Question 41
A $2 \times 100$ square grid is given. Suppose we color each of the $200$ square boxes of the grid either red or blue. How many possible ways of coloring these boxes are there that satisfy the following conditions? Regard two patterns of colored boxes to be distinct if one of the patterns can be obtained from the other by rotating or flipping.
* There is at least one box colored red and at least one box colored blue.
* The region determined by all the red colored boxes forms a connected region and the same is true for blue colored boxes. (Here we consider $2$ boxes to be connected if they share a side).
* There is at least one box colored red and at least one box colored blue.
* The region determined by all the red colored boxes forms a connected region and the same is true for blue colored boxes. (Here we consider $2$ boxes to be connected if they share a side).