11.3. Petya and Vasya live in neighboring houses (see the plan in Figure 1). Vasya lives in the fourth entrance. It is known that for Petya to run to Vasya by the shortest path (not necessarily along the sides of the cells), it does not matter which side of his house he runs around. Determine which entrance Petya lives in. (A. V. Khachaturyan)

![](https://cdn.mathpix.com/cropped/2024_05_06_14ba1dd16c5040548da9g-2.jpg?height=388&width=560&top_left_y=1145&top_left_x=702)

Fig. 1
Answer: in the sixth entrance.

Solution. The shortest path from point $A$ to Vasya's entrance is the segment $A D$ (Fig. 2). The shortest path from point $B$ to Vasya's entrance is the path along segment $B C$, and then along segment $C D$. Since triangles $A E D$ and $C E B$ are equal, $A D = B C$. Therefore, the paths from points $A$ and $B$ to Vasya's entrance differ by 4 cells. Since the paths from Petya's entrance through the "upper" corner (i.e., through point $A$) and through the "lower" corner (i.e., through point $B$) are equal, the path from Petya's entrance to point $A$ must be 4 cells longer than the path to point $B$.

![](https://cdn.mathpix.com/cropped/2024_05_06_14ba1dd16c5040548da9g-2.jpg?height=380&width=506&top_left_y=1643&top_left_x=1343)

Fig. 2 This means that Petya lives in the sixth entrance.