The wizard Albus and Brian are playing a game on a square of side length $2n+1$ meters surrounded by lava. In the centre of the square there sits a toad. In a turn, a wizard chooses a direction parallel to a side of the square and enchants the toad. This will cause the toad to jump $d$ meters in the chosen direction, where $d$ is initially equal to $1$ and increases by $1$ after each jump. The wizard who sends the toad into the lava loses. Albus begins and they take turns. Depending on $n$, determine which wizard has a winning strategy.
1. **Initial Setup and Definitions:**
   - The square has a side length of \(2n+1\) meters.
   - The toad starts at the center of the square.
   - The toad's initial jump distance \(d\) is 1 meter, and it increases by 1 meter after each jump.
   - Albus starts the game, and they take turns.

2. **Game Dynamics:**
   - Each wizard chooses a direction parallel to a side of the square.
   - The toad jumps \(d\) meters in the chosen direction.
   - The wizard who causes the toad to jump into the lava loses.

3. **Strategy Analysis:**
   - Brian mirrors Albus's moves. This means that for every move Albus makes, Brian makes an equivalent move in the opposite direction.
   - By mirroring, Brian ensures that the toad's path remains symmetric with respect to the center of the square.

4. **Path and Distance Considerations:**
   - The toad starts at the center of the square, which is at coordinates \((n+1, n+1)\).
   - The maximum distance the toad can move in any direction without falling into the lava is \(n\) meters.
   - After \(n\) moves, the toad will have moved a total distance of \(1 + 2 + 3 + \ldots + n = \frac{n(n+1)}{2}\) meters in one direction.

5. **Critical Move Analysis:**
   - After \(n\) moves, the toad will be at the edge of the square.
   - On the \((n+1)\)-th move, the toad will jump \(n+1\) meters.
   - Since the side length of the square is \(2n+1\) meters, the toad will fall into the lava on the \((n+1)\)-th move.

6. **Winning Strategy:**
   - Since Albus starts the game, he will make the \((n+1)\)-th move.
   - Therefore, Albus will be the one to cause the toad to jump into the lava.

Conclusion:
\[
\boxed{\text{Brian has a winning strategy.}}
\]