Zaslavsky A.A.

Two ants each crawled along their own closed path on a $7 \times 7$ board. Each ant only crawled along the sides of the cells and visited each of the 64 vertices of the cells exactly once. What is the minimum possible number of such sides that both the first and the second ants crawled along?
An example where the number of "common" sides equals 16 is shown in the figure (one path is black, the other is red).

![](https://cdn.mathpix.com/cropped/2024_05_06_5dbee4d95f83bceb3891g-08.jpg?height=300&width=306&top_left_y=905&top_left_x=882)

Estimate. Each ant visited 64 different sides. In total, there are 7$\cdot$8$\cdot$2 = 112 sides. Therefore, the number of sides visited at least twice is

$64+64-112=16$

## Answer

16 sides.