\section*{Exercise 2 - \(340912=341012\)}

The figure shows a grid composed of line segments. These segments are the edges of the smaller cubes obtained by dividing the cube \(A B C D E F G H\) into eight equal smaller cubes.

An ant, which can only move along these segments, is to travel from \(A\) to \(G\) on the shortest possible path. How many different paths are there in total,

a) if all segments of the grid are allowed.

b) if only those segments of the grid are allowed that belong to the surface of the cube \(A B C D E F G H\)?

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1147.jpg?height=445&width=514&top_left_y=1548&top_left_x=1225)
}

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1147.jpg?height=434&width=525&top_left_y=2082&top_left_x=297)

a) To uniquely identify the shortest path from \(A\) to \(G\), the direction of the next segment must be specified 6 times, with 2 moves to the right, 2 moves backward, and 2 moves upward. Therefore, there are as many different paths as there are different sequences of the letters \(\mathbf{r}, \mathbf{r}, \mathbf{h}, \mathbf{h}\), o, o.

The number of these sequences is known to be

\[
\frac{6!}{2!\cdot 2!\cdot 2!}=\frac{720}{2 \cdot 2 \cdot 2}=90
\]

b) A path remains on the surface of the cube \(A B C D E F G H\) only if it does not pass through the point \(M\) (see figure). There are exactly \(3!=6\) paths from \(A\) to \(M\) (sequences of \(\mathbf{r}\), \(\mathbf{h}, \mathbf{o}\)), and similarly from \(M\) to \(G\). Therefore, the number of paths to be excluded is \(6 \cdot 6=36\). The number of paths that remain on the surface is thus 54.