## Task 15/73

How many different line segments are necessary to draw the pattern shown in the figure without tracing any segment more than once?

![](https://cdn.mathpix.com/cropped/2024_06_06_1d2be4837fb839c2b31dg-234.jpg?height=306&width=437&top_left_y=972&top_left_x=1252)
The pattern represents a graph with a total of 18 nodes and 41 edges. Since 10 nodes have an odd number of edges converging, at least 5 routes are necessary; each of these 10 nodes is either the starting or endpoint of a route.

To prove that no more than 5 routes are required, we demonstrate this by providing 5 routes that trace the pattern. For this, we imagine the nodes numbered row by row from left to right and from top to bottom. The routes are represented by listing the node numbers.

1. Route: 1-5-9-13-17-14-11-7-3-6-9-12-15-16-17-18-11-4-3-2-1-8-9-10-11

2. Route: 4-7-10-13-16-12-8-5-2-6-10-14-18

3. Route: 2-9-16

4. Route: 3-10-17

5. Route: 8-15

Of course, other routes can also achieve the desired result. Therefore, exactly 5 routes are required.