Fibonacci

Fibonacci on the checkerboard (part 3)


For a relation between Fibonacci numbers and Bridge see Jim Loy's site.

Two players, A(nn) and B(ill), have put on the table a pile of biermats.
In turn, Ann first, they take a number of biermats from the heap. He or she who takes the last biermat wins.
They have to obey the following rules:
1. Ann is not allowed, in her first turn, to take all biermats.
2. They are not allowed to take more than two times the number of biermats, that the opponent has taken in his last move.
Is there a winning strategy for one of the players?

We will show that Bill can win if the heap consists of F_n biermats (for certain n).
Notice that this implies that Ann can win if the heap does not consist of F_n biermats,
for in that case Ann first takes so many mats from the heap (less than half!), that there are F_k mats left (for certain k). After that the situation is reversed. (proof).