## Task 4 - 170924

The following figure shows 11 cube nets.

a) Determine which of these can be drawn in one stroke, i.e., as a continuous line segment, where each line segment in the net is traversed exactly once!

b) For these nets, provide a starting and ending point for such a line segment!

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0999.jpg?height=146&width=120&top_left_y=1492&top_left_x=294)

(1)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0999.jpg?height=143&width=120&top_left_y=1499&top_left_x=431)

(2)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0999.jpg?height=149&width=120&top_left_y=1493&top_left_x=568)

(3)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0999.jpg?height=139&width=117&top_left_y=1501&top_left_x=707)

(4)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0999.jpg?height=160&width=120&top_left_y=1482&top_left_x=842)

(5)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0999.jpg?height=160&width=120&top_left_y=1482&top_left_x=979)

(6)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0999.jpg?height=178&width=238&top_left_y=1464&top_left_x=1114)

(7)
(8)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0999.jpg?height=137&width=120&top_left_y=1502&top_left_x=1371)

(9)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0999.jpg?height=139&width=120&top_left_y=1501&top_left_x=1508)

(10)

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0999.jpg?height=139&width=120&top_left_y=1501&top_left_x=1642)

(11)
Such a path can be drawn exactly when there are either exactly zero or exactly two points in the network where an odd number of lines meet.

(At such a point, one can only start or end, since for each line one reaches a point, one also needs another line to leave it again. The number of lines that meet at a point that is neither the start nor the endpoint of the path must therefore be even. On the other hand, a connected structure where, except for possibly two points, an even number of lines meet at all points, can always be drawn as a continuous path.)

In network (1), there are 6 points where 3 lines meet, in network (2) there are 4, in network (3) also 4, in network (4) there are 6, in networks (5) and (6) there are exactly 2 each, in network (7) there are 6, in network (8) there are 4, in network (9) exactly 2, in network (10) there are 4, and in network (11) again exactly 2. Therefore, only the networks (5), (6), (9), and (11) can be drawn in a single path.

In each of these networks, the two points where three lines meet are the starting and ending points of the corresponding path.

Remark: In the context of graph theory, the lines are considered as edges and their endpoints as nodes. The question of whether all edges can be combined into a path that contains each edge exactly once is also referred to as the question of whether the graph contains an Eulerian path. (See the Seven Bridges of Königsberg problem.)