Question 11
The diagram below shows a hexagonal room. Its floor is covered with $19$ numbered rugs of four different colors. Two rugs with a common segment along their borders must have different colors. Only two colors are used in the middle layer, and at most one rug in the outer layer has the same color as the central rug. Two coloring patterns are different if the color of at least one rug has changed. How many different coloring patterns are there?
