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.
By the way, it is interesting that I misjudged the extreme case. I thought that 49 and 50 would be the worst case, but actually (as Charlie has demonstrated) the problems occur when the two selected numbers are far apart. So, choosing 1 and 50 is the worst case. I now realize that this is because the Fibonacci alternatively underestimates and overestimates the contribution of the smaller number, at the same time that it is alternately overestimating and underestimating the contribution of the larger number. The farther apart the two are, the less the errors (which always go in different directions) will offset.
Edited on December 27, 2013, 9:16 am