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.