Question 56
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 necessarily 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.
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.