Question 97
Alice plays a game using a board with 20 rows and 25 columns. Initially, no number is written in any of the $20 \times 25$ cells. The game proceeds in several turns, and on the $n$-th turn, the following operation is performed:
Choose a positive integer $k$ and $k$ empty cells $A_1, A_2, \dots, A_k$ such that for every integer $i$ with $1 \le i \le k-1$, the cell $A_{i+1}$ is adjacent to $A_i$ either to the right or above. Write the number $n$ in each of these $k$ cells.
The game ends when all cells on the board are filled with some numbers. When Alice plays optimally to minimize the number of turns until the game ends, how many different possible numberings of the board can be obtained when the game ends?
Note: Numberings that are identical under rotation or reflection are considered distinct and counted as different.
Choose a positive integer $k$ and $k$ empty cells $A_1, A_2, \dots, A_k$ such that for every integer $i$ with $1 \le i \le k-1$, the cell $A_{i+1}$ is adjacent to $A_i$ either to the right or above. Write the number $n$ in each of these $k$ cells.
The game ends when all cells on the board are filled with some numbers. When Alice plays optimally to minimize the number of turns until the game ends, how many different possible numberings of the board can be obtained when the game ends?
Note: Numberings that are identical under rotation or reflection are considered distinct and counted as different.