# Five Points on a Line

There are five points on a line. You measure the distances between every pair of points and you find:

2, 4, 5, 7, 8, __, 13, 15, 17, 19

This list is ordered from shortest to longest. What is the missing distance?

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12

Let’s call the points A, B, C, D, and E, in order of left to right, and the missing distance x.

The biggest number, 19, must be the distance AE. We know that this distance is also the sum of other distances: AB + BE, AC + CE, and AD + DE.

We see that 2 + 17 = 19 and 4 + 15 = 19, but no other pair of distances add to 19. This means x must be part of the last pair of distances. Because the list is ordered, we know 8 < x < 13, so the last pair of distances must be x +7 or x + 8, meaning x is 12 or 11.

Repeat this logic for the second biggest number, 17, which must be either AD or BE. If it’s AD, we know AB + BD = 17 and AC + CD = 17; if it’s BE, we know BC + CE = 17 and BD + DE = 17. Either way, 17 must be the sum of two pairs of distances. Of the numbers provided, only 4 + 13 = 17. This means x must be part of the other pair of distances. Again, 8 < x < 13, so the other pair of distances must be x +5 or x + 7 or x + 8, meaning x is 12, 10, or 9.

Since only one of the possibilities satisfies both conditions above, it must be 12.