Five points are located on a certain straight line
Nine pairwise distances are known:
2,4,5,7,12,13,15,17,19
Find the 10th.
Based on Hoshino & McCurdy puzzle published in Crux Mathematicorum
Similarly, I just did the following, involving some guessing:
I guessed 19 was the whole span. If the points along the line are
0, a, b, c, d, then (a-0) + (b-a) + (c-b) + (d-c) =19
where the ( ) terms must all be present in the complete list of
10 distances. But the first 4 given add to 18. (We can't go further than the 1st 4 because the sum gets too great or there's an endless recursive generation of smaller terms). Therefore we may substitute out each one, one at a time, and replace it with a number 1 larger as the missing distance. This makes the 4 "adjacent point" distances
(3,4,5,7) or (2,5,5,7) or (2,4,6,7) or (2,4,5,8). Only the last set generates the other distances: 12,13,15,17
Edited on September 6, 2021, 12:36 pm
semi-analytic
