"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:**

**Adam Stinchcombe** wrote:

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

***

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.

***

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.

***