4. What is the greatest length that a closed, non-self-intersecting broken line can have, which runs along the grid lines of a $6 \times 10$ cell field?

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# Answer: 76.

Solution. We will color the nodes of the grid in a checkerboard pattern, with black and white colors. The length of a closed non-self-intersecting broken line is equal to the number of nodes it passes through. Each segment of the broken line connects a black and a white node. When traversing the broken line, the colors of the nodes alternate, so the length of the closed broken line is an even number. Since there are 77 nodes in total in the grid, the length of the broken line is at most 76. A corresponding example is easily constructed (Fig. 2).

![](https://cdn.mathpix.com/cropped/2024_05_06_c7a5bb7bd09987b989ebg-4.jpg?height=386&width=625&top_left_y=1703&top_left_x=721)

Fig. 2

Remark. Another reasoning is possible. Let's start constructing a closed broken line from some node. Since we need to return to it, the number of horizontal steps to the right is the same as the number of steps to the left. Therefore, the number of horizontal steps is even. The same is true for vertical steps. Hence, we again obtain that the length of the closed broken line (whose segments follow the grid lines) is an even number.

Evaluation. 13 points for a correct solution. 1 point for only the answer. 4 points if an example of a broken line of length 76 is provided, but there is no estimation of the length of the broken line.