1. Initial Setup and Definitions:
- Consider a $2007 \times 2007$ chessboard where each unit square contains a coin with heads facing upward.
- We can flip four consecutive coins in the same row or column.
- We aim to make all coins show tails except for one coin at position $(i, j)$.

2. Analyzing the Parity of Rows and Columns:
- Let $r_1, r_2, r_3, r_4$ be the number of heads in rows that are $1 \mod 4$, $2 \mod 4$, $3 \mod 4$, and $0 \mod 4$ respectively.
- Initially, $r_1 = r_2 = r_3 = r_4 = 2007$ since all coins are heads up.

3. Effect of Flips on Parity:
- Each flip of four consecutive coins in a row or column changes the number of heads in that row or column by $\pm 4$.
- This means the parity (odd or even nature) of the number of heads in each row or column is preserved modulo 2.

4. Parity Constraints:
- After any sequence of flips, the parities of $r_1, r_2, r_3, r_4$ must remain equal modulo 2.
- Therefore, it is impossible to have a lone head in a row or column that is $1, 2,$ or $3 \mod 4$ because it would violate the parity condition.

5. Constructing the Desired Configuration:
- To show the converse, we need to demonstrate that it is possible to achieve the configuration with a lone head at $(i, j)$ where both $i$ and $j$ are divisible by 4.
- Start by making all coins tails except for the coin at $(4, 4)$:
- Flip all rows from 4 to 2007 using vertical flips.
- For the first three rows, flip all columns from 4 to 2007 using horizontal flips.
- This leaves a $3 \times 3$ square of heads in the upper left corner.
- Use four vertical flips to leave three heads at $(4, 1)$, $(4, 2)$, and $(4, 3)$.
- One horizontal flip will leave a single head at $(4, 4)$.

6. Generalizing the Position of the Lone Head:
- From the position $(4, 4)$, we can move the lone head a distance of four using two adjacent flips.
- This allows us to move the lone head to any $(i, j)$ where $i \equiv 0 \mod 4$ and $j \equiv 0 \mod 4$.

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