Two players start with a pile of stones.
1st player takes any number of stones from the pile, but not all of them.
Then the players alternate, taking up to twice as many stones
as their opponent just took.
The player taking the last stone (or stones) is the winner.

Question: If you and your friend start with 100 stones, what is the smallest amount of stones you should take on your 1st turn, and how will you proceed to win the game?

Interestingly, and surely not coincidentally, each of the Fibonacci terms 3, 5, 8, 13, 21, 34, 55, and 89 are "safe spots" where you can leave your opponent en route to a victory.  I can't yet come up with a simple way to explain why this is the case, though. 

Even more specifically, while the Fibonacci terms are not the only "safe spots", they are the only spots from which the only winning move is to take ALL the remaining stones. 

Edited on June 30, 2014, 4:04 pm

Posted by tomarken on 2014-06-30 15:39:39