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        riddles >> hard >> Betting on the Digits of Pi
        
(Message started by: william wu on Jan 15^th, 2004, 6:02am)

Title: Betting on the Digits of Pi
Post by william wu on Jan 15^th, 2004, 6:02am

You are playing a game with a supercomputer from the future. The game is about betting on digits in the decimal expansion of [pi]. It works as follows:

1. You have one dollar.

2. The supercomputer chooses a ridiculously large natural number N. This corresponds to the Nth digit in the decimal expansion of [pi].

3. You are now allowed to place bets on the next k digits in the decimal expansion. You can place as many bets as you want, and slice your dollar in any fashion you like. (E.g. "I bet 2/3 of my dollar that the next five digits will be "12345", and 1/3 of my dollar that the next three digits will be "678".)

4. For each bet for which all digits are correctly predicted, the return is 10^k, where k is the length of the subsequence you predicted. (So for correctly predicting "678", you'd get 1000 times what you invested.)

The assumption is that N is so large, it is not possible using today's technology for you to simply compute [pi]'s digits up to the Nth decimal place and make perfect predictions. So you'll have to rely on other ideas. However, the supercomputer "from the future" is apparently able to quickly compute [pi]'s decimal expansion to such great lengths, and thus can properly verify the correctness of your predictions.

Problem: Design a betting strategy that will provably return more than a dollar. Yes, at least one does exist.

Hint: You'll most likely need to use some really obscure theorems you've never ever ever ever heard of. So this will probably end up being more of a miniature math lesson than a decent puzzle. In any case, I think it's comforting to know that you can win at this game.

Source: rec.puzzles

Title: Re: Betting on the Digits of Pi
Post by Benoit_Mandelbrot on Jan 15^th, 2004, 10:24am

If [pi] is a normal number, there is about a 1/10 probability that you will get a 1 digit number that you pick [0-9], and 1/90 that the number you pick will be 2 digits [10-99] and correct (this is because 0-9 are one digit), and 1/900 if you pick 3 digits [100-999]. "..., although the first 30 million digits of [pi] are very uniformly distributed (Bailey 1988 )."
I would start picking 2 digit numbers.

But this is today, and the next 30 _illion digits of tomorrow could not be. Computers of the future could get 30*10^{29857243985734985729850729574} digits or more.

This is on the basis of [pi] being normal up to the rediculous number.

By the way, a normal number is http://mathworld.wolfram.com/NormalNumber.html

Title: Re: Betting on the Digits of Pi
Post by willy in a lab on Jan 15^th, 2004, 12:32pm

I don't see how the conjectured normality of [pi] helps you provably make money. If anything, the normality is depressing to a gambler, because it says that's there's no difference in the expected limiting frequency of one k-digit sequence over another k-digit sequence.

Title: Re: Betting on the Digits of Pi
Post by towr on Jan 15^th, 2004, 1:29pm

It seems to me you have to gamble on an infinite number of sequences.. I'm not sure how I should explain this instinct of mine though..

Title: Re: Betting on the Digits of Pi
Post by Quetzycoatl on Jan 15^th, 2004, 2:54pm

on 01/15/04 at 13:29:34, towr wrote:

It seems to me you have to gamble on an infinite number of sequences.. I'm not sure how I should explain this instinct of mine though..

Well, if you take 10^k number of k length sequences(the bigger k is the better), and then you bet $1.00/(10^k-1) on all but one of them, you have a really good chance at making a very small amount of money, and a very small chance of losing.

So does making S infinite mean you are guaranteed to make an infinitely small amount of money? Or does it mean your betting $0?

Title: Re: Betting on the Digits of Pi
Post by SWF on Jan 15^th, 2004, 5:29pm

I bet that digits k through 2*k-2 never match digits 1 through k-1 (just a guess). So I would choose to bet equally on every k-1 digit number except for the first k-1 digits of pi. (unless the computer picked k=1 ::) )

Title: Re: Betting on the Digits of Pi
Post by william wu on Jan 15^th, 2004, 6:40pm

Nice ideas guys; it turns out the solution is identical in spirit -- you bet on all possible sequences of a certain length, except for one of them. Use the following theorem, proven by Mahler in 1953:

For all integers p,q > 1,

| [pi] - p/q | > q^-42

So for a given N chosen by the supercomputer, demonstrate that a certain sequence of subsequent digits cannot occur because it will contradict the theorem. Then bet equally on all sequences of that length, except for the one that cannot occur. You will be guarantted to make a minuscule amount of profit ... :)

Title: Re: Betting on the Digits of Pi
Post by Eigenray on Jan 16^th, 2004, 12:09am

Using the well known result above, it follows that the sequence of [hide]41N 0s[/hide] can never occur, starting immediately after the N-th digit (it will most likely occur later, though).
For otherwise, [hide]
let X = floor(10^N[pi]). Then we'd have
10^-42N < | [pi] - X/10^N | < 1/10^N+41N,
a contradiction.
In fact, a similar argument shows that given any sequence of length r, it cannot repeat more than 41N/r + 42 times (starting immediately after the N-th digit)[/hide].

Title: Re: Betting on the Digits of Pi
Post by Quetzycoatl on Jan 16^th, 2004, 8:08am

on 01/15/04 at 18:40:21, william wu wrote:

| [pi] - p/q | > q^-42

I looked on Mathworld and I assume that this comes from Liouville's Approximation Theorem:
| x - p/q | > 1/qⁿ
But I don't understand why are using n = 42? Can someone help me out here?

Title: Re: Betting on the Digits of Pi
Post by Rezyk on Jan 17^th, 2004, 12:01am

I'm going to disagree with the given solution. It is less feasible to compose the necessary 369N bets of average size N/2 each than it is to just determine the Nth digit.

Title: Re: Betting on the Digits of Pi
Post by Barukh on Jan 17^th, 2004, 1:49am

on 01/16/04 at 08:08:24, Quetzycoatl wrote:

I looked on Mathworld and I assume that this comes from Liouville's Approximation Theorem:
| x - p/q | > 1/qⁿ But I don't understand why are using n = 42?

What Liouville has proved in 1840 was that in order for a number x to be algebraic the aforementioned inequality must hold for any n (with suitable choices of p and q). Using this, he proved that the number L = 10^-1! + 10^-2! + 10^-3! + ... is transcendential.

The results proved by Mahler shows that [pi] cannot be a root of an equation of degree [ge] 42 with integer coefficients.

All this, however, is not directly related to this thread.

Title: Re: Betting on the Digits of Pi
Post by mikedagr8 on Jul 15^th, 2007, 6:32am

nice puzzle, problem is that i know about 500 numbers of pi from the beginning off by heart, and i know that around the 700-900 mark, there is a sequence 5 9's in a row. asuming i understodd the question, i should be able to win every time? i am almost certain i didn't, because someone would have said this.

Title: Re: Betting on the Digits of Pi
Post by pex on Jul 15^th, 2007, 7:03am

on 07/15/07 at 06:32:38, mikedagr8 wrote:

Probably 500-900 is not considered "ridiculously large" here...

Title: Re: Betting on the Digits of Pi
Post by mikedagr8 on Jul 15^th, 2007, 5:08pm

no it isn't, when you consider hte world record is above 80 thousand digits in a row, all correct, or when the computer knows all the numbers we currently do plus more. I am just saying that, the only time where 5 consecutive digits are the same, is here, and they happen to all be 9 (well to my knowledge at least).

Title: Re: Betting on the Digits of Pi
Post by ThudanBlunder on Jul 15^th, 2007, 6:15pm

on 07/15/07 at 17:08:02, mikedagr8 wrote:

I am just saying that, the only time where 5 consecutive digits are the same, is here,

Considering how many digits?

Title: Re: Betting on the Digits of Pi
Post by mikedagr8 on Jul 15^th, 2007, 6:20pm

not enough :P, sorry, bad reasoning, i need to start using more of this (reasoning) , and not worry about the final answer and an answer, but the way to work it out.

Title: Re: Betting on the Digits of Pi
Post by towr on Jul 16^th, 2007, 12:34am

I think it has been proved that every finite sequence of integers occurs somewhere in the decimal expansion of pi.
So somewhere down the line there will be at least 6 times the same digit in a row. And you're name will be spelled out in ascii somewhere as well. Which is a nice thought.

Title: Re: Betting on the Digits of Pi
Post by mikedagr8 on Jul 16^th, 2007, 1:49am

yay, i am immortalised in pi! :D Nice thought.

Title: Re: Betting on the Digits of Pi
Post by Hippo on Jul 24^th, 2007, 11:57pm

Immortalised but lost among others ;)

Title: Re: Betting on the Digits of Pi
Post by Earendil on Aug 8^th, 2007, 12:52am

on 01/17/04 at 01:49:12, Barukh wrote:

Or maybe just

http://en.wikipedia.org/wiki/The_Answer_to_Life,_the_Universe,_and_Everything ;D

Title: Re: Betting on the Digits of Pi
Post by Grimbal on Aug 8^th, 2007, 4:52am

The answer to that question is a piece of pie...
3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679