At the beginning, three boxes contain $m$, $n$, and $k$ pieces, respectively. Ayşe and Burak are playing a turn-based game with these pieces. At each turn, the player takes at least one piece from one of the boxes. The player who takes the last piece will win the game. Ayşe will be the first player. They are playing the game once for each $(m,n,k)=(1,2012,2014)$, $(2011,2011,2012)$, $(2011,2012,2013)$, $(2011,2012,2014)$, $(2011,2013,2013)$. In how many of them can Ayşe guarantee to win the game?

$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$
To determine in how many of the given scenarios Ayşe can guarantee a win, we need to analyze each initial configuration and determine if Ayşe can force a win using a strategy based on the properties of the game.

1. **Game Analysis for $(m,n,k) = (1,2012,2014)$:**
   - Ayşe can make the first move to $(1,2012,2013)$.
   - This configuration $(1,2012,2013)$ is of the form $(1,a,a+1)$.
   - If Burak touches the box with 1 piece, Ayşe can equalize the other two boxes, ensuring a win.
   - If Burak makes a move to $(1,1,a)$, Ayşe will make $(1,1,0)$ and win.
   - For other moves, Ayşe can always make moves like $(1,b,b+1)$ or $(1,b,b-1)$, eventually forcing Burak into a losing position.
   - Therefore, Ayşe can guarantee a win in this game.

2. **Game Analysis for $(m,n,k) = (2011,2011,2012)$:**
   - Ayşe can make the first move to $(2011,2011,0)$.
   - This configuration $(2011,2011,0)$ is of the form $(a,a,0)$.
   - Ayşe can always make a symmetric move against Burak's move, ensuring that she will take the last piece.
   - Therefore, Ayşe can guarantee a win in this game.

3. **Game Analysis for $(m,n,k) = (2011,2012,2013)$:**
   - Ayşe can make the first move to $(1,2012,2013)$.
   - This configuration $(1,2012,2013)$ is of the form $(1,a,a+1)$.
   - As analyzed in the first game, Ayşe can guarantee a win from this position.
   - Therefore, Ayşe can guarantee a win in this game.

4. **Game Analysis for $(m,n,k) = (2011,2012,2014)$:**
   - Ayşe can make the first move to $(2011,2012,1)$.
   - This configuration $(2011,2012,1)$ is of the form $(a,a+1,1)$.
   - If Burak touches the box with 1 piece, Ayşe can equalize the other two boxes, ensuring a win.
   - If Burak makes a move to $(1,1,a)$, Ayşe will make $(1,1,0)$ and win.
   - For other moves, Ayşe can always make moves like $(1,b,b+1)$ or $(1,b,b-1)$, eventually forcing Burak into a losing position.
   - Therefore, Ayşe can guarantee a win in this game.

5. **Game Analysis for $(m,n,k) = (2011,2013,2013)$:**
   - Ayşe can make the first move to $(0,2013,2013)$.
   - This configuration $(0,2013,2013)$ is of the form $(0,a,a)$.
   - Ayşe can always make a symmetric move against Burak's move, ensuring that she will take the last piece.
   - Therefore, Ayşe can guarantee a win in this game.

Conclusion:
Ayşe can guarantee a win in all five games.

The final answer is $\boxed{5}$.