tchebytchev wrote: On the board is written an integer $N \geq 2$. Two players $A$ and $B$ play in turn, starting with $A$. Each player in turn replaces the existing number by the result of performing one of two operations: subtract 1 and divide by 2, provided that a positive integer is obtained. The player who reaches the number 1 wins. Determine the smallest even number $N$ requires you to play at least $2015$ times to win ($B$ shifts are not counted). Hmmm, If $A$ start with even number then $B$ have winning strategy by "always subtract by $1$" , this force $A$ to subtract by $1$ again (Otherwise it not provide positive integer), so $B$ always write down smaller odd integer until $B$ reach $1$