André is hiking on the paths shown in the map. He is planning to visit sites $A$ to $M$ in alphabetical order. He can never retrace his steps and he must proceed directly from one site to the next. What is the largest number of labelled points he can visit before going out of alphabetical order?
(A) 6
(D) 10
(B) 7
(E) 13
(C) 8

![](https://cdn.mathpix.com/cropped/2024_04_20_46ead6524a8d61e21c51g-100.jpg?height=374&width=340&top_left_y=1919&top_left_x=1337)
André is hiking on the paths shown in the map. He is planning to visit sites $A$ to $M$ in alphabetical order. He can never retrace his steps and he must proceed directly from one site to the next. What is the largest number of labelled points he can visit before going out of alphabetical order?
(A) 6
(B) 7
(C) 8
(D) 10
(E) 13

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-440.jpg?height=529&width=483&top_left_y=207&top_left_x=1363)

## Solution

If we trace André's route we can see that he can travel to site $J$ travelling in alphabetical order.

Once he reaches site $J$, it is not possible to reach $K$ without passing through $G$ or retracing his steps. Since $J$ is the tenth letter in the alphabet, he can visit ten sites before going out of order.

ANSWER: (D)