Problems & Puzzles: Puzzles

Puzzle 277.  Pi again

Cino Hilliard sent the following claim related to an old puzzle of these pages:

"I have a solution to puzzle 50 that is 100%. 833719/265381 = 3.141592653581.."

The claim is not true according to the rules of the Puzzle 50 (in order to calculate the rank associated to a solution we take into account only the good decimal digits of the approximation to Pi).

The real rank of the Cino's solution is (11/12)*100 = 91.6...%.

Nevertheless this rank is the highest rank already obtained for the Puzzle 50 by the simple quotient of two primes!

Apparently Cino's has found a systematic approach in order to get good approximations and in his email he challenges us to find what his method can be.

Questions:

1. Can you get a better solution than the Cino's one using only the quotient of two primes and a systematic approach?

Solution: I believe I can identify Cino's method and would counter-challenge him to find another high performer. The approximation 833719/265381 is one of the continued fraction approximants to pi.  I have not found another ratio of two primes in the continued fraction approximants to pi, through the 100th convergent.  The anticipated performance of such a convergent would be right around 100% : convergent p/q, d digits for q, approximately d digits for p (either d or d+1, depending on how big pi*q was), so total of d+d=2d digits for fraction, continued fraction theory says the error is approximately 1/q^2, in other words 2*d digits of accuracy in the fraction, so a rating of about 2d/2d*100=100%.

If you remove the requirement in the (current) problem of "simple quotient of two primes," then you can generate around 100% ratings for puzzle 50.  For example, take the convergent 430010946591069243/136876735467187340, and write it as (430010946591069241+2)/(3*45625578489062449-7).  This is accurate to 35 decimal places and uses 38 decimal digits in the expression, for a rating of 92.1

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Interesting the Stinchcombe's guess about the Cino's method. Regarding the quotient calculated by Adam is interesting too, but not yet I will remove the requirement of the current puzzle. I still think that a better quotient could come soon.

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Nevertheless I don't remove the requirement to this puzzle, I want to report my 'best' result trying to hunt a better result than the Cino's one:

2795342701/889785217 = (7727 * 361763)/(277 * 383 * 8387) = 3.14159265358979323 Rank = 17/20 = 0.85

Form the result reported it's evident what was my approach. Anyway, no better result in any sense. Sorry.

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