An old puzzle, extended precision - and logs

[sa-penguin]

(07-28-2016 12:16 PM)Dieter Wrote:  I don't think this is a secret, but here is how you can do it:

Code:

  (99/100)^30 
= 99^30 / 100^30
= 99^30 / 10^60

      99x = 100x - x

    99*99 = 9900 - 99         = 9801   (=99^2)
  9801*99 = 980100 - 9801     = 970299   (=99^3)
970299*99 = 97029900 - 970299 = 96059601   (=99^4)
      ...   ...

Do this until 99^30, then shift the decimal point 60 digits left to get 0,7397...
1 minus this is 0,2602996... so the answer (times 100) is 26 cups.

Dieter

Very elegant!
A re-reading of the answer mentions a "simplified process of calculation".
Probably the same as you have shown.

Quote:and after the thirtieth theft there would remain in the cask the thirtieth power of 99 divided by the twenty-ninth power of 100.
This by the ordinary method of calculation gives us a number composed of 59 figures to be divided by a number composed of 58 figures!
But by the use of logarithms it may be quickly ascertained that the required quantity is
very nearly 73-97/100 pints of wine left in the cask. Consequently the cellarer stole nearly 26.03 pints.
The monks doubtless omitted the answer for the reason that they had no tables of logarithms, and did not care to face the task of making that long and tedious calculation in order to get the quantity "to a nicety," as the wily cellarer had stipulated.

By a simplified process of calculation, I have ascertained that the exact quantity of wine stolen would be:
26.0299626611719577269984907683285057747323737647323555652999