Overlapping of unit length real lines?

Question

X, Y and Z are closed intervals of unit length on the real line. The overlap of X and Y is half a unit. The overlap of Y and Z is also half a unit. Let the overlap of X and Z be k units. Which of the following is true?

1. k must be 1
2. k must be 0
3. k can take any value between 0 and 1
4. None of the above

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My attempt :

-------------    z
       ---------------- y
------------- x
in above case k is 1.
                 ----------------- z
       ----------------- y
------------- x
now k is 0.

value of k would be either 0 or 1.

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Can you give an explanation in formal way please ?

Answer 1

WLOG let $Y = [0,1]$. Let $X = [a, a+1]$. The over lap of $X$ and $Y$ is of length $1/2$, so $X\cap Y$ is some interval of length $1/2$. Now say $a > 0$. Then $a < 1$ other wise there would be no intersection. So $a+1 > 1$ and the overlap is $[a,1]$. The only way that can have length $1/2$ is if $a = 1/2$. Likewise you get only one possibility when $a< 0$, namely the interval $[-1/2], 1/2]$. Apply the same argument to $Z$ and you see that both $X$ and $Z$ have to be of the form $[-1/2, 1/2]$, or $[1/2, 3/2]$. Going through the four options you see that $k=0$ and $k=1$ are the only options.