A 15-inch-long stick has four marks on it, dividing it into five segments of length 1,2,3,4, and 5 inches (although not neccessarily in that order) to make a “ruler.” Here is an example.
Using this ruler, you could measure $8$ inches (between the marks $B$ and $D$ ) and $11$ inches (between the end of the ruler at $A$ and the mark at $E$ ), but there's no way you could measure $12$ inches.
Prove that it is impossible to place the four marks on the stick such that the five segments have length 1,2,3,4, and 5 inches, and such that every integer distance from 1 inch through 15 inches could be measured.
