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 problem involves filling in a grid of cells, where each row and column must contain all the digits 1, 2, 3, 4, 5, and 6, and the outer cells (those on the boundary) have already been filled in. To solve this, we need to apply the concept of **Latin squares** and permutation counting, as well as the constraints from the given outer boundary.

Here's a step-by-step breakdown:

1. **Grid Configuration and Size**: We are dealing with a 6x6 grid, where each row and each column must contain the digits 1 to 6 exactly once (no repetition in any row or column).

2. **Outer Cells Constraint**: Since the digits in the outer cells are already given and cannot be changed, these fixed cells impose significant constraints on the arrangement of the digits in the inner 16 cells.

3. **Latin Square Counting**: A Latin square of size \(n \times n\) (here \(n = 6\)) is a square arrangement where each row and column contains exactly one occurrence of each symbol (in this case, the digits 1 to 6). However, because some of the cells (outer ones) are already filled in, the number of ways to fill in the remaining 16 cells depends on the fixed outer digits.

4. **Symmetry and Permutation Impact**: The number of ways to fill the inner cells will depend on how much freedom we have, which in turn depends on how the fixed outer digits are distributed. If the outer digits break a lot of symmetry or pre-determine some rows/columns, fewer configurations will be possible for the inner cells.

The problem is similar to finding how many Latin squares can be formed given a partial filling of the grid. This is a computationally complex problem, typically solved using backtracking or algorithms designed for constraint satisfaction problems.

### Approximate Approach:
- **No Outer Digits Constraint**: If there were no fixed digits (outer cells were not filled), the total number of 6x6 Latin squares is known to be **812,851,200**.
  
- **With Outer Digits Fixed**: Fixing the outer digits substantially reduces the number of valid Latin squares. However, the exact number depends on the specific configuration of those fixed digits.

Calculating this exactly requires solving the specific constraint problem, which is often done using computer algorithms designed for this purpose. Tools like **backtracking search** or **constraint programming** are typically used to explore all possible configurations of the remaining 16 cells.

### Final Answer:
Without the exact configuration of the outer digits, we cannot compute the exact number of ways to fill the remaining cells. However, solving this problem would involve counting the number of valid Latin squares that fit the fixed boundary conditions, which is computationally feasible but requires algorithmic support.

Would you like to see a detailed example of how this kind of problem might be solved using backtracking, or would you prefer further clarification on the general theory?

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Here are 5 related questions:
1. What is a Latin square, and how is it relevant to grid problems like this?
2. How do backtracking algorithms work in solving constraint satisfaction problems?
3. How many distinct Latin squares exist for grids of different sizes (e.g., 3x3, 4x4)?
4. What are some common techniques for solving Sudoku-like puzzles algorithmically?
5. Can the problem of counting Latin squares be generalized for larger grids?

**Tip**: For grid-filling problems, always look for fixed cells or constraints that reduce the number of possibilities, as this can simplify the solution approach significantly.

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 6x6 Latin square puzzle where outer digits are fixed, and each row and column must contain the digits 1-6 exactly once. Learn key concepts like combinatorics, backtracking algorithms, and permutation symmetry. Suitable for advanced high school to early college students.

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