This is a combinatorics problem that involves a coloring condition for a grid. Let's analyze and solve it step by step.

Problem Restatement:

You have an $n \times n$ grid, where each square is either colored black or white. For every pair of rows and every pair of columns, the four squares at the intersections of those rows and columns must not all be the same color (i.e., not all black or all white). The task is to determine the largest possible value of $n$ that satisfies this condition.

Step-by-Step Solution:

1. Key Observation:

   • For any two rows and two columns, the four intersection points form a 2×2 subgrid. The condition means that in every such 2×2 subgrid, there must be at least one square that is a different color from the others. In other words, it is not allowed to have all four squares black or all four squares white.

2. Connection to Latin Squares:

   • The problem is closely related to the concept of Latin squares in combinatorics. A Latin square is an arrangement of $n \times n$ cells, where each cell contains one of $k$ symbols, such that no symbol repeats in any row or column. However, our problem is not exactly a Latin square because we are only concerned about avoiding uniform coloring in subgrids.

3. Constructing the Grid:

   • A known result in combinatorics states that for such coloring conditions, the grid size $n \times n$ cannot grow arbitrarily large while maintaining the condition for all 2×2 subgrids. A specific construction using finite projective planes (from geometry) can be used to maximize $n$.

4. Result from Finite Geometry:

   • The largest possible $n$ is determined by the existence of a finite projective plane of order $k$. A finite projective plane of order $k$ has $k + 1$ points in each row and column, and $k + 1$ total rows and columns. The largest $n$ satisfying this problem's conditions is equal to $k + 1$, where $k$ is a power of a prime number.

   For example:

   • If $k = 2$: $n = k + 1 = 3$,
   • If $k = 4$: $n = k + 1 = 5$,
   • If $k = 8$: $n = k + 1 = 9$, and so on.

5. Largest $n$:

   • To ensure the condition is satisfied, the largest $n$ depends on the existence of finite projective planes. The largest known projective planes are related to powers of primes. For most practical purposes, the maximum $n$ is a power of a prime + 1, determined by construction methods from geometry.

6. Conclusion:

   • The exact upper limit of $n$ depends on whether the grid construction aligns with a finite projective plane. For small examples, it has been shown that $n = 3, 5, 7, 9, 11$, etc., are feasible for such configurations.

Final Answer:

The largest possible $n$ depends on the context of the problem and the availability of finite projective planes. A common answer in standard combinatorics literature is $n = 3$, as larger values require more advanced constructions that are not universally valid.

Would you like a detailed explanation of the finite projective plane connection or more examples of constructing such grids?

Related Questions:

1. What are finite projective planes, and how are they constructed?
2. How does the coloring condition relate to Latin squares or combinatorics problems?
3. Can you provide an example of a $3 \times 3$ grid satisfying this condition?
4. What is the general form of a combinatorial proof for this problem?
5. What are some practical applications of these grids in coding theory?

Tip: Visualizing small examples (e.g., $n = 3, n = 4$) helps in understanding how the condition works and why larger grids become difficult.