Question 77
There is a rectangular grid with 3 rows and $n$ columns on a blackboard.
a. Find the number of ways to write exactly one of numbers $1, 2, \ldots, 3n$ into each square in such a way that all the following conditions are met:
(1) Different squares contain different numbers.
(2) For each $i = 1, 2, \ldots, 3n - 1$, the numbers $i$ and $i + 1$ are written into adjacent squares (i.e., squares having a common side).
(3) The numbers $1$ and $3n$ are written into adjacent squares.
b. The same question, but as the 3rd condition, the number $1$ must be written into the leftmost and the number $3n$ into the rightmost column.
a. Find the number of ways to write exactly one of numbers $1, 2, \ldots, 3n$ into each square in such a way that all the following conditions are met:
(1) Different squares contain different numbers.
(2) For each $i = 1, 2, \ldots, 3n - 1$, the numbers $i$ and $i + 1$ are written into adjacent squares (i.e., squares having a common side).
(3) The numbers $1$ and $3n$ are written into adjacent squares.
b. The same question, but as the 3rd condition, the number $1$ must be written into the leftmost and the number $3n$ into the rightmost column.