We say that a sequence x1, x2, . . . , xn is increasing if xi ≤ xi+1 for all 1 ≤ i < n. How many ways are
there to fill an 8 × 8 table by numbers 1, 2, 3, and 4 such that:
• The numbers in each row are increasing from left to right,
• The numbers in each column are increasing from top to bottom,
• and there is no pair of adjacent cells such that one is filled with 2 and the other one is filled with
3. (We say two distinct cells are adjacent if they share a side)

To solve this problem, we need to find the number of ways to fill an \(8 \times 8\) table using the numbers 1, 2, 3, and 4 under the specified conditions. Here's how we can approach the solution:

1. **Understanding the constraints**:
   - Each row must be non-decreasing from left to right.
   - Each column must be non-decreasing from top to bottom.
   - The numbers 2 and 3 cannot be adjacent in the grid.

2. **Applying combinatorial methods**:
   We can think of the arrangement of the numbers in the grid as a combinatorial problem involving multisets, where we must account for the adjacency constraint between 2 and 3.

3. **Using a generating function**:
   The generating function can help represent the arrangements:
   - Define the states based on the last number in the current row. For example, if we fill a row with the last number as 1, 2, 3, or 4, we can define the number of ways to fill the remaining cells based on that choice.
   - We can create a dynamic programming table where we keep track of how many configurations can lead to each ending digit under the constraints.

4. **Dynamic programming formulation**:
   Let \(dp[i][j][k]\) represent the number of valid configurations for the first \(i\) rows and \(j\) columns where the last number in the last filled cell is \(k\) (where \(k\) can be 1, 2, 3, or 4).

5. **State transition**:
   - For each number we choose to fill, we must ensure the adjacency constraint is maintained. Specifically, if we place a 2 in a cell, we cannot place a 3 in the directly adjacent cells.

6. **Base case and Iteration**:
   We initialize the base case with the first row and iterate through filling each subsequent row, applying the transition rules based on the previous row's configuration.

7. **Final count**:
   The final answer will be the sum of all valid configurations in the \(8 \times 8\) table.

### Result
However, calculating the exact number of configurations requires careful bookkeeping of the states and transitions, particularly respecting the adjacency constraint. This is a complex combinatorial problem typically tackled through programming or advanced combinatorial techniques.

If you'd like to dive deeper into any of these specific steps or require further elaboration on a particular point, let me know! 

### Questions for Further Exploration:
1. What specific dynamic programming approach can be implemented for this problem?
2. How can adjacency constraints affect combinatorial counting in different scenarios?
3. What are the applications of similar combinatorial problems in real-world scenarios?
4. How does changing the size of the table affect the number of configurations?
5. Can this problem be solved using a different combinatorial technique, like recursion?

### Tip
When dealing with adjacency constraints in combinatorial problems, visualizing the problem or sketching possible configurations can help clarify the relationships and constraints between different elements.

How many ways are there to fill an 8 × 8 table by numbers 1, 2, 3, and 4 such that the numbers in each row are increasing from left to right, the numbers in each column are increasing from top to bottom, and there is no pair of adjacent cells such that one is filled with 2 and the other one is filled with 3?

Explore the combinatorial problem of filling an 8x8 table with the numbers 1, 2, 3, and 4, while adhering to specific constraints on arrangement. This topic covers dynamic programming, generating functions, and adjacency constraints. Ideal for advanced high school students.

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