Question 82
Consider filling each square of an $100 \times 100$ table with one of the letters J, M, or O. A $2 \times 2$ square of the table is called a **good block** if it satisfies one of the following conditions:
* The four squares have exactly one type of letter.
* The four squares have exactly two types of letters, and each letter appears twice.
* The four squares have exactly three types of letters, and the lower-left and upper-right squares have the same letter.
Find the number of ways of filling the 10000 squares of the table satisfying the following two conditions:
* Every $2 \times 2$ square of the table is a good block.
* Among pairs of two adjacent squares (sharing an edge) of the table, there are exactly 10000 pairs whose two squares have different letters. Here, two pairs where the order of squares is just swapped are considered as the same pair and are counted only once.
Note that we distinguish two ways of filling which can be obtained from each other by rotations or reflections.
* The four squares have exactly one type of letter.
* The four squares have exactly two types of letters, and each letter appears twice.
* The four squares have exactly three types of letters, and the lower-left and upper-right squares have the same letter.
Find the number of ways of filling the 10000 squares of the table satisfying the following two conditions:
* Every $2 \times 2$ square of the table is a good block.
* Among pairs of two adjacent squares (sharing an edge) of the table, there are exactly 10000 pairs whose two squares have different letters. Here, two pairs where the order of squares is just swapped are considered as the same pair and are counted only once.
Note that we distinguish two ways of filling which can be obtained from each other by rotations or reflections.