Kira can draw a connected path from $M$ to $N$ by drawing arrows along only the diagonals of the nine squares shown. One such possible path is shown. A path cannot pass through the interior of the same square twice. In total, how many different paths can she draw from $M$ to $N$ ?
(A) 5
(B) 6
(C) 7
(D) 8
(E) 9

![](https://cdn.mathpix.com/cropped/2024_04_20_46ead6524a8d61e21c51g-062.jpg?height=350&width=380&top_left_y=2118&top_left_x=1317)
Labeling the diagram as shown below, we can describe paths using the points they pass through.

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-278.jpg?height=426&width=426&top_left_y=1088&top_left_x=882)

The path $M A D N$ is the only path of length 3 (traveling along 3 diagonals).

Since the diagram is symmetrical about $M N$, all other paths will have a reflected path in the line $M N$ and therefore occur in pairs. This observation alone allows us to eliminate (B) and (D) as possible answers since they are even.

The following table lists all possible paths from $M$ to $N$ traveling along diagonals only.

| Path length | Path Name | Reflected Path |
| :---: | :--- | :--- |
| 3 | $M A D N$ | same |
| 5 | $M A B C D N$ | MAFEDN |
| 9 | $M A B C D E F A D N$ | MAFEDCBADN |
|  | $M A B C D A F E D N$ | MAFEDABCDN |
|  | $M A D C B A F E D N$ | MADEFABCDN |

At this point we have listed 9 valid paths. Since paths occur in pairs (with the exception of $M A D N$ ), the next possible answer would be 11. Since 11 is not given as an answer (and 9 is the largest possible answer given), we can be certain that we have found them all.

ANSWER: (E)