7
accepted An approximate relationship between the totient function and sum of divisors
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6
asked An approximate relationship between the totient function and sum of divisors
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