A6. A garden with a pond (the black square) will be tiled with hexagonal tiles, as shown in the figure. We have three colors of tiles: red, green, and blue. It is not allowed to place two tiles of the same color next to each other.

In how many ways can we tile the garden?
A) 3
B) 6
C) 12
D) 18
E) 24

![](https://cdn.mathpix.com/cropped/2024_04_17_9e808ca4ac6657f5611fg-2.jpg?height=363&width=399&top_left_y=1252&top_left_x=1451)
A6. D) 18 We first color the two indicated tiles at the bottom with two different colors. This can be done in six ways: three possibilities for the first tile and then for each choice two possibilities for the second tile. In the figure, the colors red $(r)$ and green $(g)$ are chosen.

$\mathrm{Now}$ that the first two tiles are colored, the colors of most of the other tiles are fixed. The tile numbered 1 can only be blue. Then tile 2 must be red and tile 3 must be blue. Consequently,

![](https://cdn.mathpix.com/cropped/2024_04_17_8a1f196dae4ba99bbc23g-2.jpg?height=366&width=411&top_left_y=505&top_left_x=1462)
the colors of all other tiles are fixed, except for the two tiles all the way to the right (white in the figure). For these last two tiles, there are three possible colorings. The top and bottom tiles can be either green and red, blue and red, or blue and green.

For each of the six allowed colorings of the first two tiles, there are exactly three ways to complete the coloring. In total, there are thus $6 \times 3=18$ possible colorings.