Problem 4. The maze for mice (see the figure) is a $5 \times 5$ meter square, and the mice can only run along the paths. Two pieces of cheese (marked with crosses) are placed at two intersections. At another intersection, a mouse (marked with a circle) is sitting. She can smell the cheese, but she needs to run the same distance to both pieces, so she doesn't know which one to choose and sits thoughtfully in place.

![](https://cdn.mathpix.com/cropped/2024_05_06_51d7eed368663d90d59bg-2.jpg?height=260&width=273&top_left_y=1152&top_left_x=1002)

a) Mark five more intersections where the mouse could sit thoughtfully (from where she would need to run the same distance to both pieces of cheese).

[2 points]

b) Come up with two intersections where you can place a piece of cheese so that there are as many suitable intersections for a thoughtful mouse as possible.

[up to 5 points]
Answer. a)

![](https://cdn.mathpix.com/cropped/2024_05_06_51d7eed368663d90d59bg-3.jpg?height=275&width=268&top_left_y=125&top_left_x=415)

b) The maximum number of places for thoughtful mice is 26:

![](https://cdn.mathpix.com/cropped/2024_05_06_51d7eed368663d90d59bg-3.jpg?height=285&width=291&top_left_y=487&top_left_x=581)

Comments. 1. In any example for part b), one end of each of the following 10 segments must be free (if both ends are marked, then the pieces of cheese must be placed on a line perpendicular to this segment, which means there will be exactly 6 suitable places.
![](https://cdn.mathpix.com/cropped/2024_05_06_51d7eed368663d90d59bg-3.jpg?height=224&width=490&top_left_y=1010&top_left_x=480)

From this, we can also conclude that there cannot be more than 26 thoughtful mice: if one end is free, then we have at least 10 unmarked intersections.