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.
Let the smaller of the two numbers be a and the larger b
The generated sequence of numbers is fibonnacci-like, with ratio of successive terms approaching the golden ratio (~1.61803398).
The 9th term is 55a + 89b
The 10th term is 89a + 144b
The 11th term is 144a + 233b
If the magician multiplies the supplied number by 1.618, and rounds, that seems good enough. Multiplying by 1.618034 would be better.
For instance, taking an extreme case, let a = 49 and b = 50.
If you supply the 9th term, 7145, then 7145*1.618 = 11560.6, which rounds to the correct next term of 11561.
If you supply the 10th term, 11561, then 11561*1.618 = 18705.7, which rounds to the correct next term of 18706.