Five points lie on the same line. When we list the 10 distances between any two of these points, from smallest to largest, we find $2,4,5$, $7,8, k, 13,15,17$ and 19. What is the value of $k$?
Solution 1: Since the greatest distance between two points is 19 and the smallest is 2, we draw a number line with the two points 0 and 19 at the ends and the point 2 two units from 0, obtaining the first three points in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-064.jpg?height=92&width=940&top_left_y=2147&top_left_x=638)

Next, we place two other points to try to meet the required distances. Since we need distances of 5 and 7, we place the point 7 on the line, which gives us distances that are not incompatible with the problem data, as the distances between these 4 points are 2, 5, 7, 12, 17, and 19, as shown in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-064.jpg?height=94&width=985&top_left_y=2580&top_left_x=615)

Thus, if our attempt to place all the points is successful, $k$ must necessarily be 12. We are lucky because to obtain the distances 4, 8, 13, and 15, it is enough to place the point 15 on the line, obtaining all the distances 2, 4, 5, 7, 8, 12, 13, 15, 17, and 19, as shown in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-065.jpg?height=100&width=991&top_left_y=521&top_left_x=567)

Choosing 4 as the third point, we obtain another distribution of points with the same distances between them, as in the following figure, where again, $k=12$.

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-065.jpg?height=103&width=988&top_left_y=851&top_left_x=565)

Solution 2: Since the greatest distance is 19, we can assume that one point is 0 and the other is 19. If a third point is equal to $a$, we will have the distances $a-0=a$ and $19-a$ in the given list of distances. Since the pairs 2 and 17, as well as 4 and 15, appear in this list of distances, we can choose the number $a=2$ or $a=4$ as the third point. Let's choose the point 2. Since 4 and 15 are in the list of distances, 4 or 15 is another point on the line; but, 4 cannot be one of the points because the distance 2, between 2 and 4, does not appear twice. Therefore, 15 is the fourth point on the line. Finally, the fifth point must be 5 units from one of the points and 7 units from another. Thus, the missing point is the point 7 and the unknown distance is $k=19-7=12$.

Choosing 4 as the third point, we would obtain the points 12 and 14 as the fourth and fifth points, and again, the unknown distance is $k=12$.