Two players, $A$ and $B$, play a game on a board which is a rhombus of side $n$ and angles of $60^{\circ}$ and $120^{\circ}$, divided into $2n^2$ equilateral triangles, as shown in the diagram for $n=4$.
$A$ uses a red token and $B$ uses a blue token, which are initially placed in cells containing opposite corners of the board (the $60^{\circ}$ ones). In turns, players move their token to a neighboring cell (sharing a side with the previous one). To win the game, a player must either place his token on the cell containing the other player's token, or get to the opposite corner to the one where he started.
If $A$ starts the game, determine which player has a winning strategy.
1. **Understanding the Board and Initial Setup:**
   - The board is a rhombus with side length \( n \) and angles of \( 60^\circ \) and \( 120^\circ \).
   - The board is divided into \( 2n^2 \) equilateral triangles.
   - Player \( A \) (Red) starts at one \( 60^\circ \) corner, and Player \( B \) (Blue) starts at the opposite \( 60^\circ \) corner.

2. **Objective of the Game:**
   - A player wins by either:
     - Placing their token on the cell containing the other player's token.
     - Reaching the opposite corner from where they started.

3. **Movement and Strategy:**
   - Each player moves their token to a neighboring cell (sharing a side with the previous one) on their turn.
   - The shortest path to the opposite corner involves moving through \( 4n-4 \) cells.

4. **Red's Winning Strategy:**
   - Red can win within \( 4n-4 \) moves by continuously moving downward towards the opposite corner.
   - If Blue attempts to block Red, Red can move onto Blue's cell and win by capturing Blue's token.
   - If Blue does not block, Red reaches the opposite corner in \( 4n-4 \) moves and wins.

5. **Blue's Constraints:**
   - Blue cannot reach the opposite corner before \( 4n-4 \) moves elapse.
   - By the time \( 4n-4 \) moves elapse, Red must have either reached the opposite corner or captured Blue's token.

6. **Conclusion:**
   - Since Red can always reach the opposite corner or capture Blue's token within \( 4n-4 \) moves, Blue has no chance of winning.

\[
\boxed{\text{Player A (Red) has a winning strategy.}}
\]