The Mictlán is an $n\times n$ board and each border of each $1\times 1$ cell is painted either purple or orange. Initially, a catrina lies inside a $1\times 1$ cell and may move in four directions (up, down, left, right) into another cell with the condition that she may move from one cell to another only if the border that joins them is painted orange. We know that no matter which cell the catrina chooses as a starting point, she may reach all other cells through a sequence of valid movements and she may not abandon the Mictlán (she may not leave the board). What is the maximum number of borders that could have been colored purple?

[i]Proposed by CDMX[/i]
1. **Identify the total number of borders:**
   - An \( n \times n \) board has \( n^2 \) cells.
   - Each cell has 4 borders, but each internal border is shared by two cells.
   - Therefore, the total number of borders is \( 2n(n+1) \).

2. **Consider the borders on the boundary:**
   - The boundary of the board consists of \( 4n \) borders (each side of the square has \( n \) borders, and there are 4 sides).

3. **Calculate the number of internal borders:**
   - The total number of borders is \( 2n(n+1) \).
   - Subtracting the boundary borders, the number of internal borders is:
     \[
     2n(n+1) - 4n = 2n^2 + 2n - 4n = 2n^2 - 2n
     \]

4. **Determine the minimum number of orange borders:**
   - The problem states that the catrina must be able to reach any cell from any starting cell, implying the underlying graph of cells connected by orange borders must be connected.
   - A connected graph with \( n^2 \) vertices (cells) must have at least \( n^2 - 1 \) edges (orange borders).

5. **Calculate the maximum number of purple borders:**
   - The total number of borders is \( 2n(n+1) \).
   - Subtracting the minimum number of orange borders, the maximum number of purple borders is:
     \[
     2n(n+1) - (n^2 - 1) = 2n^2 + 2n - n^2 + 1 = n^2 + 2n + 1 = (n+1)^2
     \]

6. **Verify the construction:**
   - We can construct a spanning tree with \( n^2 - 1 \) orange borders, ensuring the graph is connected.
   - The remaining borders can be purple, and their count will be \( (n+1)^2 \).

The final answer is \(\boxed{(n+1)^2}\).