A magician asks you to choose two integers between 1 and 50 and add them.
Then add the largest two of the three integers at hand.
Then add the largest two again.
Repeat this around ten times.
Disclose to the magician your final number n.
The magician then tells you the next number.
(In reply to re: I would need a calculator (spoiler)
Excellent! thanks, Charlie.
Just to recap, you have found that using 1.618 as a multiplier suffices if the volunteer from the audience does 9, 10 or 11 calculations, but can be wrong if he does more than 11 or less than 9. Using 1.618034 extends the upper range, but is still inaccurate if the volunteer does less than 9 calculations.
I guess the magician needs to insist that the volunteer does at least 9 calculations ...