Each row and each column of this arrangement of cells has to
contain all of the digits 1, 2, 3, 4, 5 and 6. The digits in the outer
cells have been entered — these cannot be changed.
In how many different ways can the remaining 16 cells be filled in?

This is a problem involving a **Latin square** — a grid where each row and each column contains distinct digits from a specific set (in this case, \( \{1, 2, 3, 4, 5, 6\} \)). You are given a 6x6 grid where the outer cells are already filled, and you are tasked with determining how many different ways the remaining 16 inner cells can be filled in such that the arrangement still satisfies the conditions of a Latin square.

### Approach to the Solution:

1. **Known Digits**: The outer 20 cells (in the four rows and columns on the perimeter) are filled in, and these digits are fixed. Since no changes are allowed to these cells, they impose constraints on the filling of the remaining 16 inner cells.

2. **Remaining Cells**: The 16 inner cells need to be filled with the digits 1 to 6, but they must follow the rules of a Latin square. Specifically:
    - Each row must contain the digits 1 to 6 exactly once.
    - Each column must contain the digits 1 to 6 exactly once.
   
3. **Counting Valid Configurations**: The number of valid ways to fill the grid depends on how the fixed outer digits restrict the possible configurations of the inner cells. This type of counting problem involves constraints from both the rows and columns, making it more complicated than simply permuting the digits.

Since this type of problem usually requires advanced combinatorial techniques or a computer algorithm (often based on constraint satisfaction or backtracking) to solve, let's outline the possible approaches:
   
   - **Constraint Satisfaction Approach**: The fixed digits in the outer cells act as constraints on the digits allowed in the inner cells. By progressively filling in the cells while maintaining the Latin square property (each digit must appear exactly once per row and column), we could explore all possible combinations.
   
   - **Backtracking Algorithm**: An algorithm can try different combinations for each cell, backtracking when a combination violates the row or column uniqueness condition.

### Conclusion:
Without the exact configuration of the outer digits, it is difficult to calculate the precise number of solutions manually. However, solving this problem computationally using a backtracking algorithm or a constraint satisfaction solver would provide the exact number of valid configurations for the remaining 16 cells.

Would you like a detailed explanation on how such algorithms work? Or feel free to upload the grid for a more specific analysis based on the given configuration.

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Here are 5 related questions that expand on this concept:
1. What is a Latin square, and how does it differ from a Sudoku puzzle?
2. How do constraint satisfaction problems work in combinatorics?
3. What are backtracking algorithms, and how are they applied in grid-filling problems?
4. How would this problem change if some inner cells were also pre-filled with digits?
5. How can computational algorithms like SAT solvers be applied to solve Latin squares?

**Tip**: When solving combinatorial problems involving constraints, it’s useful to consider the problem's symmetry and use algorithms that exploit these symmetries to reduce the number of cases you need to consider.

Each row and each column of this arrangement of cells has to contain all of the digits 1, 2, 3, 4, 5 and 6. The digits in the outer cells have been entered — these cannot be changed. In how many different ways can the remaining 16 cells be filled in?

Explore how to solve a combinatorial problem involving a 6x6 grid, where the outer cells are fixed, and the goal is to fill the remaining 16 inner cells while maintaining the Latin square property. Learn about constraint satisfaction, Latin square theory, and backtracking algorithms.

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