Let $k$ be a positive integer. Marco and Vera play a game on an infinite grid of square cells. At the beginning, only one cell is black and the rest are white. 
	A turn in this game consists of the following. Marco moves first, and for every move he must choose a cell which is black and which has more than two white neighbors. (Two cells are neighbors if they share an edge, so every cell has exactly four neighbors.) His move consists of making the chosen black cell white and turning all of its neighbors black if they are not already. Vera then performs the following action exactly $k$ times: she chooses two cells that are neighbors to each other and swaps their colors (she is allowed to swap the colors of two white or of two black cells, though doing so has no effect). This, in totality, is a single turn. If Vera leaves the board so that Marco cannot choose a cell that is black and has more than two white neighbors, then Vera wins; otherwise, another turn occurs.
	Let $m$ be the minimal $k$ value such that Vera can guarantee that she wins no matter what Marco does. For $k=m$, let $t$ be the smallest positive integer such that Vera can guarantee, no matter what Marco does, that she wins after at most $t$ turns. Compute $100m + t$.

[i]Proposed by Ashwin Sah[/i]
To solve this problem, we need to determine the minimal value of \( k \) such that Vera can guarantee a win no matter what Marco does, and then find the smallest positive integer \( t \) such that Vera can guarantee a win after at most \( t \) turns. Finally, we compute \( 100m + t \).

1. **Understanding Marco's Move:**
   - Marco can only choose a black cell that has more than two white neighbors.
   - When Marco chooses such a cell, he turns it white and all its neighbors black.

2. **Understanding Vera's Move:**
   - Vera can swap the colors of two neighboring cells exactly \( k \) times.
   - Vera's goal is to prevent Marco from having any valid moves.

3. **Initial Configuration:**
   - The game starts with one black cell surrounded by white cells.

4. **Analyzing Marco's Moves:**
   - Marco's first move will turn the initial black cell white and its four neighbors black.
   - This results in a cross pattern of black cells.

5. **Vera's Strategy:**
   - Vera needs to use her \( k \) swaps to ensure that Marco cannot find a black cell with more than two white neighbors.
   - By carefully choosing her swaps, Vera can manipulate the board to prevent Marco from making any valid moves.

6. **Determining Minimal \( k \):**
   - We need to find the smallest \( k \) such that Vera can always win.
   - Through analysis and trial, we find that \( k = 2 \) is sufficient for Vera to guarantee a win.

7. **Determining Smallest \( t \):**
   - With \( k = 2 \), we need to find the smallest number of turns \( t \) such that Vera can guarantee a win.
   - By examining the sequence of moves, we find that Vera can guarantee a win in at most 3 turns.

8. **Final Calculation:**
   - Given \( m = 2 \) and \( t = 3 \), we compute \( 100m + t \).

\[
100m + t = 100 \cdot 2 + 3 = 200 + 3 = 203
\]

The final answer is \(\boxed{203}\)