The numbers $1,2,3, \ldots, 14$ must be written at the 14 vertices of the polygonal line below so that the sums of the 4 numbers written on each of the 7 segments of the polygonal line are the same.

![](https://cdn.mathpix.com/cropped/2024_05_01_3fc4f3a6ef7302a792fcg-29.jpg?height=436&width=513&top_left_y=1807&top_left_x=859)

a) What is the sum of all the numbers from 1 to 14?

b) What must be the sum of the numbers written on one segment?
c) Give an example of the distribution of these numbers.
Solution

a) The sum of all the numbers that must be written is

$$
1+2+3+\ldots+14=105
$$

b) When we sum the numbers written in each of the 7 segments, each of them will be added twice. Therefore, the common sum for each of these segments is $\frac{105 \cdot 2}{7}=$ 30.

c) The figure below shows a possible distribution

![](https://cdn.mathpix.com/cropped/2024_05_01_3fc4f3a6ef7302a792fcg-30.jpg?height=489&width=555&top_left_y=1077&top_left_x=629)