MathDB

On a blackboard a positive integer

n_0

is written. Two players,

A

and

B

are playing a game, which respects the following rules:

-

acting alternatively per turn, each player deletes the number written on the blackboard

n_k

and writes instead one number denoted with

n_{k+1}

from the set \left\{n_k-1, \dsp \left\lfloor\frac {n_k}3\right\rfloor\right\};

-

player

A

starts first deleting

n_0

and replacing it with n_1\in\left\{n_0-1, \dsp \left\lfloor\frac {n_0}3\right\rfloor\right\};

-

the game ends when the number on the table is 0 - and the player who wrote it is the winner. Find which player has a winning strategy in each of the following cases: a)

n_0=120

; b) n_0=\dsp \frac {3^{2002}-1}2; c) n_0=\dsp \frac{3^{2002}+1}2.

Problem Statement