This article is from the Puzzles FAQ, by Chris Cole firstname.lastname@example.org and Matthew Daly email@example.com with numerous contributions by others.
Are the digits of pi random (i.e., can you make money betting on them)?
No, the digits of pi are not truly random, therefore you can win money
playing against a supercomputer that can calculate the digits of pi far
beyond what we are currently capable of doing. This computer selects a
position in the decimal expansion of pi -- say, at 10^100. Your job is
to guess what the next digit or digit sequence is. Specifically, you
have one dollar to bet. A bet on the next digit, if correct, returns
10 times the amount bet; a bet on the next two digits returns 100 times
the amount bet, and so on. (The dollar may be divided in any fashion,
so we could bet 1/3 or 1/10000 of a dollar.) You may place bets in any
combination. The computer will tell you the position number, let you
examine the digits up to that point, and calculate statistics for you.
It is easy to set up strategies that might win, if the supercomputer
doesn't know your strategy. For example, "Always bet on 7" might win,
if you are lucky. Also, it is easy to set up bets that will always
return a dollar. For example, if you bet a penny on every two-digit
sequence, you are sure to win back your dollar. Also, there are
strategies that might be winning, but we can't prove it. For example,
it may be that a certain sequence of digits never occurs in pi, but we
have no way of proving this.
The problem is to find a strategy that you can prove will always win
back more than a dollar.
The assumption that the position is beyond the reach of calculation
means that we must rely on general facts we know about the sequence of
digits of pi, which is practically nil. But it is not completely nil,
and the challenge is to find a strategy that will always win money.
A theorem of Mahler (1953) states that for all integers p, q > 1,
-42 |pi - p/q| > q